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Stochastic multi-objective optimization: a survey on non-scalarizing methods

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Abstract

Currently, stochastic optimization on the one hand and multi-objective optimization on the other hand are rich and well-established special fields of Operations Research. Much less developed, however, is their intersection: the analysis of decision problems involving multiple objectives and stochastically represented uncertainty simultaneously. This is amazing, since in economic and managerial applications, the features of multiple decision criteria and uncertainty are very frequently co-occurring. Part of the existing quantitative approaches to deal with problems of this class apply scalarization techniques in order to reduce a given stochastic multi-objective problem to a stochastic single-objective one. The present article gives an overview over a second strand of the recent literature, namely methods that preserve the multi-objective nature of the problem during the computational analysis. We survey publications assuming a risk-neutral decision maker, but also articles addressing the situation where the decision maker is risk-averse. In the second case, modern risk measures play a prominent role, and generalizations of stochastic orders from the univariate to the multivariate case have recently turned out as a promising methodological tool. Modeling questions as well as issues of computational solution are discussed.

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Notes

  1. Expressed in the terms used in Ben Abdelaziz (2012), this means that we focus on the “multi-objective method” instead of the “stochastic method”.

  2. The sample space Ω is the first component of the probability space \((\varOmega,\mathcal{A},P)\). Note that in stochastic optimization, the probability model given by \((\varOmega,\mathcal{A},P)\) is always assumed as known, and therefore in particular also the distributions of the considered random variables.

  3. The axioms proposed, in the convex setting, are essentially

    1. (i)

      Monotonicity: ρ(X)≤ρ(Y) provided that X≥Y almost everywhere,

    2. (ii)

      Convexity: ρ((1−λ)Y 0+λY 1)≤(1−λ)ρ(Y 0)+λρ(Y 1),

    3. (iii)

      Positive Homogeneity: ρ(λY)=λρ(Y) for all λ>0 and

    4. (iv)

      Translation Invariance: ρ(Y+c)=ρ(Y)−c.

  4. In Sects. 4.1.3 and 4.2.3, also problem formulations encompassing stochastic dominance constraints will be discussed. It is clear that these constraints cannot be considered anymore as “deterministic”. This case will nevertheless be included here, since stochastic dominance constraints are not so much externally defined conditions restricting the feasible set, but rather constitute an alternative way to express the preferences of a decision maker. We admit that the distinction is not sharp.

  5. An alternative way to judge risk-neutrality is to apply the Arrow-Pratt risk aversion measure defined by −u″(⋅)/u′(⋅) to each random variable f j . Evidently, the values vanish for the utility u given by (13). Using a more general observation (Theorem 1 in Gutjahr 2012a), it can be shown that on the additional constraint that u is nondecreasing and concave, (13) is the only form u can have for a decision maker who is risk-neutral toward all properly stochastic objective functions. This does not hold anymore if the restriction to concave utilities is dropped.

  6. Let us also mention the s-convex class given by , where u (s) is the s-th derivative of u. This class contains the function x↦x k as well as the function x↦−x k for every k<s. Notably, the relation thus ensures that the moments of order less than s coincide: \(\mathbb{E}[X^{k}] = \mathbb{E}[Y^{k}]\) for all k<s. The special case  of the 2-convex class is closely related to the class defined above, but note that it has an equality constraint for expected values instead of an inequality constraint.

  7. For example, one criterion for each , but also other representations of the criteria can be chosen, as it will be seen later. Under some special circumstances, some of these criteria may coincide in a way making the problem finite-dimensional even for infinite , but in general, this does not hold.

  8. The last part of the sentence says that an efficient solution x needs not necessarily to be stochastically nondominated itself, but it can only be stochastically dominated by a solution y with \(\mathbb{E}[f(x,\omega )] = \mathbb{E} [f(y,\omega)] \mbox{ and } \rho(f(x,\omega)) = \rho(f(y,\omega))\). Therefore, if the pre-image of a Pareto-optimal point of the mean-risk model (4) is finite, it contains at least one stochastically nondominated solution.

  9. Analogously as in the case mentioned in Sect. 2.2, it is often sufficient to have only one representative from the pre-image of each stochastically nondominated random vector in the solution set.

  10. Actually, the authors use the term CVaR instead of \(\mathsf {AV@R}\).

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Vienna, Universitaetsstrasse 5/9, 1010, Vienna, Austria

    Walter J. Gutjahr

  2. Norwegian University of Science and Technology, Trondheim, Norway

    Alois Pichler

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  1. Walter J. Gutjahr
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  2. Alois Pichler
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Correspondence to Walter J. Gutjahr.

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Invited Survey Article for Annals of Operations Research: Special Volume.

Second author gratefully acknowledges support of the Norwegian grant 207690/E20.

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Gutjahr, W.J., Pichler, A. Stochastic multi-objective optimization: a survey on non-scalarizing methods. Ann Oper Res 236, 475–499 (2016). https://doi.org/10.1007/s10479-013-1369-5

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