Advances in Multiobjective Optimization Dealing with computational cost

Transcription

1 Advances in Multiobjective Optimization Dealing with computational cost Jussi Hakanen

2 Contents Motivation: why approximation? An example of approximation: Pareto Navigator Software demo

3 Computational cost Typically, optimization requires high number of function evaluations especially multiobjective optimization Key issue: how long does simulating the problem take? Examples no computational cost: functions having analytical expressions probably high computational cost: e.g. numerical simulation of PDEs, experiments, simulators

4 Illustrative figures 1 simulation 1 ms 1000 simulations = 1second 1 simulation 1 s 1000 simulations 17 minutes 1 simulation 1 minute 1000 simulations 17 hours 1 simulation 1 hour 1000 simulations 42 days What can be done?

5 Surrogate models Simple models for approximating a given data Low computational cost Requires training data that is generated by using the original high accuracy model Surrogate model is trained with the training data (find parameters) need for validation! Many different surrogate models Polynomials Radial basis functions Artificial neural networks Kriging Support vector regression

6 Surrogate models in single objective optimization Used to approximate the objective function Two main approaches: One shot: optimize with approximated objective function, evaluate original model in the optimal solution, stop Adaptive: optimize with approximated objective function, evaluate original model in the optimal solution, update the surrogate model and repeat until solution good enough

7 Question How could surrogate models be used to reduce computational cost in MO? Ideas? Approximate objective functions Approximate PO set in the objective space Approximate PO set in the decision space Approximate fitness function evaluation in EMO

8 Pareto Navigator: An example of utilizing approximation P. Eskelinen, K. Miettinen, K. Klamroth, J. Hakanen, Pareto Navigator for Interactive Nonlinear Multiobjective Optimization, OR Spectrum, 32, 2010

9 Motivation for Pareto Navigator Assume that evaluation of the objective functions is time consuming Idea is to enable fast navigation in the PO set in the objective space through approximation Interactive method for nonlinear, convex multiobjective optimization problems Has been influenced by the Pareto Race (Korhonen & Wallenius, 1988)

10 Decision Making Decision process can often be divided into two phases Learning phase: DM learns about the possibilities and limitations of the problem as well as his/her own preferences Decision phase: making the final decision based on information obtained from the learning phase Pareto Navigator concentrates on the learning phase

11 Multiobjective Optimization Methods (reminder) MOO methods can be classified according to the role of the DM into No-preference methods A posteriori methods (e.g. approximation algorithms, EMO methods) A priori methods (e.g. lexicographic ordering, goal programming) Interactive methods

12 Interactive Methods (reminder) Iterative interaction between the DM and the method During the solution process the DM expresses preferences on how the current solution should be improved DM s preferences are taken into account when new compromise solution(s) are computed Well suited for solving real-world problems because DM learns about the behaviour of the problem and from his/her preferences DM can utilize his/her experiences in the solution procedure only the solutions of interest to the DM are computed

13 Pareto Navigator (PN) Idea: to enable convenient examination of tradeoffs between the objectives using an approximation of the PO set, that is, to navigate in an approximation of the PO set Approximation allows real-time generation and consideration of desirable PO solutions. Consists of an initialization phase and a navigation phase During navigation, the DM can learn about the interdependencies between the conflicting objectives Interesting solutions found during navigation can be projected into the actual PO set

14 Pareto Navigator Algorithm

15 PN: Initialization phase An approximation of the PO set is formed based on a small set of PO solutions in the objective space computed with some other MO method if EMO used, then the set described by the final population is approximated (not necessarily PO set) can be computed before the DM is involved and may take long time Polyhedral approximation (convex problem) e.g. convex hull of PO solutions Az b Approximate ideal and nadir objective vectors Select a starting point (DM involved), e.g. one of the solutions used to build the approximation

16 PN: Navigation phase The DM can navigate around the approximation and direct the search for the most promising regions Approximated PO solutions are feasible for convex problems Preferences of the DM are used to define a search direction in the approximation from the starting solution e.g. based on a reference point z R k given by the DM search direction d = z z c where z c is the current solution

17 Polyhedral approximation for two objectives

18 PN: Navigation phase Progress towards the specified search direction approximated solutions can be computed e.g. by using the reference point method α is a step length parameter moving the reference point z α to the specified search direction and w i are the weights Nondifferentiable due to min-max term can be smoothened

19 Smooth formulation Problem linear w.r.t. z!

20 PN: Navigation phase Approximated solutions can be obtained by solving linear problems (minimization w.r.t. z ) good for computationally demanding problems Parametric linear programming produces approximated solutions real-time for any problem Drawback: connection to the decision variables is lost Approximated solutions can be projected into the PO set original model used, may be time consuming

21 Illustration PO set and initial PO solutions (7) Polyhedral approximation and a navigation path from solution A to D

22 Implementation of Pareto Navigator IND-NIMBUS multiobjective optimization framework (developed in JyU) Contains several interactive methods in the same platform an implementation of the NIMBUS method for solving complex (industrial) problems Pareto Navigator Pareto Front Interpolation (PAINT)

23 Pareto Navigator Demo with an example

24 PhD thesis related to approximation in our group Markus Hartikainen (2011): Approximation through interpolation in nonconvex multiobjective optimization, 05 Tomi Haanpää (2012): Approximation method for computationally expensive nonconvex multiobjective optimization problems, 01

25 Acknowledgements Dr. Petri Eskelinen, Kela The Social Insurance Institution of Finland (previously University of Jyväskylä) PhD Tomi Haanpää, PhD Markus Hartikainen, Prof. Kaisa Miettinen and MSc Suvi Tarkkanen, University of Jyväskylä Prof. Kathrin Klamroth, University of Wuppertal, Germany

26 Thank You! PhD Jussi Hakanen Industrial Optimization Group Department of Mathematical Information Technology P.O. Box 35 (Agora) FI University of Jyväskylä, Finland