From 179908bca35a3e56a6ddaefb0c0f5e3891ab1959 Mon Sep 17 00:00:00 2001
From: Inako <baptiste.signolle@free.fr>
Date: Wed, 31 Mar 2021 23:10:08 +0200
Subject: [PATCH] ouerhvou
---
.../heldAndKarp-checkpoint.ipynb | 1511 +++++++--
heldAndKarp.ipynb | 2711 +++--------------
2 files changed, 1783 insertions(+), 2439 deletions(-)
diff --git a/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb b/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
index bb23119..50f404e 100644
--- a/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
+++ b/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
@@ -53,8 +53,10 @@
},
{
"cell_type": "code",
- "execution_count": 2,
- "metadata": {},
+ "execution_count": 1,
+ "metadata": {
+ "tags": []
+ },
"outputs": [
{
"name": "stdout",
@@ -67,18 +69,32 @@
"source": [
"using LinearAlgebra # Basic algebra structures & functions\n",
"using JuMP # Modeling language\n",
- "using GLPK, Cbc # LP and MIP solvers\n",
+ "using Cbc # MIP solver\n",
"using LightGraphs # Efficient graph structures & functions\n",
- "using GraphPlot # Graph visualization \n",
+ "using GraphPlot # Graph visualization\n",
+ "using Combinatorics # Combinatorials\n",
"\n",
"println(\"Modules loaded\")"
]
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
+ "execution_count": 2,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "checkTermination (generic function with 2 methods)"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"function checkTermination(m, verbose = true)\n",
" if !( \n",
@@ -100,12 +116,42 @@
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
+ "execution_count": 3,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "display_S (generic function with 2 methods)"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
- "function display(S, W = Nothing)\n",
- " gplot(Graph(S), nodelabel=1:size(S,1))\n",
+ "# Graph display function\n",
+ "# S and W are assumed to be square matrices\n",
+ "# S the adjacency matrix of a graph\n",
+ "# W the weights of its edges\n",
+ "function display_S(S, W = nothing)\n",
+ " edgelabel = []\n",
+ " n = size(S,1)\n",
+ " if W != nothing\n",
+ " m = size(W, 1)\n",
+ " if m != n\n",
+ " error(\"display_S : S and W don't match ; S is of size \", n, \", W of size \", m)\n",
+ " end\n",
+ " for i in 1:n, j in i:n\n",
+ " if X[i, j] == 1\n",
+ " push!(edgelabel, W[i, j])\n",
+ " end\n",
+ " end\n",
+ " end\n",
+ " display(gplot(Graph(S), nodelabel=1:n, edgelabel = edgelabel))\n",
"end"
]
},
@@ -121,14 +167,14 @@
"\n",
"However on such small graphs column generation techniques will not show their full potential, as these methods aim to tackle large scale problems.\n",
"\n",
- "\n",
- "\n",
- ""
+ "\n",
+ "\n",
+ ""
]
},
{
"cell_type": "code",
- "execution_count": 11,
+ "execution_count": 4,
"metadata": {
"tags": []
},
@@ -220,10 +266,27 @@
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
+ "execution_count": 5,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "IPM (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
+ "# Integer Programming Master problem\n",
+ "# C is a vector such that C[t] = dot(c, x^t) for t in 1:T_1\n",
+ "# where T_1 is the number of columns of the master problem\n",
+ "# D[i, t] is d_i^t the degree of vertex i in column number t\n",
"function IPM(C, D)\n",
" ipm = Model(Cbc.Optimizer)\n",
" set_silent(ipm)\n",
@@ -260,14 +323,22 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 6,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "LPM (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"function LPM(C, D)\n",
" lpm = Model(Cbc.Optimizer)\n",
@@ -291,7 +362,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "In (LPM) there is a column $ ( c^T x^t , d_1^t , \\ldots , d_n^t , 1 )^T$ for every $x^t \\in X^1 \\ (t = 1, \\ldots, T_1)$ \n",
+ "In (LPM) there is a column $ ( c^{\\dagger} x^t , d_1^t , \\ldots , d_n^t , 1 )^{\\dagger}$ for every $x^t \\in X^1 \\ (t = 1, \\ldots, T_1) $ ($\\dagger$ means transpose in this context) \n",
+ "\n",
"The second equation is a convexity constraint\n",
"\n",
"Associating the second equation with dual variable $\\mu$ and the degree constraints with dual variables $\\{\\pi_i\\}_{i=1}^m$ the dual of (LPM) is\n",
@@ -299,26 +371,46 @@
"$$\\text{(DM)} \\quad\n",
"\\begin{array}{r l}\n",
"\\max_{\\mu, \\pi} & \\sum_{i=1}^m 2 \\pi_i + \\mu \\\\\n",
- "\\text{s.t.} & \\pi^T d^t + \\mu \\leq c^T x^t \\quad (t = 1 \\ldots T_1)\n",
+ "\\text{s.t.} & \\pi^{\\dagger} d^t + \\mu \\leq c^{\\dagger} x^t \\quad (t = 1 \\ldots T_1)\n",
"\\end{array}$$"
]
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
+ "execution_count": 7,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "DM (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
+ "# Dual Master problem\n",
+ "# C is a vector such that C[t] = sum_e∈E(c_e * x_e^t) for t in 1:T_1\n",
+ "# where T_1 is the number of columns of the master problem\n",
+ "# D[i, t] is d_i^t the degree of vertex i in column number t, for i in 1:n, t in 1:T_1\n",
+ "\n",
"function DM(C, D)\n",
" dm = Model(Cbc.Optimizer)\n",
" set_silent(dm)\n",
" \n",
- " @variable(dm, π[1:size(D,1)])\n",
+ " n, T_1 = size(D) #number of vertices, number of columns\n",
+ " \n",
+ " @variable(dm, π[1:n])\n",
" @variable(dm, μ)\n",
" \n",
" @objective(dm, Max, 2*sum(π) + μ)\n",
" \n",
- " @constraint(dm, oneTree[t in 1:size(D,2)], dot(π, D[:,t]) + μ <= C[t])\n",
+ " @constraint(dm, oneTree[t in 1:T_1], dot(π, D[:,t]) + μ <= C[t])\n",
" \n",
" optimize!(dm)\n",
" checkTermination(dm)\n",
@@ -332,26 +424,53 @@
"metadata": {},
"source": [
"The associated subsidiary problem is the following\n",
- "\n"
+ "\n",
+ "\n",
+ "$$\\begin{array}{r l}\n",
+ "\\min c_x - \\pi^{\\dagger} D_x - \\mu \\\\\n",
+ "0 \\le x_e \\le 1 & e \\in E \\\\\n",
+ "x_e \\in \\mathbb{Z} & e \\in E \\\\\n",
+ "\\sum_{e \\in \\delta(1)} x_e = 2 \\\\\n",
+ "\\sum_{e \\in E [X] } x_e \\le |X| - 1 & X \\subseteq \\{2,...,n\\}, X \\ne \\emptyset \\\\\n",
+ "\\sum_{e \\in E} x_e = n \\\\\n",
