Combinatorial Optimization
Chapter 8, Essentials of Metaheuristics, 2013
Spring, 2014
Metaheuristics
Byung-Hyun Ha
R2
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
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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
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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
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