TIES598 Nonlinear Multiobjective Optimization Methods to handle computationally expensive problems in multiobjective optimization

Transcription

1 TIES598 Nonlinear Multiobjective Optimization Methods to hle computationally expensive problems in multiobjective optimization Spring 2015 Jussi Hakanen Markus Hartikainen

2 Outline 1. Computational cost what is it? 2. Hing computational cost in single objective optimization 3. Hling computational cost in multiobjective optimization 4. Example methods 2

3 Computational cost in optimization What comes to your mind? 3

4 What is computational cost? In optimization, in general, it means that finding the optimal solution takes a long time In engineering desing, often results from black-box (e.g., simulators) or experiment-based objective functions or constraints Can also result from a large feasible set, very bad multimodality, large number of discrete decision variables, etc. Value-focused problem formulation Mathematical Black-box Experiment Design of experiments Multiobjective optimization Interactive Non- Interactive Optimal solution(s) Product design Knowledge discovery Design engineer 4

5 Illustrative figures about simulation time Optimization typically requires large number of simulations 1 simulation 1 ms simulations = 10 seconds 1 simulation 1 s simulations 170 minutes 1 simulation 1 minute simulations 170 hours 1 simulation 1 hour simulations 417 days 5

6 Computational cost with black-box experiment-based functions The computational cost of the functions is multiplied in optimization For example, let s assume that we want to optimize the shape of an exhaust pipe to a snowmobile A single simulation with the MOTA* simulator takes a couple of minutes function calls optimization time of min (ca. 5.5 h) Computationally expensive function Function values Delay Function calls Optimization algorithm Optimal solution(s) >>>Delay *) T. Aittokoski K. Miettinen. Cost Effective Simulation-Based Multiobjective Optimization in the Performance of an Internal Combustion Engine. Engineering Optimization, Volume 40, pages

7 Hling computational cost in singleobjective optimization There are three ways of reducing the delay caused by computationally expensive function Reducing the cost of the simulation/experiment Parallelization Approximation Active research topic in the Industrial Optimization Group In this presentation we assume that simulation or experiment time cannot be reduced further 7

8 Parallelization in single-objective optimization In general, parallelization can happen on function side, or Computationally expensive function Computationally expensive function Computationally expensive function Optimization algorithm Optimal solution(s) >>Delay 8

9 Parallelization in single-objective optimization In general, parallelization can happen on function side, or on optimization side (through decomposition) In optimization side parallelization, the optimal solutions from one side can also be passed to the other side Evolutionary methods are very suitable for optimization-based parallelization Different approaches e.g., the isl model Computationally expensive function Almost optimal solution(s) >Delay Optimization algorithm Combining Computationally expensive function Optimization algorithm Almost optimal solution(s) >Delay Optimal solution(s) >>Delay 9

10 Approximation in single objective optimization Different surrogates are used Metamodels: RBF, NN, Kriging, etc. Problem specific approximations Computationally expensive function Function values Delay Function values Computationally inexpensive surrogate Approximate function values Optimization algorithm Optimal solution(s) >>Delay 10

11 Metamodeling Simple models for approximating a given data Low computational cost Requires training data that is generated by using the original high accuracy model Design of experiments Metamodel is trained with the training data (find parameters) need for validation! Different approaches One shot vs. adaptive 11

12 Computational cost in multiobjective optimization (a posteriori methods) One more layer of multiplication added Similar methods than in singleobjective optimization parallelization approximation Single objective optimization of the scalarization 1 Pareto optimal solution >>>Delay Computationally expensive function a posteriori method Single objective optimization of the scalarization 2 Pareto optimal solution >>>Delay Pareto front >>>>>Delay 12

13 Computational cost in multiobjective optimization (interactive methods) The DM can direct the search towards more interesting solutions Less computational cost Must also consider human aspects How much are we asking from the DM? How long does the DM have to wait in between solutions? Computationally expensive function Function values Delay Function calls Single objective optimization of the scalarization PO solutions >>>Delay DM Preferences Preferred solution(s) >>>>Delay 13

14 Hling computational cost in interactive methods Approximation of the Pareto front Allows the DM to direct the search quicker to interesting areas Less waiting for the DM ( real-time optimization) For example Pareto Navigator the PAINT or PAINT-SiCon methods Interactive HyperBox Exploration Computationally expensive function PO solution(s) Single objective optimization of the scalarization PO solution(s) >>>Delay PO solution(s) DM Approximation of the Pareto front Single objective optimization of the scalarization Preferred solution(s) >>>Delay Approximate PO solution(s) 14