+ "%\\sum_{e \\in \\delta(v)} x_e = 2 & v \\in \\{2,...,n\\} \\\\\n",
+ "\\end{array}$$"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 8,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "SP (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
- "function SP()\n",
+ "function SP(W, π, μ)\n",
+ " n = size(W,1)\n",
+ "\n",
" sp = Model(Cbc.Optimizer)\n",
" set_silent(sp)\n",
" \n",
- " @variable(sp, )\n",
+ " @variable(sp, X[u in 1:n, v in 1:n], Bin)\n",
" \n",
- " @objective(sp, Min, (c - π^T A ) x - μ)\n",
+ " @objective(sp, Min, sum(W .* X)/2 - dot(π, [sum(X[u,:]) for u ∈ 1:n]) - μ)\n",
" \n",
- " @constraint(sp, degree1)\n",
- " @constraint(sp, nbEdges, sum(x[2:]) == )\n",
- " @constraint(sp, spanningTree)\n",
+ " @constraint(sp, degree1, sum(X[1,:]) == 2)\n",
+ " @constraint(sp, nbEdges, sum(X) == 2*n)\n",
+ " for np in 2:n, X2N in collect(permutations(collect(2:np),np-1))\n",
+ " @constraint(sp, sum(X[X2N[i], X2N[i+1]] for i in 1:length(X2N)-1) + X[X2N[length(X2N)], X2N[1]] <= length(X2N) - 1)\n",
+ " end\n",
+ " @constraint(sp, symmetry[u in 1:n, v in 1:n], X[u,v] == X[v,u])\n",
+ " @constraint(sp, diagonal[u in 1:n], X[u,u] == 0)\n",
" \n",
" optimize!(sp)\n",
" checkTermination(sp)\n",
@@ -368,7 +487,7 @@
"source": [
"Our algorithm will follow this flowchart, classic for column generation algorithm\n",
"\n",
- ""
+ ""
]
},
{
@@ -379,12 +498,12 @@
"source": [
"Our first \"dummy\" basis will consist in a collection of \"wheel\"-like 1-trees, as described in Held and Karp's paper\n",
"\n",
- ""
+ ""
]
},
{
"cell_type": "code",
- "execution_count": 6,
+ "execution_count": 9,
"metadata": {
"tags": []
},
@@ -392,26 +511,32 @@
{
"data": {
"text/plain": [
- "wheel_like (generic function with 1 method)"
+ "wheel_like_1trees (generic function with 1 method)"
]
},
- "execution_count": 6,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
+ "# Returns an array of n-1 trivial 1-trees of a complete graph with n vertices\n",
"function wheel_like_1trees(n)\n",
" res = []\n",
+ " \n",
+ " # k is the \"center\" vertex of a wheel like tree : all other vertices are connected to it\n",
" for k in 2:n\n",
" one_tree = zeros(n,n)\n",
" \n",
- " for v in union(2:k-1, k+1:n)\n",
+ " for v in union(1:k-1, k+1:n)\n",
" one_tree[k,v] = 1 ; one_tree[v,k] = 1\n",
" end\n",
" \n",
- " one_tree[1,k] = 1 ; one_tree[1, n==k ? 2 : k+1] = 1\n",
- " one_tree[k,1] = 1 ; one_tree[n==k ? 2 : k+1, 1] = 1 \n",
+ " # After the center is connected to all vertices,\n",
+ " # vertex 1 is connected to k+1 (or 2 if k = n), denoted as k_next,\n",
+ " # to form a 1-tree\n",
+ " k_next = n==k ? 2 : k+1\n",
+ " one_tree[1, k_next] = 1 ; one_tree[k_next, 1] = 1 \n",
" \n",
" push!(res, one_tree)\n",
" end\n",
@@ -423,47 +548,69 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {},
+ "metadata": {
+ "tags": []
+ },
"outputs": [],
"source": [
- "function column_generation(W)\n",
+ "wl = wheel_like_1trees(6)\n",
+ "\n",
+ "for t in wl\n",
+ " display_S(t)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "function column_generation(W; max_iter = 1000)\n",
" n = size(W, 1)\n",
" one_trees_init = wheel_like_1trees(n)\n",
" \n",
" columns = one_trees_init[1:n-2]\n",
" tree_degrees = [sum(columns[t][u,:]) for u ∈ 1:n, t ∈ 1:n-2]\n",
- " tree_weights = [dot(columns[t], W) for t ∈ 1:n-2]\n",
+ " tree_weights = [dot(columns[t], W)/2 for t ∈ 1:n-2]\n",
" \n",
" reduced_cost = -1\n",
" new_column = one_trees_init[n-1]\n",
+ " k = 1\n",
" \n",
- " while reduced_cost < 0\n",
+ " while reduced_cost < 0 && k <= max_iter\n",
" \n",
" push!(columns, new_column)\n",
" tree_degrees = hcat(tree_degrees, [sum(new_column[u,:]) for u ∈ 1:n]) \n",
- " push!(tree_weights, dot(new_column, W))\n",
+ " push!(tree_weights, dot(new_column, W)/2)\n",
" \n",
" dm = DM(tree_weights, tree_degrees)\n",
" π = value.(dm[:π])\n",
" μ = value(dm[:μ])\n",
" \n",
" # Solve subproblem\n",
- " sp = SP()\n",
- " new_column = value.(sp[:??????])\n",
+ " sp = SP(W, π, μ)\n",
+ " new_column = value.(sp[:X])\n",
" new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
- " new_weight = dot(new_column, W)\n",
+ " new_weight = dot(new_column, W)/2\n",
" reduced_cost = objective_value(sp)\n",
+ " \n",
+ " k += 1\n",
" end\n",
" \n",
" lpm = LPM(tree_weights, tree_degrees)\n",
" # ipm = IPM(tree_weights, tree_degrees)\n",
" \n",
" λ = value.(lpm[:λ])\n",
- " println(\"Lambda_t : \", λ)\n",
+ " #println(\"Lambda_t : \", λ)\n",
" println(\"Minimal 1-tree \")\n",
" println(\"Lambda : \", maximum(λ))\n",
" println(\"Index : \", argmax(λ))\n",
- " println(\"1-tree : \", columns[argmax(λ)])\n",
+ " #println(\"1-tree : \", columns[argmax(λ)])\n",
+ " println(\"Weight : \", tree_weights[argmax(λ)])\n",
+ " println(\"k : \", k)\n",
" \n",
" return columns[argmax(λ)]\n",
"end"
@@ -475,8 +622,16 @@
"metadata": {},
"outputs": [],
"source": [
- "display(column_generation(W))\n",
- "display(column_generation(W2))"
+ "display_S(column_generation(example))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "display_S(column_generation(exercise))"
]
},
{
@@ -485,236 +640,325 @@
"source": [
"## Held and Karp algorithm - attempt 1\n",
"\n",
- "This is the part where we implement the Held and Karp algorithm as described in J.E. Mitchell's slides using Kruskal's algorithm to solve the minimum spanning tree problem.\n",
+ "This is the part where we implement the Held and Karp algorithm as described in J.E. Mitchell's slides using Kruskal's algorithm to solve the minimum spanning tree problem. Since the goal of this algorithm is allegedly to provide a new column to the master problem, there is no guarantee that it gives an optimal solution.\n",
"\n",
- "However this approach seems to be flawed as the example is easily solved, however the exercise seems to alternate between two identical states. \n",
+ "Even though this approach seems to be flawed, the example is easily solved ; however the exercise seems to alternate between two identical states. \n",
"This may be due to the update rule of $\\lambda$, results may vary by modifying it."
