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Combinatorial Optimization

Combinatorial Optimization. Chapter 8 Luke, Essentials of Metaheuristics, 2011 Byung-Hyun Ha. R2. Outline. Introduction Greedy Randomized Adaptive Search Procedures (GRASP) Ant Colony Optimization (ACO) Guided Local Search (GLS) Summary. Introduction. Combinatorial optimization

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Combinatorial Optimization

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  1. Combinatorial Optimization Chapter 8 Luke, Essentials of Metaheuristics, 2011 Byung-Hyun Ha R2

  2. Outline • Introduction • Greedy Randomized Adaptive Search Procedures (GRASP) • Ant Colony Optimization (ACO) • Guided Local Search (GLS) • Summary

  3. 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 qaulity • To assign infeasible solutions a poor quality • Hamming cliff?

  4. 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)

  5. 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)

  6. 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

  7. 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?

  8. 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?

  9. 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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