4. Fitness landscape analysis for multi-objective optimization - PowerPoint PPT Presentation

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL 4. Fitness landscape analysis for multi-objective optimization problems Fitness landscape analysis for understanding and designing local search heuristics S´ ebastien Verel LISIC - Universit´ e du Littoral Cˆ ote d’Opale, Calais, France http://www-lisic.univ-littoral.fr/~verel/ The 51st CREST Open Workshop Tutorial on Landscape Analysis University College London 27th, February, 2017

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Outline of this part Basis of fitness landscape (Done) Geometries of fitness landscapes (Done) Local optima network : (Done) Definition inspired by complex systems science (Done) Features of the network, design and performance (Done) Performance prediction and portfolio (Done) Fitness landscape analysis for multi-objective optimization problems Brief introduction on MO and MO algorithms Fitness landscapes features Performance prediction based on MO features Set-based fitness landscape

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Multiobjective optimization Multiobjective optimization problem • Search space : decision space X R m • Objective function : objective space f : X → I component f i : objective, criterion. x 2 f 2 x 1 f 1 Decision space Objective space

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Definition Pareto dominance relation A solution x ∈ X dominates a solution x ′ ∈ X ( x ′ ≺ x ) iff ∀ i ∈ { 1 , 2 , . . . , M } , f i ( x ′ ) � f i ( x ) ∃ j ∈ { 1 , 2 , . . . , M } such that f j ( x ′ ) < f j ( x ) x 2 f 2 non- dominated dominated vector non- dominated solution vector x 1 f 1 Decision space Objective space

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Pareto set, Pareto front x 2 f 2 Pareto front efficient solution Efficient set x 1 f 1 Decision space Objective space Goal of multi-objective optimization Find the whole Pareto Optimal Set , or a good approximation of the Pareto Optimal Set

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Why multiobjective optimization ? Position of multiobjective optimization : Decision making No a priori on the importance of the different objectives a posteriori selection by a decision marker : Selection of one Pareto optimal solution after a deep study of the possible solutions.

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Multi-objective, many-objective optimization Approximative definition Multi-objective : 2, 3 or 4 objectives Many-objective : 4, 5 and more objectives Number of Pareto optimal solutions Suppose that : Probability to improve : p for all objective, Objective are independent. Probability to be non-dominated for m objectives is :

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Multi-objective, many-objective optimization Approximative definition Multi-objective : 2, 3 or 4 objectives Many-objective : 4, 5 and more objectives Number of Pareto optimal solutions Suppose that : Probability to improve : p for all objective, Objective are independent. Probability to be non-dominated for m objectives is : 1 − p m

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Multi-objective, many-objective optimization Approximative definition Multi-objective : 2, 3 or 4 objectives Many-objective : 4, 5 and more objectives Number of Pareto optimal solutions Suppose that : Probability to improve : p for all objective, Objective are independent. Probability to be non-dominated for m objectives is : 1 − p m Intuitive goals Convergence toward the front, and diversity of the solutions. Many-objective : convergence ”easy”, diversity ”hard” Note : objective correlation is also important.

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Performance assessment Quality of the approximation of the Pareto front [FKTZ05] Goal : indicator function related to the quality of the set. No universal indicator Indicator functions Hypervolume indicator Epsilon indicator Atteinment function :

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Performance assessment Quality of the approximation of the Pareto front [FKTZ05] Goal : indicator function related to the quality of the set. No universal indicator Indicator functions Hypervolume indicator Epsilon indicator Atteinment function : Probability to reach a point in objective space with algo. objective 1 0.5 0.55 0.6 0.65 0.7 0.75 [0.8, 1.0] [0.6, 0.8) 0.75 0.75 [0.4, 0.6) [0.2, 0.4) [0.0, 0.2) 0.7 0.7 objective 2 objective 2 0.65 0.65 0.6 0.6 0.55 0.55 0.5 0.5 0.5 0.55 0.6 0.65 0.7 0.75 objective 1 DLBS OD HEMO