]
},
{
"cell_type": "code",
- "execution_count": 88,
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "update_rule_3! (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function update_rule!(λ, D, Ï::T) where T <: Integer\n",
+ " λ .= λ .+ Ï .* (2 .- D)\n",
+ "end\n",
+ "\n",
+ "update_rule_1!(λ, D) = update_rule!(λ, D, 1)\n",
+ "update_rule_2!(λ, D) = update_rule!(λ, D, 2)\n",
+ "update_rule_3!(λ, D) = update_rule!(λ, D, 3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "held_and_karp_iter! (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function held_and_karp_iter!(W, W_updated, π, X, D, g, update_rule)\n",
+ " n = size(W, 1)\n",
+ " X .= falses(n, n) #X is reset every iteration because a different 1-tree is computed\n",
+ " W_updated .= convert(Matrix{Float64}, deepcopy(W)) #we need to modify the weights according to λ\n",
+ " \n",
+ " # Minimum spanning tree (span_tree : st)\n",
+ " for u in 2:n, v in 2:n\n",
+ " W_updated[u,v] -= π[u] + π[v]\n",
+ " end\n",
+ " st = kruskal_mst(g, W_updated[2:n,2:n])\n",
+ " \n",
+ " # Conversion of the spanning tree to an adjacency matrix\n",
+ " for i ∈ 1:size(st, 1)\n",
+ " X[src(st[i])+1, dst(st[i])+1] = 1\n",
+ " X[dst(st[i])+1, src(st[i])+1] = 1\n",
+ " end\n",
+ " \n",
+ " # We connect vertex 1 to two vertices to obtain a 1-tree\n",
+ " W_updated[1,2:n] .-= π[2:n]\n",
+ " connect_v1!(X, W_updated[1,:], n)\n",
+ " \n",
+ " #then we compute the objective value\n",
+ " z = 2*sum(π) + sum((W[u,v] - π[u] - π[v])*X[u,v] for u ∈ 1:n for v ∈ 1:u)\n",
+ " \n",
+ " #the degrees of all vertices in graph X\n",
+ " D = [sum(X[u,:]) for u ∈ 1:n]\n",
+ " \n",
+ " if all(D .== 2) # If not a tour, update\n",
+ " return true, z\n",
+ " else\n",
+ " update_rule(Ï€, D)\n",
+ " end\n",
+ " \n",
+ " return false, z\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "multiargmin (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "#returns the nbmin indexes for smallest values of itr\n",
+ "#size is the length of itr\n",
+ "function multiargmin(itr, nbmin, size)\n",
+ " if nbmin > size\n",
+ " error(\"multiargmin : size of array is \", size, \" < nbmin = \", nbmin)\n",
+ " end\n",
+ " \n",
+ " argmins = [i for i in 1:nbmin]\n",
+ " itr_copy = deepcopy(itr)\n",
+ " maxitr = maximum(itr)\n",
+ " \n",
+ " #we find the nbmin smallest values of itr\n",
+ " for i in 1:nbmin\n",
+ " argmins[i] = argmin(itr_copy)\n",
+ " #once a min is found, it is set to maxitr to find a new minimum\n",
+ " itr_copy[argmins[i]] = maxitr\n",
+ " end\n",
+ " \n",
+ " return argmins\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
- "hk (generic function with 1 method)"
+ "connect_v1! (generic function with 1 method)"
]
},
- "execution_count": 88,
+ "execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "function hk(W)\n",
+ "#connects the vertex 1 (of graph X) to the two vertices v1, v2 with smallest W1[v1], W1[v2]\n",
+ "#X is an adjacency matrix, W1 the weights of edges starting from 1, n the number of vertices\n",
+ "function connect_v1!(X, W1, n) \n",
+ " argmin1, argmin2 = multiargmin(W1[2:n], 2, n-1)\n",
+ " \n",
+ " X[1, argmin1+1] = 1\n",
+ " X[1, argmin2+1] = 1\n",
+ " \n",
+ " for u ∈ 2:n\n",
+ " X[u, 1] = X[1, u]\n",
+ " end \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "function hk_mitchell(W; disp=true, maxiter=10000, update_rule=update_rule_2!)\n",
" \n",
" # Initialization\n",
" n = size(W, 1)\n",
" \n",
- " if size(W, 2)!=n\n",
- " error(\"Weight matrice must be square\")\n",
- " end\n",
- " # \n",
+ " if size(W, 2)!=n ; error(\"Weight matrice must be square\") ; end\n",
" \n",
" λ = zeros(n)\n",
"\n",
- " W_updated = W\n",
- " g = complete_graph(n-1)\n",
- " z = -1\n",
- " S = zeros(n,n)\n",
- " D = zeros(n)\n",
- " while D != fill(2, n) # z!=sum(W .* S) # While not a tour\n",
- " \n",
- " # print(z, \" \")\n",
- " \n",
- " # Minimum spanning tree (span_tree : st)\n",
- " for u in 1:n\n",
- " for v in 1:n\n",
- " W_updated[u,v] -= λ[u] + λ[v]\n",
- " end\n",
- " end\n",
- " st = kruskal_mst(g, W_updated[2:n,2:n])\n",
+ " g = complete_graph(n-1) #in the TSP the graph is complete\n",
+ " z = -1 #the objective function\n",
+ " \n",
+ " k = 0 \n",
+ " X = Matrix{Bool}(undef, (n,n)) #the adjacency matrix of the 1-tree solutions\n",
+ " D = Array{Integer}(undef, n)\n",
+ " W_updated = Matrix{Integer}(undef, (n,n))\n",
+ " \n",
+ " converge = false #whether the algorithm has converged or not\n",
"\n",
- " # println(st)\n",
- " \n",
- " # Connect vertex 1\n",
- " m = Model(Cbc.Optimizer) # PL(NE) Model\n",
- " set_silent(m)\n",
- " \n",
- " @variable(m, X[1:n,1:n], Bin)\n",
+ " while !converge && k < maxiter # While not a tour\n",
" \n",
- " @objective(m, Min, \n",
- " sum(W .* X) + \n",
- " sum(λ[j]*(2-sum(X[1,:])) for j in 2:n)\n",
- " )\n",
+ " converge, z = held_and_karp_iter!(W, W_updated, λ, X, D, g, update_rule)\n",
" \n",
- " @constraint(m, degreeRoot, sum(X[1,:]) == 2)\n",
- " for i ∈ 1:size(st, 1)\n",
- " @constraint(m, X[src(st[i])+1, dst(st[i])+1] == 1)\n",
- " end\n",
- " #@constraint(m, spanningTree[i ∈ 1:size(st,1)], X[src(st[i]), dst(st[i])] == 1)\n",
- " for u ∈ 1:n\n",
- " for v ∈ 1:n\n",
- " @constraint(m, X[u,v] == X[v,u])\n",
+ " # We display the current 1-Tree\n",
+ " if disp && (k == 0 || k == maxiter || converge)\n",
+ " println(\"Iteration\", k)\n",
+ " \n",
+ " one_tree = SimpleGraph(X)\n",
+ " edgelabel = []\n",
+ " for i in 1:n, j in i:n\n",
+ " if X[i, j] == 1\n",
+ " push!(edgelabel, W_updated[i, j])\n",
+ " end\n",
" end\n",
- " @constraint(m, X[u,u] == 0)\n",
- " end\n",
- " \n",
- " #@constraint(m, diagonalMatrix[u ∈ 1:n, v ∈ 1:n], X[u,v] == X[v,u])\n",
- " \n",
- " #@constraint(m, diagonalNulle[i ∈ 1:n], X[i,i] == 0)\n",
- " \n",
- " optimize!(m)\n",
- " \n",
- " if !( \n",
- " ( termination_status(m) == MOI.OPTIMAL ) || \n",
- " ( termination_status(m) == MOI.TIME_LIMIT \n",
- " && has_values(m) ) \n",
- " )\n",
- " println(termination_status(m))\n",
- " println(termination_status(m))\n",
- " println(has_values(m))\n",
- " println(m)\n",
- " error(\"Couldn't connect vertex 1 (PLNE failed).\")\n",
+ " display(gplot(one_tree, nodelabel = 1:n, edgelabel = edgelabel, edgelabelc=\"red\"))\n",
" end\n",
" \n",
- " z = objective_value(m)\n",
- " S = value.(X)\n",
- " D = [sum(S[u,:]) for u ∈ 1:n]\n",
- " # For the degree, we suppose a null diagonal, to be checked later on\n",
- " \n",
- " if D != fill(2, n) # z!=sum(W .* S) # If not a tour, update\n",
- " λ .= λ .+ 2 .* (2 .- D) # Experimenter d'autres règles\n",
- " end\n",
- " \n",
- " println(D)\n",
- " \n",
+ " k+=1\n",
+ "\n",
" end\n",
" \n",
- " return S\n",
+ " return (z, X, λ)\n",
" \n",
"end"
]
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": null,
"metadata": {
- "collapsed": true,
- "jupyter": {
- "outputs_hidden": true
- },
"tags": []
},
- "outputs": [
- {
- "ename": "UndefVarError",
- "evalue": "UndefVarError: hk not defined",
- "output_type": "error",
- "traceback": [
- "UndefVarError: hk not defined",
- "",
- "Stacktrace:",
- " [1] top-level scope at In[9]:1"
- ]
- }
- ],
+ "outputs": [],
"source": [
- "columnGeneration(W)\n",
- "columnGeneration(W2)\n",
- "display(hk(W))\n",
- "display(hk(W2))"
+ "z, X, λ = hk_mitchell(example; disp=true, maxiter=100000)"
]
},
{
- "cell_type": "markdown",
- "metadata": {},
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
+ "tags": []
+ },
+ "outputs": [],
"source": [
- "## Held and Karp algorithm - attempt 2\n",
- "\n",
- "Here we will work on the same Master Problem as seen in \"STSP using column generation\". \n",
- "However the subsidiary problem will be solved using the algorithm presented by Held and Karp in the section 4 \"An ascent method\" of their paper (which correspond to the one J.E. Mitchell presents in his slides)."
