Empirical Convergence Analysis Of Genetic Algorithm For Solving Unit - PowerPoint PPT Presentation

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Empirical Convergence Analysis Of Genetic Algorithm For Solving Unit Commitment Problem Domen Butala Coauthors: doc. dr. Dejan Veluˇ sˇ cek doc. dr. Gregor Papa Ljubljana, September 13th, 2014 1

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography 1 Introduction 2 Convergence analysis An Upper Bound on the Convergence Speed Convergence of Homogenous Algorithm Combination of Both Approaches 3 Implementation Problem formulation Algorithm 4 Results and comparisons One-point crossover Multi-point crossover 5 Appendix 6 Authors 7 Bibliography 2

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Introduction Power system 3

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Introduction Power system Unit Commitment problem? 3

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Introduction Power system Unit Commitment problem? Motivation for an optimization approach 3

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Introduction Power system Unit Commitment problem? Motivation for an optimization approach Techniques as MIP, MILP, LR, Benders Decomposition, Dynamic Programming, . . . 3

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography An Upper Bound on the Convergence Speed An Upper Bound on the Convergence Speed Theorem [1] Let the size of population of the GA be n ≥ 1 , coding length l > 1 , mutation probability 0 < p m ≤ 1 2 and let { � X t , t ≥ 0 } be the Markov chain population, π ( t ) distribution of t th generation of � X t and π be the stationary distribution. Then it holds || π ( k ) − π || ≤ (1 − (2 p m ) nl ) k . 4

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Convergence of Homogenous Algorithm Convergence of Homogenous Algorithm Theorem [2] Let a , b , c > 0 be constants and i intensity perturbations of algorithm. If it holds m > an + c ( n − 1)∆ ⊗ min( a , b / 2 , c δ ) , (1) then ∀ x ∈ S N : t →∞ P ([ X i t ] ⊂ f ∗ | X i i →∞ lim lim 0 = x ) = 1 . 5

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Combination of Both Approaches Combination of Both Approaches Idea, to get the best algorithm possible, is to set a sequence of parameters { ( n t , p m ( t )) , t ≥} that it holds n t < n t +1 and p m ( t ) > p m ( t + 1). A Genetic algorithm set like this could be called a variable-structure GA [1]. 6

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Problem formulation Problem formulation � T n � � � ( mp i , t x type min + max { s i , t − s i , t − 1 , 0 } sc i ) i , t x type t =1 i =1 i , t n � x type ≥ PDP t ( price ), ∀ t (2) i , t i =1 � 1 , ”if x type � > 0 , i , t s i , t = , ∀ t , i (3) 0 , otherwise. st i , t = ( − 1) 1 − s i , t � � (4) 1 � I =[ t − a , t + b ] ∧ a , b ≥ 0: t ∀ ¯ s i , t = s i , ¯ t ∈ I ∧ s i , t − a − 1 = s i , t + b +1 =1 − s i , t st i , t ≥ tup i ∨ st i , t ≤ − tdown i , ∀ t , i (5) x i , t = xmax i , t (6) 7

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm Algorithm 1: t = 0 2: P ( t ) = SetInitialPopulation( P ) 8

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm Algorithm 1: t = 0 2: P ( t ) = SetInitialPopulation( P ) 3: Evaluate( P ( t )) 4: while not EndingCondition() do t + = 1 5: P ( t ) = Selection( P ( t − 1)) 6: 8

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm Algorithm 1: t = 0 2: P ( t ) = SetInitialPopulation( P ) 3: Evaluate( P ( t )) 4: while not EndingCondition() do t + = 1 5: P ( t ) = Selection( P ( t − 1)) 6: P ( t ) = Crossover( P ( t )) 7: 8

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm Algorithm 1: t = 0 2: P ( t ) = SetInitialPopulation( P ) 3: Evaluate( P ( t )) 4: while not EndingCondition() do t + = 1 5: P ( t ) = Selection( P ( t − 1)) 6: P ( t ) = Crossover( P ( t )) 7: P ( t ) = Mutation( P ( t )) 8: 8

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Algorithm Algorithm 1: t = 0 2: P ( t ) = SetInitialPopulation( P ) 3: Evaluate( P ( t )) 4: while not EndingCondition() do t + = 1 5: P ( t ) = Selection( P ( t − 1)) 6: P ( t ) = Crossover( P ( t )) 7: P ( t ) = Mutation( P ( t )) 8: Evaluate( P ( t )) 9: 10: end while 8

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography One-point crossover Results 9

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography One-point crossover 10

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Multi-point crossover 11

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Appendix Algorithm was implemented in the programming language Python. Numpy Cython Numba On some parts of the code performance comparisons were made to implementations in other languages (R, Matlab and C#). 12

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Domen Butala Financial Mathematics, Faculty of Mathematics and Physics, Ljubljana, Slovenia domen.butala@yahoo.com Dejan Veluˇ sˇ cek Department of Mathematics, Faculty of Mathematics and Physics, Ljubljana, Slovenia dejan.veluscek@fmf.uni-lj.si Gregor Papa Computer Systems Department, Joˇ zef Stefan Institute, Ljubljana, Slovenia gregor.papa@ijs.si 13

Introduction Convergence analysis Implementation Results and comparisons Appendix Authors Bibliography Y. Gao, An Upper Bound on the Convergence Rates of Canonical Genetic Algorithms . Complexity International, Vol. 5, 1998. R. Cerf, Asymptotic Convergence of Genetic Algorithms. CNRS, Universit´ e d’Orsay, Paris, 1997. 14