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Hypervolume ( I H ) Properties f 2 Compliant with the (weak) Pareto dominance relation → A ≺ B ⇒ I H ( A ) � I H ( B ) A single parameter : the reference point Minimal solution-set maximizing I H reference f 1 point Objective space → subset of the Pareto optimal set Volume cover by the set. arg max σ ∈ Σ I H ( σ )

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Epsilon indicator ( I ε ) Definition Smallest coefficient ε to translate the set A to ”cover” each point of the set B ∀ z b ∈ B , ∃ z a ∈ A such that z b ≺ (1 + ε ) . z a Properties Compliant with the (weak) Pareto dominance relation (using Pareto front) → A ≺ B ⇒ I ε ( A ) � I ε ( B ) A parameter : the reference set

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Main types of MO local search algorithms Three main classes : Pareto-based approaches : directly or indirectly focus the search on the Pareto dominance relation. Pareto Local Search (PLS), Global SEMO, NSGA-II, etc. Indicator approaches : Progressively improvement the indicator function : IBEA, SMS-MOEA, etc. Scalar approaches : multiple scalarized aggregations of the objective functions : MOEA/D, etc. f f 2 2 supported solution Accept No accept f 1 f 1 Objective space Objective space

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Pareto-based approaches Simple dominance-based EMO with unbounded archive local search : global search : Pareto Local Search (PLS) Global-Simple EMO (G-SEMO) Pick a random solution x 0 ∈ X Pick a random solution x 0 ∈ X A ← { x 0 } A ← { x 0 } repeat repeat Select a non-visited x ∈ A Select x ∈ A at random Create x ′ by flipping each bit of Create N ( x ) by flipping each bit of x in turns x with a rate 1 / N Flag x as visited A ← non-dom. from A ∪ { x ′ } A ← non-dom. from A ∪ N ( x ) until all-visited ∨ maxeval until maxeval [Paquete et al. 2004][PCS04] [Laumanns et al. 2004][LTZ04]

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL A Pareto-based approach : Pareto Local Search • Archive solutions using Dominance relation • Iteratively improve this archive by exploring the neighborhood f f 2 2 Accept neighbor No Accept current current archive archive f 1 f 1 Objective space Objective space f f 2 2 Accept current current archive archive f 1 f 1 Objective space Objective space

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL SMS-MOEA : S metric selection- EMOA [Beume et al. 2007][BNE07] P ← initialization() repeat q ← Generate( P ) P ← Reduce( P ∪ { q } ) until maxeval Reduction Remove the worst solution according to non-dominated sorting, and S metric

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL IBEA : Indicator-Based Evolutionary algorithm [Zitzler et al. 2004][ZK04] P ← initialization() repeat ′ ← selection( P ) P ′ ) Q ← random variation( P Evaluation of Q P ← replacement( P , Q ) until maxeval Fitness assignment Pairwise comparison of solutions in a population w.r.t. indicator I Fitness value : ”loss in quality” in the population P if x was removed ′ , x ) /κ ) � ( − e − I ( x f ( x ) = x ′ ∈ P \{ x } Often the ε -indicator is used

Multiobjective Optimization Optimization algorithms Features in MO Prediction Set-based FL Original MOEA/D [ZL07] (minimization) /* µ sub-problems defined by µ directions */ ( λ 1 , . . . , λ µ ) ← initialization direction() Initialize ∀ i = 1 ..µ B ( i ) the neighboring sub-problems of sub-problem i /* one solution for each sub-problem */ ( x 1 , . . . , x µ ) ← initialization solution() repeat for i = 1 ..µ do ′ randomly in { x j : j ∈ B ( i ) } Select x and x ′ ) y ← mutation crossover( x , x for j ∈ B ( i ) do if g ( y | λ j , z ⋆ j ) < g ( x j | λ j , z ⋆ j ) then x j ← y end if end for end for until max eval B ( i ) is the set of the T closest neighboring sub-problems of sub-problem i g ( | λ i , z ⋆ i ) : scalar function of sub-pb. i with λ i direction, and z ⋆ i reference point