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 آذر ماه 1395

Multi-Objective Optimization (MOO) Cost Comfort

Multi-Objective Optimization (MOO) Solution space c f2 Objective space Pareto-optimal f1

Multi-Objective Optimization (MOO) f1 f2

General Formulation of MOO

Goals in MOO f2 f2 f1 f1 Pareto-Optimality Diversity

Weighted Sum method

Weighted Sum method f2 f1

Weighted Sum method f2 f1

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 …. Benchmark Problems and Comparison Metrics

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

Approximation Algorithms Guaranteed to run in polynomial time for all instances. Guaranteed to find solution within ratio 𝛼 of optimum. Challenge. Need to prove a solution's value is close to optimum, without even knowing what optimum value is!

Approximation Algorithms-MCC

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

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

Schematic view of approximation factor Minimization objective f f(Opt) 𝑓 𝑥 ≤𝛼. 𝑓(𝑂𝑝𝑡) Maximization objective f 𝑓 𝑥 ≥𝛼. 𝑓(𝑂𝑝𝑡) f(Opt)

Approximation of MOO What is the meaning of “MOO approximation” ? What is the difficulty of “MOO approximation” ? f2 f2 f1 f1

Approximation of MOO 𝑓 1 𝑥′ = 𝑓 1 𝑥 𝛼 𝑓 2 𝑥′ = 𝑓 2 𝑥 𝛽 Def. Let 𝛱 be a bi-objective minimization problem with objectives 𝑓 1 and 𝑓 2 . Solution x is an 𝛼,𝛽 -approximation solution for 𝛱 if there is no solution y such that 𝑓 1 𝑥 >𝛼 𝑓 1 𝑦 and 𝑓 2 𝑥 ≥𝛽 𝑓 2 𝑦 , or 𝑓 1 𝑥 ≥𝛼 𝑓 1 𝑦 and 𝑓 2 𝑋 >𝛽 𝑓 2 𝑦 , f2 f1 𝑓 1 𝑥′ = 𝑓 1 𝑥 𝛼 𝑓 2 𝑥′ = 𝑓 2 𝑥 𝛽 x 𝛽 𝛼 x'

Approximation of MOO D𝐞𝐟. 𝜖-approximate Pareto curve is a set of solutions which are not dominated by any other by a ratio of more than 1+ 𝜖. f2 f1 𝜖-approximate Pareto curve P𝜖 is a set of solutions s such that there is no other solution 𝑠 ′ such that, for all s∈P𝜖, 𝑓 𝑖 𝑠 ′ ≥ 𝑓 𝑖 𝑠 1+𝜖 for some 𝑖 PType equation here.

Multi-objective optimization (MOO)

Multi-objective optimization (MOO) Optimal algorithm for Manhattan and 2 , 2 -approximation for the Euclidean metric 𝜃(n log n)

Bi-objective Path Planning

Bi-objective Path Planning

Bi-objective Path Planning

Bi-objective Path Planning

Bi-objective Path Planning 2 ,1 -approximation of the Euclidean path

Challenges … . Clustering Locating Planning Networking The real world is a trade-off world and it is full of conflict objectives. Any application of MOO approximation is welcome. Basic definitions of approximation Pareto-optimal curve. Complexity of MOO approximation. … . Clustering Locating Planning Networking

References I. Diakonikolas, Approximation of Multiobjective Optimization Problems, PhD 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.

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