OmniOptimizer A Procedure for Single and Multiobjective Optimization
Omni-Optimizer A Procedure for Single and Multi-objective Optimization Prof. Kalyanmoy Deb and Santosh Tiwari
Motivation Generic Programming Practices Ø Unified algorithm for all types of optimization problems Ø An efficient implementation of NSGA-II framework (procedure) Ø Developing an efficient and self-adaptive optimization paradigm Ø
Literature Survey Ø Ø CHC (Cross generation elitist selection, Heterogeneous recombination, Cataclysmic mutation) – Explicit Diversity GENITOR (Steady state GA), more like (µ+1)-ES so far as selection mechanism is concerned. – High selection pressure Ø NPGA (Niched Pareto Genetic Algorithm), uses sharing parameter σshare – # of niches obtained depend on the sharing parameter
Literature Survey contd… Ø Ø Ø PESA (Pareto Envelope-based Selection Algorithm), Hypergrid division of phenotypic space, selection based on crowding measure NSGA-II (Non-dominated Sorting Genetic Algorithm) SPEA 2 (Strength Pareto Evolutionary Algorithm), Fine grained fitness assignment mechanism utilizing density information, Only archive members participate in mating – Excellent Diversity in phenotypic space Ø Ø Ø NCGA (Neighborhood Cultivation Genetic Algorithm), used neighborhood crossover, based on NSGA-II and SPEA 2 RPSGAe (Reduced Pareto Set Genetic Algorithm with elitism) ENORA (Evolutionary Algorithm of Non-dominated Sorting with Radial Slots)
Salient Features of the Algorithm Ø Based on NSGA-II framework Ø Ø Ø Based on the concept of Pareto dominance Incorporates elitism Explicit diversity preserving mechanism Can be used for single-objective as well as multiobjective problems Can be used for uni-global as well as multi-global problems Independent of the number of niches that an optimization problems exhibits
Moving beyond NSGA-II Ø Restricted Selection Scheme Ø Ø Ø Crowding Distance Assignment Ø Ø Tournament selection based on usual domination Non-dominated sorting based on epsilon dominance Genotypic as well as Phenotypic space niching Choose best members from above average population Remove worst members from below average population More robust recombination and variation operators Ø Ø Two point crossover for binary variables Highly disruptive real variable mutation
Restricted Selection Ø Ø Helps in preserving multi-modality Experiments show that it gives faster overall convergence
Epsilon Domination Principle n n n A finite percentage (based on function value) of solutions assigned a particular rank Allows somewhat inferior solutions to remain in the population Provides guaranteed diversity Helps to obtain multi-modal solutions in case of single objective problems Epsilon is generally taken in the range 10 -3 ~ 10 -6
Modified Crowding Distance Ø Genotypic as well as Phenotypic space niching
Highly Disruptive Mutation Operator
Computational Complexity Restricted selection O (n. N 2) Ø Ranking procedure O (MN 2) Ø Crowding procedure max{ O (MN log N), O (n. N log N) } Ø Overall iteration-wise complexity max {O (n. N 2), O (MN 2), O (n. N log N)} Ø
Implementation Details Ø Ø Code written in simple C and strictly conforms to ANSI/ISO standard Data structure used is a custom doubly linked list (gives O(1) insertion and deletion) Randomized quick sort used for sorting Almost all the functions perform in-place operation (addresses are passed, significantly decreases stack overheads)
Simulation Results Ø GA parameters for all problems chosen as follows Ø Ø Ø ηc = 20 ηm = 20 P (crossover) = 0. 8 P (mutation) = 1/n, where n = # of real variables δ = 0. 001 Population size and number of generations different for different problems
Simulation Results contd… Ø 20 variable Rastrigin function Ø # of function evaluation Ø Ø 20 variable Schwefel function Ø # of function evaluation Ø Ø Ø Least = 19260 Median = 24660 Worst = 29120 Least = 54950 Median = 69650 Worst = 103350 Other single objective problems can be found in the paper In all cases, better results are found in comparison to previous reported studies
Single objective multi-modal function f(x) = sin 2 (πx) x є [0, 20]
Single objective multi-modal function Unconstrained Himmelblau’s function
Multi-objective Uni-Global Test Problems 30 variable ZDT 2 (100× 100)
Multi-objective Uni-Global Test Problems 10 variable ZDT 4 (100× 250)
Multi-objective Uni-Global Test Problems CTP 4 (100× 7000)
Multi-objective Uni-Global Test Problems CTP 8 (100× 100)
Multi-objective Uni-Global Test Problems DTLZ 4 (300× 100)
Multi-objective Multi-Global Test Problem F 1 (x) = summation (sin (πxi) ) F 2 (x) = summation (cos (πxi) ) xi є [0, 6] Efficient points in phenotypic space
Multi-objective Multi-Global Test Problem Genotypic space plots
Few Sample Simulations F(x) = sin 2 (10, 000*pi*x) Ø Himmelblau’s Functions Ø ZDT Test Problems Ø CTP Test Problems Ø Test Problem TNK Ø Multi-global Multi-objective Test Problem Ø
Further Ideas and Future Work Incorporating PCX instead of SBX for crossover Ø Automatically fine-tuning mutation index so as to achieve arbitrary precision Ø Self-adaptation of parameter δ Ø Segregating population into niches without the introduction of DM Ø Dynamic population sizing Ø Using hierarchical NDS for the crowding distance assignment Ø
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