New Challenges for Memetic Algorithms on Continuous Multi-objective Problems MOEAs + Gradient-based Line Search in Multi-objective Adriana Lara López Oliver Schütze, Carlos A. Coello Coello. CINVESTAV, México. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 1 / 24
Outline Goal Efficiently combining gradient-based line search procedures with MOEAs. Introduction 1 Memetic Algorithms Continuous Multi-objective Local Search 2 Search Direction Step Length Control 3 Memetic Issues Balance, Archiving Conclusions and Future Work 4 Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 2 / 24
Outline of the Presentation Introduction 1 Memetic Algorithms Continuous Multi-objective Local Search 2 Memetic Issues 3 Conclusions and Future Work 4 Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 3 / 24
Introduction Multi-objective Memetic Algorithm Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 4 / 24
Introduction Memetic algorithms: H. Ishibuchi, Murata, et al. (since 2003). A. Jaszkiewick (since 2004). J. Knowles, Corne (since 2004). Talbi et al. Wanner et al. Yin, Sendhoff. Among others. Specific for continuous spaces: Hu, Huang, Wang et. al. (CEC 2003) Shukla (EMO 2005) Sindhya, Deb, Miettinen (PPSN 2008) Brown, Smith (2005) Bosman, deJong (GECCO 2005-2006) Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 5 / 24
Some Issues Combining several gradients: ( ∇ f 1 , . . . , ∇ f m ) . Compromise between the cost and the benefits of the local-search procedure. Efficient balance between the global and the local procedures (It has a mayor impact in the overall efficiency). Choose the solutions to apply the local search. Most benchmarks have intrinsic difficulties for methods that use gradient- based information. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 6 / 24
Outline of the Presentation Introduction 1 Continuous Multi-objective Local Search 2 Search Direction Step Length Control Memetic Issues 3 Conclusions and Future Work 4 Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 7 / 24
Continuous Multi-objective Local Search Let x ∈ R n be a certain solution for a problem. The line search approach aims to iteratively improve x using x new = x + t ν, (1) where ν ∈ R n is a certain search direction and t ∈ R is the step length. Efficiency issues: The computation of the search direction ν. The computation of the step size value t . Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 8 / 24
Single-objective Descent Direction Considering a point x , the negative of the gradient −∇ f i ( x ) := � T � ∂ x 1 ( x ) , . . . , ∂ f i ∂ f i − ∂ x n ( x ) gives us the maximum decreasing direction for f i . Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 9 / 24
Descent Cone For each f i , with i ∈ { 1 , . . . , m } we define: � ∇ f i ( x ) � � � v ∈ R n : H x , i = ||∇ f i ( x ) || , v = 0 � ∇ f i ( x ) � � � v ∈ R n : H − x , i = ||∇ f i ( x ) || , v ≤ 0 and call the Descent Cone of x to the set m � C x ( − , − , . . . , − ) = H − x , i . \ { x } . i = 1 Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 10 / 24
Which is a good search direction in multi-objective? A common direction v such that decreases all the functions simultaneously. A vector ν ∈ R n is called a descent direction of the point x ∈ R n if ν ∈ C x ( − , − , . . . , − ) . In other words, a multi-objective descent direction is such that the directional derivatives with respect to ν in x are non positive, i.e. �∇ f i ( x ) , ν � ≤ 0 for all i ∈ 1 , . . . , m without allowing them to be all equal to zero. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 11 / 24
How getting a Descent Direction? It is a multi-objective problem itself! Previous Work: Compute disc. directions for unconstrained MOPs (Bosman and deJong 2005-2006, Fliege 2001, Schäffler, Schultz, Weinzierl 2002). Compute them when box-constraints exist (Harada, Sakuma, Kobayahi 2006). HCS (Lara, Sánchez, Coello, Schütze, 2010) Directed Search (Schütze, Lara, Coello, 2010) Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 12 / 24
Directed Search (Schütze, Lara, Coello 2010) See for details: The Directed Search Method for Unconstrained Multi-objective Optimization Problems Oliver Schütze, Adriana Lara, Carlos Coello. Technical report. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 13 / 24
Open questions How to efficiently integrate them into a population-based context (MOEAs algorithms)? What is an effective search direction? Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 14 / 24
Step Length Control Once the multi-objective descent direction ν is set for a specific point x ∈ R n , we can just define the new line search function as f i : R − → R ν t �− → f i ( x + t ν ) . The computation of the step length of f i ν is again a multi-objective problem. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 15 / 24
Step Length Control Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 16 / 24
Step Length Control A practical choice: Since an exact step length calculation is not possible in this case, the use of inexact methods is a good option. The widely known Armijo-Goldstein rule to the multi-objective case, accepting any step length t that holds F ( x + t ν ) ≤ F ( x ) − c t JF x ν, where F : R n → R m is the multi-objective function and JF x : R n → R n is the Jacobian matrix of F at x . With a suitable initial step length, this method is easily applicable; however, this is not an efficient approach in general. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 17 / 24
Step Length Control Open questions (experimental and theoretical) The application, on the multi-objective context, of the widely studied inexact methods (Wolfe conditions, Armijo, etc). Ensuring convergence of these methods in the multi-objective framework. Studing the speed of convergence, if possible. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 18 / 24
Outline of the Presentation Introduction 1 Continuous Multi-objective Local Search 2 Memetic Issues 3 Balance, Archiving Conclusions and Future Work 4 Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 19 / 24
Memetic Issues: Balance How to balance the computational resources between the local and the global search procedures? Fixed amount for each one. Two stages algorithms. Adaptive mechanism. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 20 / 24
Memetic Issues: Selection and Archiving Special archiving strategies to: Avoid loosing solutions that have already been refined by the local search Open questions: Interplay of the archiving strategy and the selecction mechanism of the MOEA. Adapting the MOEA operators to the special archive. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 21 / 24
Outline of the Presentation Introduction 1 Continuous Multi-objective Local Search 2 Memetic Issues 3 Conclusions and Future Work 4 Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 22 / 24
Conclusions Open Issues: How to efficiently integrate line search into a population-based context (MOEAs algorithms)? What is an effective search direction? Make an efficient step length control in multi-objective. Balance of the computational resources. Archiving and selection. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 23 / 24
Conclusions From practice: Since the high cost of gradient-based local searchers, the biggest part of the search effort should be made by the MOEA. We should apply the local search procedure on some of the best individuals each certain number of generations. We have few clues about in which problems the use of gradient information is effective and those problems that cause troubles to these memetic algorithms. Lara, Coello, Schütze (CINVESTAV, México.) Theoretical EMO Workshop, GECCO 2010 July 8, 2010 24 / 24
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