+ "z, X, λ = hk_mitchell(exercise; disp=true, maxiter=100000, update_rule=update_rule_1!)"
]
},
{
- "cell_type": "code",
- "execution_count": null,
+ "cell_type": "markdown",
"metadata": {},
- "outputs": [],
"source": [
- "function held_and_karp_algo()\n",
- " \n",
- "end"
+ "Linear changes in the update rule do not seem to impact the results"
]
},
{
- "cell_type": "code",
- "execution_count": null,
+ "cell_type": "markdown",
"metadata": {},
- "outputs": [],
"source": [
- "function update_rule!(λ, D, Ï)\n",
- " λ .= λ .+ Ï .* (2 .- D)\n",
- "end\n",
+ "## Held and Karp algorithm - attempt 2\n",
"\n",
- "update_rule_1!(λ, D) = update_rule!(λ, D, 1)\n",
- "update_rule_2!(λ, D) = update_rule!(λ, D, 2)\n",
- "update_rule_3!(λ, D) = update_rule!(λ, D, 3)\n",
- "update_rule_05!(λ, D) = update_rule!(λ, D, 0.5)"
+ "Here we will work on the same Master Problem as seen in \"STSP using column generation\". \n",
+ "However the subsidiary problem will be solved using the algorithm presented by Held and Karp in the section 4 \"An ascent method\" of their paper (which correspond to the one J.E. Mitchell presents in his slides)."
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 14,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "held_and_karp_2 (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
- "function held_and_karp_2(W)\n",
+ "function held_and_karp_2(W; update_rule = update_rule_1!, max_iter = 1000)\n",
" n = size(W, 1)\n",
" one_trees_init = wheel_like_1trees(n)\n",
" \n",
" columns = one_trees_init[1:n-2]\n",
" tree_degrees = [sum(columns[t][u,:]) for u ∈ 1:n, t ∈ 1:n-2]\n",
- " tree_weights = [dot(columns[t], W) for t ∈ 1:n-2]\n",
+ " tree_weights = [dot(columns[t], W)/2 for t ∈ 1:n-2]\n",
" \n",
" reduced_cost = -1\n",
" new_column = one_trees_init[n-1]\n",
" new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
- " new_weight = dot(new_column, W)\n",
+ " new_weight = dot(new_column, W)/2\n",
" \n",
- " while reduced_cost < 0\n",
+ " g = complete_graph(n-1)\n",
+ " k = 0\n",
+ " \n",
+ " while reduced_cost < 0 && k < max_iter\n",
" \n",
" push!(columns, new_column)\n",
" tree_degrees = hcat(tree_degrees, new_degree)\n",
- " push!(tree_weights, new_weights)\n",
+ " push!(tree_weights, new_weight)\n",
" \n",
" dm = DM(tree_weights, tree_degrees)\n",
" π = value.(dm[:π])\n",
" μ = value.(dm[:μ])\n",
" \n",
- " new_column = held_and_karp_algo(W, , , update_rule_2!)\n",
- " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
- " new_weight = dot(new_column, W)\n",
" \n",
+ " held_and_karp_iter!(W, convert(Matrix{Float64}, deepcopy(W)), π, new_column, new_degree, g, update_rule)\n",
+ " #new_column = held_and_karp_algo(W, , , update_rule_2!)\n",
+ " #reduced_cost, new_column, _ = hk_mitchell(W; disp=false, maxiter=100000, update_rule=update_rule_1!)\n",
+ " #new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
+ " new_weight = dot(new_column, W)/2\n",
" reduced_cost = new_weight - dot(π, new_degree) - μ\n",
+ " \n",
+ " k += 1\n",
" end\n",
" \n",
" lpm = LPM(tree_weights, tree_degrees)\n",
" # ipm = IPM(tree_weights, tree_degrees)\n",
" \n",
" λ = value.(lpm[:λ])\n",
- " println(\"Lambda_t : \", λ)\n",
+ " #println(\"Lambda_t : \", λ)\n",
" println(\"Minimal 1-tree \")\n",
" println(\"Lambda : \", maximum(λ))\n",
" println(\"Index : \", argmax(λ))\n",
- " println(\"1-tree : \", columns[argmax(λ)])\n",
+ " #println(\"1-tree : \", columns[argmax(λ)])\n",
" \n",
" return columns[argmax(λ)] \n",
"end"
@@ -722,12 +966,768 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 15,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
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+ "Minimal 1-tree \n",
+ "Lambda : 0.20000000000000057\n",
+ "Index : 6"
+ ]
+ },
+ {
+ "data": {
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+ " stroke=\"none\"\n",
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Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}}(Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}[Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.6821788259538699cx, -1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((1.0cx, -0.024579038352964933cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.015391914719724453cx, -0.7078754749312393cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.3237098892056093cx, 0.44397345427767054cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.44654244691936396cx, 0.7108758132873532cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, 1.0cy), 0.04082482904638631w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}}(Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}[Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.6821788259538699cx, -1.0cy), \"1\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((1.0cx, -0.024579038352964933cy), \"2\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.015391914719724453cx, -0.7078754749312393cy), \"3\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.3237098892056093cx, 0.44397345427767054cy), \"4\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.44654244691936396cx, 0.7108758132873532cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-1.0cx, 1.0cy), \"6\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\n",
+ "1-tree : [0.0 1.0 1.0 0.0 0.0 0.0; 1.0 0.0 0.0 0.0 1.0 0.0; 1.0 0.0 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 0.0 1.0 1.0; 0.0 1.0 0.0 1.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0]\n"
+ ]
+ }
+ ],
"source": [
- "display(held_and_karp_2(W))\n",
- "display(held_and_karp_2(W2))"
+ "display_S(held_and_karp_2(example))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
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+ "Minimal 1-tree \n",
+ "Lambda : 0.16666666666666707\n",
+ "Index : 3\n",
+ "1-tree : [0.0 0.0 0.0 1.0 1.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0 0.0; 1.0 1.0 1.0 0.0 1.0 1.0 1.0; 1.0 0.0 0.0 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0 0.0; 0.0"
+ ]
+ },
+ {
+ "data": {
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+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ " 0.0 0.0 1.0 0.0 0.0 0.0]\n"
+ ]
+ }
+ ],
+ "source": [
+ "display_S(held_and_karp_2(exercise))"
]
},
{
@@ -739,7 +1739,7 @@
"Whilst column generation based techniques work well on wide graphs, on smaller ones we may want to use a different method.\n",
"\n",
"Here we use dynamical programming to implement the Bellman-Held-Karp algorithm. \n",
- "It works well on small graph, but scales terribly because of its exponential complexity $O(2^n n^2)$ in time (and $O(2^n n)$ in space)"
+ "It works well on small graphs, but scales terribly because of its exponential complexity $O(2^n n^2)$ in time (and $O(2^n n)$ in space)."
