An algorithmic approach to multiobjective optimization with decision uncertainty

Abstract

In real life applications optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one variable is a whole set, which includes all possible outcomes of this variable. We choose a robust approach and thus these sets have to be compared using the so called upper-type less order relation.We propose a numerical method to calculate a covering of the set of optimal solutions of such an uncertain multiobjective optimization problem. We use a branchand-bound approach and lower and upper bound sets for being able to compare the arising sets. The calculation of these lower and upper bound sets uses techniques known from global optimization as convex underestimators as well as techniques used in convex multiobjective optimization as outer approximation techniques. We also give first numerical results for this algorithm.

A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality

“…• Algorithms based on scalarization [11,12,19,20,27,44]. The methods in this group follow a scalarization approach and are derived for problems where the set-valued objective mapping has a particular structure that comes from the so-called robust counterpart of a vector optimization problem under uncertainty, see [20].…”

“…Weighted Chebyshev scalarization and some of its variants (augmented, min-ordering) were also studied in [19,27,44]. • Branch and bound [12].…”

“…The algorithm in [12] is also designed for uncertain vector optimization problems, but in particular it is assumed that only the decision variable is the source of uncertainty. There, the authors propose a branch and bound method for finding a box covering of the solution set.…”

Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets

“…Current important applications of set optimization can be found for instance in the areas of socio economics [1], welfare economics [2], and finance [3]. Such problems arise also for instance when a vector-valued objective map is considered, but (relative) errors are allowed [4], in case of robust approaches to uncertain multiobjective optimization problems [5,6], or in bilevel optimization, if neither the optimistic nor the pessimistic approach is used [7,8]. For a detailed introduction to set optimization see the extensive book by Khan, Tammer and Zǎlinescu [9].…”

“…Which of these set order relations is suitable depends on the considered application. The set approach with the u-less order relation allows for instance the treatment of decision uncertainty in multiobjective optimization [5,6]. Moreover, the -less order relation and the u-less order relation correspond to the optimistic and pessimistic approach in bilevel optimization [8], respectively.…”