Department of Computer Science and Information Technology MultiObjective

Department of Computer Science and Information Technology ﺑﻬیﻨﻪیﺎﺑی ﻣﺴﺎﺋﻞ چﻨﺪﻫﺪﻓﻪ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻟگﻮﺭیﺘﻢﻫﺎی ﺗﻘﺮیﺒی Multi-Objective Optimization Using Approximation Algorithms ﻣﻨﺼﻮﺭ ﺩﺍﻭﺩﻱ ﻣﻨﻔﺮﺩ Mansoor Davoodi mdmonfared@iasbs. ac. ir 1395 آﺬﺭ ﻣﺎﻩ

Multi-Objective Optimization (MOO) Cost Comfort

Multi-Objective Optimization (MOO) Solution space c f 2 Pa ret o-o pti Objective space ma l f 1

Multi-Objective Optimization (MOO) f 1 f 2

General Formulation of MOO

Goals in MOO f 2 f 1 Pareto-Optimality f 1 Diversity

Weighted Sum method

Weighted Sum method f 2 f 1

Multi-Objective Evolutionary Algorithms �Non-dominated Sorting Genetic Algorithm-II �Strength Pareto EA �Rudolph’s Elitist MOEA �Distance-Based Pareto GA �Thermodynamical GA �Pareto-Archived Evolution Strategy �…. s m e l b o r hmark P Benc and ics r t e M n o s i r a p m Co

Advantages and disadvantages of EAs Advantages General framework Reasonable resource consumption Parallel evolution Disadvantages No guarantee in finding optimal solutions Need to user parameter setting

Approximation Algorithms �

Approximation Algorithms-MCC

Approximation Algorithms-TSP Input: Graph G, a complete weighted graph. Output: Minimum weight tour that visits all the nodes A D E B C F G

Approximation Algorithms-TSP A A D D E E B B G F F C C MST Double MST A D E MST<OPTTSP B TSP tour ≤ 2 MST C F G TSP Tour G

Schematic view of approximation factor Minimization objective f 0 f(Opt) Maximization objective f 0 f(Opt)

Approximation of MOO What is the meaning of “MOO approximation” ? What is the difficulty of “MOO approximation” ? f 2 f 1

Approximation of MOO f 2 x x' f 1

Approximation of MOO f 2 f 1

Multi-objective optimization (MOO) 20

Multi-objective optimization (MOO) 21

Bi-objective Path Planning

Challenges � The real world is a trade-off world and it is full of conflict objectives. Any application of MOO approximation is …. ng ki ng or w et N ni an Pl g tin ca ng ri te us Lo Cl welcome. � Basic definitions of approximation Pareto-optimal curve. � Complexity of MOO approximation.

References � I. Diakonikolas, Approximation of Multiobjective Optimization Problems, Ph. D thesis, Columbia University, 2011. � M. Yannakakis, CH. Papadimitriou, On the approximability of offs - trade and optimal access of web source, 2000. � V. Roostapour, I. Kiarazm, M. Davoodi, Deterministic Algorithm for 1 -Median 1 - Center Two-Objective Optimization Problem. TTCS, Springer, 2016. � M. Davoodi, Bi-objective Path Planning using Deterministic Algorithms, submitted to Robotics and Autonomous Systems, March 2016. � C. C. A. Coello, D. A. Van Veldhuizen, G. B. Lamont, Evolutionary algorithms for solving multi-objective problems. New York, 2002.

Thanks… �� Any Question ? ? ?