Włodzimierz Ogryczak

Dual Stochastic Dominance and Related Mean-Risk Models

We consider the problem of constructing mean-risk models which are consistent with the second degree stochastic dominance relation. By exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions. This allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation. We also provide stochastic linear programming formulations of these models.

From Stochastic Dominance to Mean-Risk Models: Semideviations as Risk Measures

Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible trade-off analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resulting mean-variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean-risk model consistent with the second degree stochastic dominance, provided that the trade-off coefficient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome-Risk diagram, which appears to be particularly useful for comparing uncertain outcomes.

On consistency of stochastic dominance and mean–semideviation models

Abstract.We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. New necessary conditions for stochastic dominance are developed. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, maximization of such a functional yields solutions which are efficient in terms of stochastic dominance. The results are illustrated graphically.

Multiple criteria linear programming model for portfolio selection

The portfolio selection problem is usually considered as a bicriteria optimization problem where a reasonable trade-off between expected rate of return and risk is sought. In the classical Markowitz model the risk is measured with variance, thus generating a quadratic programming model. The Markowitz model is frequently criticized as not consistent with axiomatic models of preferences for choice under risk. Models consistent with the preference axioms are based on the relation of stochastic dominance or on expected utility theory. The former is quite easy to implement for pairwise comparisons of given portfolios whereas it does not offer any computational tool to analyze the portfolio selection problem. The latter, when used for the portfolio selection problem, is restrictive in modeling preferences of investors. In this paper, a multiple criteria linear programming model of the portfolio selection problem is developed. The model is based on the preference axioms for choice under risk. Nevertheless, it allows one to employ the standard multiple criteria procedures to analyze the portfolio selection problem. It is shown that the classical mean-risk approaches resulting in linear programming models correspond to specific solution techniques applied to our multiple criteria model.

Conditional value at risk and related linear programming models for portfolio optimization

Many risk measures have been recently introduced which (for discrete random variables) result in Linear Programs (LP). While some LP computable risk measures may be viewed as approximations to the variance (e.g., the mean absolute deviation or the Gini’s mean absolute difference), shortfall or quantile risk measures are recently gaining more popularity in various financial applications. In this paper we study LP solvable portfolio optimization models based on extensions of the Conditional Value at Risk (CVaR) measure. The models use multiple CVaR measures thus allowing for more detailed risk aversion modeling. We study both the theoretical properties of the models and their performance on real-life data.

On solving linear programs with the ordered weighted averaging objective

Abstract The problem of aggregating multiple criteria to form overall objective functions is of considerable importance in many disciplines. The most commonly used aggregation is based on the weighted sum. The ordered weighted averaging (OWA) aggregation, introduced by Yager, uses the weights assigned to the ordered values (i.e. to the worst value, the second worst and so on) rather than to the specific criteria. This allows to model various aggregation preferences, preserving simultaneously the impartiality (neutrality) with respect to the individual criteria. In this paper we analyze solution procedures for linear programs with the OWA objective functions. Two alternative linear programming formulations are introduced and their computational efficiency is analyzed.

Equitable aggregations and multiple criteria analysis

Abstract In the past decade, increasing interest in equity issues resulted in new methodologies in the area of operations research. This paper deals with the concept of equitably efficient solutions to multiple criteria optimization problems. Multiple criteria optimization usually starts with an assumption that the criteria are incomparable. However, many applications arise from situations which present equitable criteria. Moreover, some aggregations of criteria are often applied to select efficient solutions in multiple criteria analysis. The latter enforces comparability of criteria (possibly rescaled). This paper presents aggregations which can be used to derive equitably efficient solutions to both linear and nonlinear multiple optimization problems. An example with equitable solutions to a capital budgeting problem is analyzed in detail. An equitable form of the reference point method is introduced and analyzed.

On the lexicographic minimax approach to location problems

Abstract When locating public facilities, the distribution of travel distances among the service recipients is an important issue. It is usually tackled with the minimax (center) solution concept. The minimax solution concept, despite the most commonly used in the public sector location models, is criticized as it does not comply with the major principles of the efficiency and equity modeling. In this paper we develop a concept of the lexicographic minimax solution (lexicographic center) being a refinement of the standard minimax approach to location problems. We show that the lexicographic minimax approach complies with both the Pareto-optimality (efficiency) principle (crucial in multiple criteria optimization) and the principle of transfers (essential for equity measures) whereas the standard minimax approach may violate both these principles. Computational algorithms are developed for the lexicographic minimax solution of discrete location problems.

Minimizing the sum of the k largest functions in linear time

Given a collection of n functions defined on Rd, and a polyhedral set Q ⊂ Rd, we consider the problem of minimizing the sum of the k largest functions of the collection over Q. Specifically we focus on collections of linear functions and several classes of convex, piecewise linear functions which are defined by location models. We present simple linear programming formulations for these optimization models which give rise to linear time algorithms when the dimension d is fixed. Our results improve complexity bounds of several problems reported recently by Tamir [Discrete Appl. Math. 109 (2001) 293-307], Tokuyama [Proc. 33rd Annual ACM Symp. on Theory of Computing, 2001, pp. 75-84] and Kalcsics, Nickel, Puerto and Tamir [Oper. Res. Lett. 31 (1984) 114-127].

Inequality measures and equitable approaches to location problems

Abstract Location problems can be considered as multiple criteria models where for each client (spatial unit) there is defined an individual objective function, which measures the effect of a location pattern with respect to the client satisfaction (e.g., it expresses the distance or travel time between the client and the assigned facility). This results in a multiple criteria model taking into account the entire distribution of individual effects (distances). Moreover, the model enables us to introduce the concept of equitable efficiency which links location problems with theories of inequality measurement. In this paper special attention is paid to solution concepts based on the bicriteria optimization of the mean distance and the absolute inequality measures. The restrictions for the trade-offs are identified which guarantee that the bicriteria approaches comply with the concept of equitable efficiency. These results are further generalized to bicriteria approaches not using directly the trade-off technique.