Stochastic Multiobjective Optimization: Sample Average Approximation and Applications

We investigate one stage stochastic multiobjective optimization problems where the objectives are the expected values of random functions. Assuming that the closed form of the expected values is difficult to obtain, we apply the well known Sample Average Approximation (SAA) method to solve it. We pr...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 151; no. 1; pp. 135 - 162
Main Authors: Fliege, Jörg, Xu, Huifu
Format: Journal Article
Language:English
Published: Boston Springer US 01.10.2011
Springer
Springer Nature B.V
Subjects:
Applications of Mathematics
Applied sciences
Approximation
Calculus of Variations and Optimal Control; Optimization
Decision theory. Utility theory
Engineering
Exact sciences and technology
Expected values
Mathematical programming
Mathematics
Mathematics and Statistics
Numerical analysis
Operational research and scientific management
Operational research. Management science
Operations Research/Decision Theory
Optimization
Random variables
Studies
Theory of Computation
Scalarization
Exponential convergence
Stochastic multiobjective programming
Efficient solution
Sample average approximation
Probabilistic approach
Sample size
Multiobjective programming
Modeling
Stochastic programming
Exact solution
Efficiency
Smoothing
Random function
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:We investigate one stage stochastic multiobjective optimization problems where the objectives are the expected values of random functions. Assuming that the closed form of the expected values is difficult to obtain, we apply the well known Sample Average Approximation (SAA) method to solve it. We propose a smoothing infinity norm scalarization approach to solve the SAA problem and analyse the convergence of efficient solution of the SAA problem to the original problem as sample sizes increase. Under some moderate conditions, we show that, with probability approaching one exponentially fast with the increase of sample size, an ϵ -optimal solution to the SAA problem becomes an ϵ -optimal solution to its true counterpart. Moreover, under second order growth conditions, we show that an efficient point of the smoothed problem approximates an efficient solution of the true problem at a linear rate. Finally, we describe some numerical experiments on some stochastic multiobjective optimization problems and report preliminary results.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-011-9859-6