]
},
{
@@ -748,8 +1748,95 @@
"metadata": {},
"outputs": [],
"source": [
- "function bellman_held_karp()\n",
+ "function dp_held_karp(W,n)\n",
+ " C = Dict{Tuple{Array,Integer},Integer}()\n",
+ "\n",
+ " for k in collect(2:n)\n",
+ " C[(Array([k]),k)]= collect(W[1,k])[1]\n",
+ " end\n",
+ " \n",
+ " for s in 2:(n-1), S in collect(combinations(collect(2:n),s)), k in S\n",
+ " Sp = [s for s in S if (s!=k)]\n",
+ " C[(S,k)] = collect(typemax(Int64))[1]\n",
+ " for m in S\n",
+ " if ( m != k ) && ( C[(S,k)] >= C[(Sp,m)] + W[m,k] )\n",
+ " C[(S,k)] = C[(Sp,m)] + W[m,k]\n",
+ " end\n",
+ " end\n",
+ " end\n",
+ " \n",
+ " opt = Inf\n",
+ " \n",
+ " for k in 2:n\n",
+ " if opt >= C[(collect(2:n),k)] + W[k,1]\n",
+ " opt = C[(collect(2:n),k)] + W[k,1]\n",
+ " end\n",
+ " end\n",
+ " \n",
+ " return opt\n",
+ "end\n",
+ "\n",
+ "function getPath(W,n,opt)\n",
+ " println(\"Valeur optimale : \",opt)\n",
+ " S = collect(permutations(collect(1:n),n))\n",
+ " idgood = Integer[]\n",
+ " Sp = [Integer[]]\n",
+ " M=Array{Integer,2}[]\n",
+ "\n",
+ " for i in 1:length(S)\n",
+ " if (sommeTour(W,S[i],opt))\n",
+ " append!(idgood,i)\n",
+ " end\n",
+ " end\n",
+ "\n",
+ " for index in idgood\n",
+ " if !(matriceAdj(S[index]) in M)\n",
+ " append!(M,[matriceAdj(S[index])])\n",
+ " append!(Sp,[S[index]])\n",
+ " end\n",
+ " end\n",
+ " println(\"Circuits optimaux : \")\n",
+ " for i in 1:length(Sp)\n",
+ " if length(Sp[i]) != 0\n",
+ " println(Sp[i])\n",
+ " end \n",
+ " end\n",
" \n",
+ " return M\n",
+ "end\n",
+ "\n",
+ "function sommeTour(W,L,opt)\n",
+ " somme = 0\n",
+ " n = length(L)\n",
+ " for i in 1:(n-1)\n",
+ " somme+= W[L[i] ,L[i+1]]\n",
+ " end\n",
+ " somme+= W[L[n],L[1]]\n",
+ " return (somme==opt)\n",
+ "end\n",
+ "function matriceAdj(L)\n",
+ " n = length(L)\n",
+ " Madj = zeros(Integer,n,n)\n",
+ " for i in 1:(n-1)\n",
+ " Madj[L[i],L[i+1]] = 1\n",
+ " Madj[L[i+1],L[i]] = 1\n",
+ " end\n",
+ " Madj[L[n],L[1]] = 1\n",
+ " Madj[L[1],L[n]] = 1\n",
+ " return Madj\n",
+ "end\n",
+ "\n",
+ "bellman_held_karp(W) = getPath(W, size(W,1), dp_held_karp(W, size(W,1)))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "for M in bellman_held_karp(example)\n",
+ " display_S(M)\n",
"end"
]
},
@@ -759,8 +1846,9 @@
"metadata": {},
"outputs": [],
"source": [
- "display(bellman_held_karp(W))\n",
- "display(bellman_held_karp(W2))"
+ "for M in bellman_held_karp(exercise)\n",
+ " display_S(M)\n",
+ "end"
]
},
{
@@ -785,27 +1873,14 @@
},
{
"cell_type": "code",
- "execution_count": 3,
+ "execution_count": null,
"metadata": {
- "collapsed": true,
"jupyter": {
- "outputs_hidden": true,
"source_hidden": true
},
"tags": []
},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "columnGeneration (generic function with 1 method)"
- ]
- },
- "execution_count": 3,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
+ "outputs": [],
"source": [
"function columnGeneration(W)\n",
" \n",
diff --git a/heldAndKarp.ipynb b/heldAndKarp.ipynb
index 3a86dce..50f404e 100644
--- a/heldAndKarp.ipynb
+++ b/heldAndKarp.ipynb
@@ -69,7 +69,7 @@
"source": [
"using LinearAlgebra # Basic algebra structures & functions\n",
"using JuMP # Modeling language\n",
- "using GLPK, Cbc # LP and MIP solvers\n",
+ "using Cbc # MIP solver\n",
"using LightGraphs # Efficient graph structures & functions\n",
"using GraphPlot # Graph visualization\n",
"using Combinatorics # Combinatorials\n",
@@ -424,12 +424,22 @@
"metadata": {},
"source": [
"The associated subsidiary problem is the following\n",
- "\n"
+ "\n",
+ "\n",
+ "$$\\begin{array}{r l}\n",
+ "\\min c_x - \\pi^{\\dagger} D_x - \\mu \\\\\n",
+ "0 \\le x_e \\le 1 & e \\in E \\\\\n",
+ "x_e \\in \\mathbb{Z} & e \\in E \\\\\n",
+ "\\sum_{e \\in \\delta(1)} x_e = 2 \\\\\n",
+ "\\sum_{e \\in E [X] } x_e \\le |X| - 1 & X \\subseteq \\{2,...,n\\}, X \\ne \\emptyset \\\\\n",
+ "\\sum_{e \\in E} x_e = n \\\\\n",
+ "%\\sum_{e \\in \\delta(v)} x_e = 2 & v \\in \\{2,...,n\\} \\\\\n",
+ "\\end{array}$$"
]
},
{
"cell_type": "code",
- "execution_count": 20,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
@@ -438,7 +448,7 @@
"SP (generic function with 1 method)"
]
},
- "execution_count": 20,
+ "execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
@@ -552,22 +562,11 @@
},
{
"cell_type": "code",
- "execution_count": 29,
+ "execution_count": null,
"metadata": {
"tags": []
},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "column_generation (generic function with 1 method)"
- ]
- },
- "execution_count": 29,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
+ "outputs": [],
"source": [
"function column_generation(W; max_iter = 1000)\n",
" n = size(W, 1)\n",
@@ -619,798 +618,63 @@
},
{
"cell_type": "code",
- "execution_count": 30,
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "display_S(column_generation(example))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "display_S(column_generation(exercise))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Held and Karp algorithm - attempt 1\n",
+ "\n",
+ "This is the part where we implement the Held and Karp algorithm as described in J.E. Mitchell's slides using Kruskal's algorithm to solve the minimum spanning tree problem. Since the goal of this algorithm is allegedly to provide a new column to the master problem, there is no guarantee that it gives an optimal solution.\n",
+ "\n",
+ "Even though this approach seems to be flawed, the example is easily solved ; however the exercise seems to alternate between two identical states. \n",
+ "This may be due to the update rule of $\\lambda$, results may vary by modifying it."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
"metadata": {},
"outputs": [
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+ "update_rule_3! (generic function with 1 method)"
]
},
+ "execution_count": 10,
"metadata": {},
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- "k : 11\n"
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"source": [
- "display_S(column_generation(example))"
+ "function update_rule!(λ, D, Ï::T) where T <: Integer\n",
+ " λ .= λ .+ Ï .* (2 .- D)\n",
+ "end\n",
+ "\n",
+ "update_rule_1!(λ, D) = update_rule!(λ, D, 1)\n",
+ "update_rule_2!(λ, D) = update_rule!(λ, D, 2)\n",
+ "update_rule_3!(λ, D) = update_rule!(λ, D, 3)"
]
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- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Minimal 1-tree \n",
- "Lambda : 1.0\n",
- "Index : 18\n",
- "Weight : 4.0\n",
- "k : 1001\n"
- ]
- }
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- "source": [
- "display_S(column_generation(exercise))"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Held and Karp algorithm - attempt 1\n",
- "\n",
- "This is the part where we implement the Held and Karp algorithm as described in J.E. Mitchell's slides using Kruskal's algorithm to solve the minimum spanning tree problem. Since the goal of this algorithm is allegedly to provide a new column to the master problem, there is no guarantee that it gives an optimal solution.\n",
- "\n",
- "Even though this approach seems to be flawed, the example is easily solved ; however the exercise seems to alternate between two identical states. \n",
