Combinatorial Optimization

Chapter 8, Essentials of Metaheuristics, 2013

Spring, 2014

Metaheuristics

Byung-Hyun Ha

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Outline

Introduction

Greedy Randomized Adaptive Search Procedures (GRASP)

Ant Colony Optimization (ACO)

Guided Local Search (GLS)

Summary

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Introduction

Combinatorial optimization Examples

• Knapsack, TSP, VRP, …

A solution consisting of components

Hard constraints Usually, in combinatorial optimization problems

• e.g., VRP with pickup and delivery time windows

General purpose metaheuristics with hard constraints Initial solution construction

• Choose component one by one that gives feasible

Tweaking• To invent a closed Tweak operator• To try repeatedly various Tweaks• To allow infeasible solutions with distance from feasible one as quality• To assign infeasible solutions a poor quality

• Hamming cliff?

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Introduction

Components of solution e.g., edges between cities for TSP, pairs of jobs for T-problem

Component-oriented methods Random selection of components

• Greedy Randomized Adaptive Search Procedures (GRASP)• Algorithm 108

Favoring good components• Ant Colony Optimization (ACO)

Punishing components related to local optima• Guided Local Search (GLS)

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Ant Colony Optimization

Two populations Set of components with pheromones as their fitness

• e.g., all edges of TSP• Pheromone: historical quality of component

Set of candidate solutions (ant trails)

Free from Tweaking, possibly

Algorithm 109 An Abstract Ant Colony Optimization Algorithm (ACO)

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Ant Colony Optimization

Ant System Algorithm 110

• The Ant System (AS)

Selection of components based on desirability

Initialization of pheromones• e.g., = 1, = popsize(1/C) where C is cost of tour constructed greedily

Evaporation and update of pheromones Hill-climbing (optional)

• Tweak, required

Algorithm 111• Pheromone Updating with a Learning Rate

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Ant Colony Optimization

Ant Colony System Changes from AS

• Elitist approach to updating pheromones• Learning rate in pheromone updates• Evaporating pheromones, slightly differently• Strong tendency to select components used in the best trail discovered

Algorithm 112• The Ant Colony System (ACS)

Elitist Component selection• With probability q, select component with highest desirability• Otherwise, do same as AS

Disregarding linkage among components• Jacks-of-all-trade problem

• c.f., N-population cooperative coevolution• Possible remedy: considering pairs of components?

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Guided Local Search

Avoiding some components for a solution Identifying components tending to cause local optima

• Components that appear too often in local optima

Penalizing solutions that use those components (toward exploration) c.f., Feature-based Tabu Search

Fitness by quality and penalty (pheromone)

Components whose pheromone is increased One with max. penalizability, in current solution

Algorithm 113 Guided Local Search (GLS) with Random Updates

• Detection of local optima?

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Summary

Combinatorial optimization

Hard constraints Difficulties in construction of initial solution and Tweaking

Component-oriented methods Randomly

• e.g., GRASP

Favoring with desirability• e.g., ACO

Punishing with penalizability• e.g., GLS