15 Pareto Navigator Idea: convenient examination of tradeoffs between the objectives using an approximation of the PO set, that is, navigate in an approximation of the PO set Approximation allows real-time generation consideration of desirable PO solutions. Consists of an initialization phase a navigation phase An approximation of the PO set is formed based on a small set of PO solutions in the objective space Polyhedral approximation (convex problem) e.g. convex hull of PO solutions Convex problem approximated PO solutions via solving linear problems Navigation on the approximation to a specified direction E.g. using reference points Interesting solutions found during navigation can be projected into the actual PO set P. Eskelinen, K. Miettinen, K. Klamroth, J. Hakanen. Pareto Navigator for Interactive Nonlinear Multiobjective Optimization. OR Spectrum, Volume 32, pages

16 Pareto Navigator Problem linear w.r.t. z! 16

17 PAINT method for approximating the Pareto front PAreto front INTerpolation Piece-wise linear interpolation between Pareto optimal solutions in the objective space Guarantees that the Pareto optimal solutions are on the interpolation, the interpolation does not dominate, nor is dominated by the Pareto optimal solutions the interpolation is not dominated by itself Implies a mixed-integer linear surrogate for the original problem Also for non-convex problems PAINT PAINT M. Hartikainen, K. Miettinen, M. M. Wiecek. PAINT: Pareto Front Interpolation for Nonlinear Multiobjective Optimization. Computational Optimization Applications, Volume 52, pages AND M. Hartikainen, A. Lovison. PAINT-SiCon: Constructing Consistent Parametric Representations of Pareto Sets in Nonconvex Multiobjective Optimization Journal of Global Optimization, To Appear, DOI: /s

18 Interactive HyperBox Exploration Set of pre-calculated PO objective vectors Z m z 1,, z m R k Simplex k 1 : convex hull of the unit vectors of the objective space R k domain space for functions used in this presentation Each of the pre-calculated PO vectors z i are mapped into the simplex k 1 by a translation ideal Z m a corresponding scalar multiplication 1 Let the mapped vectors be z s i k j=1 z j i ideal j Z m Training data for a surrogate function consists of pairs: z s i, k j=1 z j i ideal j Z m k 1, R Then, a surrogate function f S : Δ k 1 R is trained based on the above data different surrogate functions can be used Finally, the Pareto front is approximated by the image of a function f a : k 1 R k, f a (z) =ideal Z m+ f s (z)z T. Haanpää, Approximation method for computationally expensive nonconvex multiobjective optimization problems, PhD Thesis, Univ. of Jyväskylä, 2012 (electronically available in 18

19 Example The image of the function f a Δ 1 (blue curve) by an RBF with a third degree polyharmonic spline Nondominated vectors in the approximation (f a Δ 1, gray dots) the true Pareto front (black dots)

20 Interactive HyperBox Exploration The DM is asked to provide aspiration reservation levels (hyperbox in the objective space) the number of sample vectors to be generated automatically inside the hyperbox (red markers) Sample vectors are first mapped into k 1, then by the function f a into the approximation (green dots) DM gets information about how his/her preferences are related to the approximated Pareto front no need to solve any optimization problems computationally efficient!

21 Hling computational cost in interactive methods Other approaches: approximation of the preferences of the DM Computationally expensive function Function calls Single objective optimization of the scalarization Preferences DM surrogate DM Preferred solution(s) >>>>Delay 21

22 Hling computational cost in interactive methods Other approaches: approximation of the preferences of the DM approximation of the scalarization Computationally expensive function PO solution(s) Single objective optimization of the scalarization Values of the scalarization Single objective optimization of the surrogate scalarization PO solution(s) >>>Delay Approximate PO solution(s) DM Preferred solution(s) >>>Delay 22

23 Hling computational cost in interactive methods Other approaches: approximation of the preferences of the DM approximation of the scalarization covering the search space with hyper-boxes then navigating on that visually etc. A. V. Lotov, V. A. Bushenkov, G. A. Kamenev. Interactive Decision Maps. Kluwer Academic Publishers, Boston,

24 Some literature R. Jin, W. Chen, T. Simpson. Comparative studies of meta-modelling techniques under multiple modelling criteria. Structural Multidisciplinary Optimization, 23, 1 13, 2001 B. Yang, Y. Yeun, W. Ruy. Managing approximation models in multiobjective optimization. Structural Multidisciplinary Optimization, 24, , 2002 J. Knowles H. Nakayama. Meta-Modeling in Multiobjective Optimization, In: Multiobjective Optimization: Interactive Evolutionary Approaches, J. Branke, K. Deb, K. Miettinen R. Slowinski (eds.), , Springer, 2008 Y. Tenne, C. Goh. Computational intelligence in expensive optimization problems, Springer, 2010 Y. Jin. Surrogate-assisted evolutionary computation: recent advances future challenges, Swarm Evolutionary Computation, 1, 61 70, 2011 M. Tabatabaei, J. Hakanen, M. Hartikainen, K. Miettinen K. Sindhya. A survey on hling computationally expensive multiobjective optimization problems using surrogates: non-nature inspired methods, Structural Multidisciplinary Optimization, To appear 24