- "This may be due to the update rule of $\\lambda$, results may vary by modifying it."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 32,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "update_rule_05! (generic function with 1 method)"
- ]
- },
- "execution_count": 32,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "function update_rule!(λ, D, Ï)\n",
- " λ .= λ .+ Ï .* (2 .- D)\n",
- "end\n",
- "\n",
- "update_rule_1!(λ, D) = update_rule!(λ, D, 1)\n",
- "update_rule_2!(λ, D) = update_rule!(λ, D, 2)\n",
- "update_rule_3!(λ, D) = update_rule!(λ, D, 3)\n",
- "update_rule_05!(λ, D) = update_rule!(λ, D, 0.5)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 33,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -1419,1027 +683,302 @@
"held_and_karp_iter! (generic function with 1 method)"
]
},
- "execution_count": 33,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function held_and_karp_iter!(W, W_updated, π, X, D, g, update_rule)\n",
- " n = size(W, 1)\n",
- " X .= falses(n, n) #X is reset every iteration because a different 1-tree is computed\n",
- " W_updated .= deepcopy(W) #we need to modify the weights according to λ\n",
- " \n",
- " # Minimum spanning tree (span_tree : st)\n",
- " for u in 2:n, v in 2:n\n",
- " W_updated[u,v] -= π[u] + π[v]\n",
- " end\n",
- " st = kruskal_mst(g, W_updated[2:n,2:n])\n",
- " \n",
- " # Conversion of the spanning tree to an adjacency matrix\n",
- " for i ∈ 1:size(st, 1)\n",
- " X[src(st[i])+1, dst(st[i])+1] = 1\n",
- " X[dst(st[i])+1, src(st[i])+1] = 1\n",
- " end\n",
- " \n",
- " # We connect vertex 1 to two vertices to obtain a 1-tree\n",
- " W_updated[1,2:n] .-= π[2:n]\n",
- " connect_v1!(X, W_updated[1,:], n)\n",
- " \n",
- " #then we compute the objective value\n",
- " z = 2*sum(π) + sum((W[u,v] - π[u] - π[v])*X[u,v] for u ∈ 1:n for v ∈ 1:u)\n",
- " \n",
- " #the degrees of all vertices in graph X\n",
- " D = [sum(X[u,:]) for u ∈ 1:n]\n",
- " \n",
- " if all(D .== 2) # If not a tour, update\n",
- " return true, z\n",
- " else\n",
- " update_rule(Ï€, D)\n",
- " end\n",
- " \n",
- " return false, z\n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 34,
- "metadata": {
- "tags": []
- },
- "outputs": [
- {
- "data": {
- "text/plain": [
- "multiargmin (generic function with 1 method)"
- ]
- },
- "execution_count": 34,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "#returns the nbmin indexes for smallest values of itr\n",
- "#size is the length of itr\n",
- "function multiargmin(itr, nbmin, size)\n",
- " if nbmin > size\n",
- " error(\"multiargmin : size of array is \", size, \" < nbmin = \", nbmin)\n",
- " end\n",
- " \n",
- " argmins = [i for i in 1:nbmin]\n",
- " itr_copy = deepcopy(itr)\n",
- " maxitr = maximum(itr)\n",
- " \n",
- " #we find the nbmin smallest values of itr\n",
- " for i in 1:nbmin\n",
- " argmins[i] = argmin(itr_copy)\n",
- " #once a min is found, it is set to maxitr to find a new minimum\n",
- " itr_copy[argmins[i]] = maxitr\n",
- " end\n",
- " \n",
- " return argmins\n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 35,
- "metadata": {
- "tags": []
- },
- "outputs": [
- {
- "data": {
- "text/plain": [
- "connect_v1! (generic function with 1 method)"
- ]
- },
- "execution_count": 35,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "#connects the vertex 1 (of graph X) to the two vertices v1, v2 with smallest W1[v1], W1[v2]\n",
- "#X is an adjacency matrix, W1 the weights of edges starting from 1, n the number of vertices\n",
- "function connect_v1!(X, W1, n) \n",
- " argmin1, argmin2 = multiargmin(W1[2:n], 2, n-1)\n",
- " \n",
- " X[1, argmin1+1] = 1\n",
- " X[1, argmin2+1] = 1\n",
- " \n",
- " for u ∈ 2:n\n",
- " X[u, 1] = X[1, u]\n",
- " end \n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 36,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "hk_mitchell (generic function with 1 method)"
- ]
- },
- "execution_count": 36,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "function hk_mitchell(W; disp=true, maxiter=10000, update_rule=update_rule_2!)\n",
- " \n",
- " # Initialization\n",
- " n = size(W, 1)\n",
- " \n",
- " if size(W, 2)!=n ; error(\"Weight matrice must be square\") ; end\n",
- " \n",
- " λ = zeros(n)\n",
- "\n",
- " g = complete_graph(n-1) #in the TSP the graph is complete\n",
- " z = -1 #the objective function\n",
- " \n",
- " k = 0 \n",
- " X = Matrix{Bool}(undef, (n,n)) #the adjacency matrix of the 1-tree solutions\n",
- " D = Array{Integer}(undef, n)\n",
- " W_updated = Matrix{Integer}(undef, (n,n))\n",
- " \n",
- " converge = false #whether the algorithm has converged or not\n",
- "\n",
- " while !converge && k <= maxiter # While not a tour\n",
- " \n",
- " converge, z = held_and_karp_iter!(W, W_updated, λ, X, D, g, update_rule)\n",
- " \n",
- " # We display the current 1-Tree\n",
- " if disp && (k == 0 || k == maxiter || converge)\n",
- " println(\"Iteration\", k)\n",
- " \n",
- " one_tree = SimpleGraph(X)\n",
- " edgelabel = []\n",
- " for i in 1:n, j in i:n\n",
- " if X[i, j] == 1\n",
- " push!(edgelabel, W_updated[i, j])\n",
- " end\n",
- " end\n",
- " display(gplot(one_tree, nodelabel = 1:n, edgelabel = edgelabel, edgelabelc=\"red\"))\n",
- " end\n",
- " \n",
- " k+=1\n",
- "\n",
- " end\n",
- " \n",
- " return (z, X, λ)\n",
- " \n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 37,
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- "outputs": [
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- ],
+ " n = size(W, 1)\n",
+ " X .= falses(n, n) #X is reset every iteration because a different 1-tree is computed\n",
+ " W_updated .= convert(Matrix{Float64}, deepcopy(W)) #we need to modify the weights according to λ\n",
+ " \n",
+ " # Minimum spanning tree (span_tree : st)\n",
+ " for u in 2:n, v in 2:n\n",
+ " W_updated[u,v] -= π[u] + π[v]\n",
+ " end\n",
+ " st = kruskal_mst(g, W_updated[2:n,2:n])\n",
+ " \n",
+ " # Conversion of the spanning tree to an adjacency matrix\n",
+ " for i ∈ 1:size(st, 1)\n",
+ " X[src(st[i])+1, dst(st[i])+1] = 1\n",
+ " X[dst(st[i])+1, src(st[i])+1] = 1\n",
+ " end\n",
+ " \n",
+ " # We connect vertex 1 to two vertices to obtain a 1-tree\n",
+ " W_updated[1,2:n] .-= π[2:n]\n",
+ " connect_v1!(X, W_updated[1,:], n)\n",
+ " \n",
+ " #then we compute the objective value\n",
+ " z = 2*sum(π) + sum((W[u,v] - π[u] - π[v])*X[u,v] for u ∈ 1:n for v ∈ 1:u)\n",
+ " \n",
+ " #the degrees of all vertices in graph X\n",
+ " D = [sum(X[u,:]) for u ∈ 1:n]\n",
+ " \n",
+ " if all(D .== 2) # If not a tour, update\n",
+ " return true, z\n",
+ " else\n",
+ " update_rule(Ï€, D)\n",
+ " end\n",
+ " \n",
+ " return false, z\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
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+ "multiargmin (generic function with 1 method)"
]
},
+ "execution_count": 12,
"metadata": {},
- "output_type": "display_data"
- },
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "#returns the nbmin indexes for smallest values of itr\n",
+ "#size is the length of itr\n",
+ "function multiargmin(itr, nbmin, size)\n",
+ " if nbmin > size\n",
+ " error(\"multiargmin : size of array is \", size, \" < nbmin = \", nbmin)\n",
+ " end\n",
+ " \n",
