Kibern. vyčisl. teh., 2015, Issue 179, pp 35-42.
Norkin Bogdan V., PhD (Phys. & Math.), Researcher of the Department of Methods for Discrete Optimization of Mathematical Modeling for Analysis of Complex Systems of the Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine, Acad.Glushkov ave., 40, Kiev, 03187, Ukraine, email: firstname.lastname@example.org
ON THE APPROXIMATION OF VECTOR OPTIMIZATION PROBLEMS
Introduction. Vector optimization has a great variety of applications. Such problems naturally appear in stochastic optimization, where the optimization problem contains random parameters. In the latter case the vector objective function may include mean value, median, variance, quantiles and other characteristics of the random objective function. The difficulty is that these quantities usually cannot be calculated exactly and are non-convex as functions of variable parameters. These circumstances set additional difficulties for solving corresponding vector optimization problems.
The purpose of this paper is to study conditions for convergence of the approximation method when the objective functions and the feasible set are replaced by their more and more fine approximations.
Results. We consider an approximation approach to solving vector optimization problems. The standard approach to such problems is to optimize one criterion under constraints on the others or to scalarize the problem, i.e. to combine all criteria into one scalar criterion. This paper describes a completely different approach, where the feasible set is approximated by a discrete grid (deterministic or random) and the vector function is approximately calculated on this grid. The obtained discrete problem is exactly solved by Pareto type optimization.
Conclusions. Sufficient conditions are established for Pareto-optimal solutions of the approximate problems to converge in set convergence sense to the Pareto optimal solution of the original problem (with some accuracy). Namely, it is required for the approximate functions to converge uniformly to the original function and for the feasible set approximations (possibly discrete) to converge to elements of the original feasible set, at least, in the vicinity of the solution. The result confirms a natural hypothesis that the approximation accuracy should increase when approaching to the solution.
Keywords: vector optimization, stochastic multicriteria optimization, Pareto optimality, discrete approximation, epsilon-dominance.
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