Lecture 6 Lin-Kernighan Heuristic. Simulated Annealing

31 Pages·2006·0.68 MB·English

by  Marco Chiarandini

DM63
HEURISTICS FOR
COMBINATORIAL OPTIMIZATION
Lecture 6
Lin-Kernighan Heuristic.
Simulated Annealing
Marco Chiarandini
Outline
1. Competition
2. Variable Depth Search
3. Simulated Annealing
DM63–HeuristicsforCombinatorialOptimizationProblems 2
Results - Boxplots of Errors
Trees Trees Trees Trees
city drilling grid unif
2812742569−MST
260581.pl−MST
090481−MST
Tours Tours Tours Tours
city drilling grid unif
2812742569−FA
260581.pl−NI
090481−RA
Fragments Fragments Fragments Fragments
city drilling grid unif
Stuetzle/tsp−test−NN
2812742569−NN
260581.pl−NN
090481−NN
20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80
err
TSP: Benchmark Instances, Examples
DM63–HeuristicsforCombinatorialOptimizationProblems 4
Results - Boxplots of Ranks
Trees Trees Trees Trees
city drilling grid unif
Stuetzle/tsp−test−NN
2812742569−NN
2812742569−MST
2812742569−FA
260581.pl−NN
260581.pl−NI
260581.pl−MST
090481−RA
090481−NN
090481−MST
Tours Tours Tours Tours
city drilling grid unif
Stuetzle/tsp−test−NN
2812742569−NN
2812742569−MST
2812742569−FA
260581.pl−NN
260581.pl−NI
260581.pl−MST
090481−RA
090481−NN
090481−MST
Fragments Fragments Fragments Fragments
city drilling grid unif
Stuetzle/tsp−test−NN
2812742569−NN
2812742569−MST
2812742569−FA
260581.pl−NN
260581.pl−NI
260581.pl−MST
090481−RA
090481−NN
090481−MST
2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10
ranks
Results - Scatter Plots: size vs time
090481−MST l
090481−NN l
090481−RA l
260581.pl−MST
260581.pl−NI
260581.pl−NN
2812742569−FA
2812742569−MST
2812742569−NN
Stuetzle/tsp−test−NN
10^2.410^2.610^2.810^3.0 10^2.410^2.610^2.810^3.0 10^2.410^2.610^2.810^3.0
Fragments Tours Trees
10^2 10^2
10^1 10^1
l
time 101^0−^10 ll lllll llll llllll lllllllllllllllllll ll lllll llll llllll lllllllllllllllllll ll lllll llll llllll llllllllllllllllll 1100^^−01
10^−2 10^−2
10^−3 10^−3
10^2.410^2.610^2.810^3.0 10^2.410^2.610^2.810^3.0 10^2.410^2.610^2.810^3.0
size
DM63–HeuristicsforCombinatorialOptimizationProblems 6
Software Framework for LS Methods
From EasyLocal++ by Schaerf and Di Gaspero (2003).
DM63–HeuristicsforCombinatorialOptimizationProblems 8
Variable Depth Search
I Key idea: Complex steps in large neighborhoods = variable-length
sequences of simple steps in small neighborhood.
I Use various feasibility restrictions on selection of simple search steps to
limit time complexity of constructing complex steps.
I Perform Iterative Improvement w.r.t. complex steps.
Variable Depth Search (VDS):
determine initial candidate solution s
tˆ:=s
While s is not locally optimal:
|| Repeat:
|| || select best feasible neighbor t
|| | If g(t)<g(tˆ): tˆ:=t
|| Until construction of complex step has been completed
b s:=tˆ
DM63–HeuristicsforCombinatorialOptimizationProblems 10
Example: The Lin-Kernighan (LK) Algorithm for the TSP (1)
I Complex search steps correspond to sequences
of 2-exchange steps and are constructed from
sequences of Hamiltonian paths
I δ-path: Hamiltonian path p + 1 edge connecting one end of p to interior
node of p (‘lasso’ structure):
u v
a)
u w v
b)
DM63–HeuristicsforCombinatorialOptimizationProblems 11
Basic LK exchange step:
I Start with Hamiltonian path (u,...,v):
u v
a)
I Obtain δ-path by adding an edge (v,w):
u w v
b)
I Break cycle by removing edge (w,v0):
u w v' v
c)
I Note: Hamiltonian path can be completed
into Hamiltonian cycle by adding edge (v0,u):
u w v' v
c)
DM63–HeuristicsforCombinatorialOptimizationProblems 12