+ " argmins = [i for i in 1:nbmin]\n",
+ " itr_copy = deepcopy(itr)\n",
+ " maxitr = maximum(itr)\n",
+ " \n",
+ " #we find the nbmin smallest values of itr\n",
+ " for i in 1:nbmin\n",
+ " argmins[i] = argmin(itr_copy)\n",
+ " #once a min is found, it is set to maxitr to find a new minimum\n",
+ " itr_copy[argmins[i]] = maxitr\n",
+ " end\n",
+ " \n",
+ " return argmins\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
{
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Iteration0\n",
- "Iteration2\n"
- ]
+ "data": {
+ "text/plain": [
+ "connect_v1! (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "#connects the vertex 1 (of graph X) to the two vertices v1, v2 with smallest W1[v1], W1[v2]\n",
+ "#X is an adjacency matrix, W1 the weights of edges starting from 1, n the number of vertices\n",
+ "function connect_v1!(X, W1, n) \n",
+ " argmin1, argmin2 = multiargmin(W1[2:n], 2, n-1)\n",
+ " \n",
+ " X[1, argmin1+1] = 1\n",
+ " X[1, argmin2+1] = 1\n",
+ " \n",
+ " for u ∈ 2:n\n",
+ " X[u, 1] = X[1, u]\n",
+ " end \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "function hk_mitchell(W; disp=true, maxiter=10000, update_rule=update_rule_2!)\n",
+ " \n",
+ " # Initialization\n",
+ " n = size(W, 1)\n",
+ " \n",
+ " if size(W, 2)!=n ; error(\"Weight matrice must be square\") ; end\n",
+ " \n",
+ " λ = zeros(n)\n",
+ "\n",
+ " g = complete_graph(n-1) #in the TSP the graph is complete\n",
+ " z = -1 #the objective function\n",
+ " \n",
+ " k = 0 \n",
+ " X = Matrix{Bool}(undef, (n,n)) #the adjacency matrix of the 1-tree solutions\n",
+ " D = Array{Integer}(undef, n)\n",
+ " W_updated = Matrix{Integer}(undef, (n,n))\n",
+ " \n",
+ " converge = false #whether the algorithm has converged or not\n",
+ "\n",
+ " while !converge && k < maxiter # While not a tour\n",
+ " \n",
+ " converge, z = held_and_karp_iter!(W, W_updated, λ, X, D, g, update_rule)\n",
+ " \n",
+ " # We display the current 1-Tree\n",
+ " if disp && (k == 0 || k == maxiter || converge)\n",
+ " println(\"Iteration\", k)\n",
+ " \n",
+ " one_tree = SimpleGraph(X)\n",
+ " edgelabel = []\n",
+ " for i in 1:n, j in i:n\n",
+ " if X[i, j] == 1\n",
+ " push!(edgelabel, W_updated[i, j])\n",
+ " end\n",
+ " end\n",
+ " display(gplot(one_tree, nodelabel = 1:n, edgelabel = edgelabel, edgelabelc=\"red\"))\n",
+ " end\n",
+ " \n",
+ " k+=1\n",
+ "\n",
+ " end\n",
+ " \n",
+ " return (z, X, λ)\n",
+ " \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "z, X, λ = hk_mitchell(example; disp=true, maxiter=100000)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "jupyter": {
+ "source_hidden": true
},
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "z, X, λ = hk_mitchell(exercise; disp=true, maxiter=100000, update_rule=update_rule_1!)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Linear changes in the update rule do not seem to impact the results"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Held and Karp algorithm - attempt 2\n",
+ "\n",
+ "Here we will work on the same Master Problem as seen in \"STSP using column generation\". \n",
+ "However the subsidiary problem will be solved using the algorithm presented by Held and Karp in the section 4 \"An ascent method\" of their paper (which correspond to the one J.E. Mitchell presents in his slides)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {},
+ "outputs": [
{
"data": {
"text/plain": [
- "(22.0, Bool[0 0 … 0 1; 0 0 … 1 0; … ; 0 1 … 0 0; 1 0 … 0 0], [0.0, 2.0, 0.0, 0.0, 0.0, -2.0])"
+ "held_and_karp_2 (generic function with 1 method)"
]
},
- "execution_count": 37,
+ "execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "z, X, λ = hk_mitchell(example; disp=true, maxiter=100000)"
+ "function held_and_karp_2(W; update_rule = update_rule_1!, max_iter = 1000)\n",
+ " n = size(W, 1)\n",
+ " one_trees_init = wheel_like_1trees(n)\n",
+ " \n",
+ " columns = one_trees_init[1:n-2]\n",
+ " tree_degrees = [sum(columns[t][u,:]) for u ∈ 1:n, t ∈ 1:n-2]\n",
+ " tree_weights = [dot(columns[t], W)/2 for t ∈ 1:n-2]\n",
+ " \n",
+ " reduced_cost = -1\n",
+ " new_column = one_trees_init[n-1]\n",
+ " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
+ " new_weight = dot(new_column, W)/2\n",
+ " \n",
+ " g = complete_graph(n-1)\n",
+ " k = 0\n",
+ " \n",
+ " while reduced_cost < 0 && k < max_iter\n",
+ " \n",
+ " push!(columns, new_column)\n",
+ " tree_degrees = hcat(tree_degrees, new_degree)\n",
+ " push!(tree_weights, new_weight)\n",
+ " \n",
+ " dm = DM(tree_weights, tree_degrees)\n",
+ " π = value.(dm[:π])\n",
+ " μ = value.(dm[:μ])\n",
+ " \n",
+ " \n",
+ " held_and_karp_iter!(W, convert(Matrix{Float64}, deepcopy(W)), π, new_column, new_degree, g, update_rule)\n",
+ " #new_column = held_and_karp_algo(W, , , update_rule_2!)\n",
+ " #reduced_cost, new_column, _ = hk_mitchell(W; disp=false, maxiter=100000, update_rule=update_rule_1!)\n",
+ " #new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
+ " new_weight = dot(new_column, W)/2\n",
+ " reduced_cost = new_weight - dot(π, new_degree) - μ\n",
+ " \n",
+ " k += 1\n",
+ " end\n",
+ " \n",
+ " lpm = LPM(tree_weights, tree_degrees)\n",
+ " # ipm = IPM(tree_weights, tree_degrees)\n",
+ " \n",
+ " λ = value.(lpm[:λ])\n",
+ " #println(\"Lambda_t : \", λ)\n",
+ " println(\"Minimal 1-tree \")\n",
+ " println(\"Lambda : \", maximum(λ))\n",
+ " println(\"Index : \", argmax(λ))\n",
+ " #println(\"1-tree : \", columns[argmax(λ)])\n",
+ " \n",
+ " return columns[argmax(λ)] \n",
+ "end"
]
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 15,
"metadata": {},
"outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Lambda_t : [0.19999999999999976, 0.20000000000000004, 0.2, 0.19999999999999962, 0.0, 0.20000000000000057, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}}(Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}[Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.6821788259538699cx, -1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((1.0cx, -0.024579038352964933cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.015391914719724453cx, -0.7078754749312393cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.3237098892056093cx, 0.44397345427767054cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.44654244691936396cx, 0.7108758132873532cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, 1.0cy), 0.04082482904638631w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}}(Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}[Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.6821788259538699cx, -1.0cy), \"1\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((1.0cx, -0.024579038352964933cy), \"2\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.015391914719724453cx, -0.7078754749312393cy), \"3\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.3237098892056093cx, 0.44397345427767054cy), \"4\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.44654244691936396cx, 0.7108758132873532cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-1.0cx, 1.0cy), \"6\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
]
},
"metadata": {},
@@ -2880,7 +1327,29 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "Iteration0"
+ "\n",
+ "1-tree : [0.0 1.0 1.0 0.0 0.0 0.0; 1.0 0.0 0.0 0.0 1.0 0.0; 1.0 0.0 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 0.0 1.0 1.0; 0.0 1.0 0.0 1.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0]\n"
+ ]
+ }
+ ],
+ "source": [
+ "display_S(held_and_karp_2(example))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
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+ "Minimal 1-tree \n",
+ "Lambda : 0.16666666666666707\n",
+ "Index : 3\n",
+ "1-tree : [0.0 0.0 0.0 1.0 1.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0 0.0; 1.0 1.0 1.0 0.0 1.0 1.0 1.0; 1.0 0.0 0.0 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0 0.0; 0.0"
]
},
{
@@ -2901,123 +1370,88 @@
" <path d=\"M0,0 L15,3.5 L0,7 z\" stroke=\"context-stroke\" fill=\"context-stroke\"/>\n",
" </marker>\n",
"</defs>\n",
- "<g stroke-width=\"1.13\" fill=\"#000000\" fill-opacity=\"0.000\" stroke=\"#D3D3D3\" id=\"img-8a9d3829-1\">\n",
- " <g transform=\"translate(59.61,16.63)\">\n",
- " <path fill=\"none\" d=\"M-10.89,-5.9 L10.89,5.9 \" class=\"primitive\"/>\n",
+ "<g stroke-width=\"1.13\" fill=\"#000000\" fill-opacity=\"0.000\" stroke=\"#D3D3D3\" id=\"img-12f8a343-1\">\n",
+ " <g transform=\"translate(100.41,49.59)\">\n",
+ " <path fill=\"none\" d=\"M23.66,-0.23 L-23.66,0.23 \" class=\"primitive\"/>\n",
" </g>\n",
- " <g transform=\"translate(28.04,14.07)\">\n",
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List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), 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Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-1.0cx, -0.7246308077740307cy), \"4\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.6978513137166641cx, -0.22060239147852467cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((1.0cx, 1.0cy), \"6\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.5794492270816001cx, 0.5533416217520393cy), \"7\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
+ "Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), Compose.UnitBox{Float64,Float64,Float64,Float64}(-1.2, -1.2, 2.4, 2.4, 0.0mm, 0.0mm, 0.0mm, 0.0mm), nothing, nothing, nothing, List([Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.LinePrimitive}(Compose.LinePrimitive[Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.9055179626946348cx, -0.015327511771497975cy), (0.10254599595431403cx, -0.004194395087874463cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.9726174984022002cx, 0.07379905786545382cy), (0.8524151020155163cx, 0.4707931125210726cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.3205965923109369cx, -0.9106557381905526cy), (0.03882390517517571cx, -0.0922286739970505cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.9081029217123578cx, 0.21633066379097596cy), (-0.08383311963869336cx, 0.01910418555682785cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.08581966463742435cx, 0.050805639170215355cy), (0.7472768944292412cx, 0.5075396137004773cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.05494558143630249cx, -0.07330010139687326cy), (-0.6350695485917968cx, -0.7216119113037657cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.01609823126742584cx, 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"metadata": {},
@@ -3323,140 +1722,10 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "\n",
- "Iteration100000\n"
- ]
- },
- {
- "data": {
- "text/plain": [
- "(3.0, Bool[0 0 … 0 0; 0 0 … 0 1; … ; 0 0 … 0 1; 0 1 … 1 0], [0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0])"
- ]
- },
- "execution_count": 38,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "z, X, λ = hk_mitchell(exercise; disp=true, maxiter=100000, update_rule=update_rule_1!)"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "Linear changes in the update rule do not seem to impact the results"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Held and Karp algorithm - attempt 2\n",
- "\n",
- "Here we will work on the same Master Problem as seen in \"STSP using column generation\". \n",
- "However the subsidiary problem will be solved using the algorithm presented by Held and Karp in the section 4 \"An ascent method\" of their paper (which correspond to the one J.E. Mitchell presents in his slides)."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 46,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "held_and_karp_2 (generic function with 1 method)"
- ]
- },
- "execution_count": 46,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "function held_and_karp_2(W; update_rule = update_rule_1!)\n",
- " n = size(W, 1)\n",
- " one_trees_init = wheel_like_1trees(n)\n",
- " \n",
- " columns = one_trees_init[1:n-2]\n",
- " tree_degrees = [sum(columns[t][u,:]) for u ∈ 1:n, t ∈ 1:n-2]\n",
- " tree_weights = [dot(columns[t], W)/2 for t ∈ 1:n-2]\n",
- " \n",
- " reduced_cost = -1\n",
- " new_column = one_trees_init[n-1]\n",
- " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
- " new_weight = dot(new_column, W)/2\n",
- " \n",
- " g = complete_graph(n-1)\n",
- " converge = false\n",
- " \n",
- " while reduced_cost < 0\n",
- " \n",
- " push!(columns, new_column)\n",
- " tree_degrees = hcat(tree_degrees, new_degree)\n",
- " push!(tree_weights, new_weight)\n",
- " \n",
- " dm = DM(tree_weights, tree_degrees)\n",
- " π = value.(dm[:π])\n",
- " μ = value.(dm[:μ])\n",
- " \n",
- " \n",
- " held_and_karp_iter!(W, deepcopy(W), vcat([μ], π), new_column, new_degree, g, update_rule)\n",
- " #new_column = held_and_karp_algo(W, , , update_rule_2!)\n",
- " #reduced_cost, new_column, _ = hk_mitchell(W; disp=false, maxiter=100000, update_rule=update_rule_1!)\n",
- " #new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
- " new_weight = dot(new_column, W)/2\n",
- " reduced_cost = new_weight - dot(vcat([μ], π), new_degree)\n",
- " end\n",
- " \n",
- " lpm = LPM(tree_weights, tree_degrees)\n",
- " # ipm = IPM(tree_weights, tree_degrees)\n",
- " \n",
- " λ = value.(lpm[:λ])\n",
- " println(\"Lambda_t : \", λ)\n",
- " println(\"Minimal 1-tree \")\n",
- " println(\"Lambda : \", maximum(λ))\n",
- " println(\"Index : \", argmax(λ))\n",
- " println(\"1-tree : \", columns[argmax(λ)])\n",
- " \n",
- " return columns[argmax(λ)] \n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 47,
- "metadata": {},
- "outputs": [
- {
- "ename": "InexactError",
- "evalue": "InexactError: Int64(12.439024390243906)",
- "output_type": "error",
- "traceback": [
- "InexactError: Int64(12.439024390243906)",
- "",
- "Stacktrace:",
- " [1] Int64 at .\\float.jl:710 [inlined]",
- " [2] convert at .\\number.jl:7 [inlined]",
- " [3] setindex! at .\\array.jl:828 [inlined]",
- " [4] held_and_karp_iter!(::Array{Int64,2}, ::Array{Int64,2}, ::Array{Float64,1}, ::Array{Float64,2}, ::Array{Float64,1}, ::SimpleGraph{Int64}, ::typeof(update_rule_1!)) at .\\In[33]:8",
- " [5] held_and_karp_2(::Array{Int64,2}; update_rule::Function) at .\\In[46]:28",
- " [6] held_and_karp_2(::Array{Int64,2}) at .\\In[46]:2",
- " [7] top-level scope at In[47]:1"
+ " 0.0 0.0 1.0 0.0 0.0 0.0]\n"
]
}
],
- "source": [
- "display_S(held_and_karp_2(example))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
"source": [
"display_S(held_and_karp_2(exercise))"
]
--
GitLab