The Markowitz model for single period portfolio optimization quantifies the problem by means of o... more The Markowitz model for single period portfolio optimization quantifies the problem by means of only two criteria: the mean, representing the expected outcome, and the risk, a scalar measure of the variability of outcomes. The classical Markowitz model uses the variance as the risk measure, thus resulting in a quadratic optimization problem. Following Sharpe's work on linear approximation to the mean-variance model, many attempts have been made to linearize the portfolio optimization problem. There were introduced several alternative risk measures which are computationally attractive as (for discrete random variables) they result in solving Linear Programming (LP) problems. The LP solvability is very important for applications to real-life financial decisions where the constructed portfolios have to meet numerous side constraints and take into account transaction costs. This paper provides a systematic overview of the LP solvable models with a wide discussion of their properties.
The Reference Point Method (RPM) is based on the so-called augmented max-min aggregation. The reg... more The Reference Point Method (RPM) is based on the so-called augmented max-min aggregation. The regularization by the average achievement is easily implementable but it may disturb the basic max-min model. The only consequent regularization of the max-min aggregation is the lexicographic max-min (nucleolar) solution concept where in addition to the worst achievement, the second worst achievement is also optimized (provided that the worst remains on the optimal level), the third worst is optimized (provided that the two worst remain optimal), and so on. Such a nucleolar regularization satisfies the addition/deleting principle thus making the corresponding nucleolar RPM not affected by any passive criteria. The nucleolar regularization is more complicated in implementation but the recent progress made in optimization methods for ordered averages allows one to implement the nucleolar RPM quite effectively. Both the theoretical and implementation issues of the nucleolar RPM are analyzed.
— The rapid growth of traffic induced by Internet services makes the simple over-provisioning of ... more — The rapid growth of traffic induced by Internet services makes the simple over-provisioning of resources not economical and hence imposes new requirements on the di-mensioning methods. Therefore, the problem of network design with the objective of minimizing the cost and at the same time solving the tradeoff between maximizing the service data flows and providing fair treatment of all demands becomes more and more important. In this context, the so-called Max-Min Fair (MMF) principle is widely considered to help finding reasonable bandwidth allocation schemes for competing demands. Roughly speaking, MMF assumes that the worst service performance is maximized, and then is the second worst performance, the third one, and so on, leading to a lexicographically maximized vector of sorted demand bandwidth allocations. It turns out that the MMF optimal solution cannot be approached in a standard way (i.e., as a mathematical programming problem) due to the necessity of lexicographic maximization of ordered quantities (bandwidth allocated to demands). Still, for convex models, it is possible to formulate effective sequential procedures for such lexicographic optimization. The purpose of the presented paper is threefold. First, it discusses resolution algorithms for a generic MMF problem related to telecommunications network design. Second, it gives a survey of network design instances of the generic formulation, and illustrates the efficiency of the general algorithms in these particular cases. Finally, the paper discusses extensions of the formulated problems into more practical (unfortunately non-convex) cases, where the general for convex MMF problems approach fails.
As an active participant of a competitive energy market, the generator (the energy supplier) chal... more As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).
As an active participant of a competitive energy market, the generator (the energy supplier) chal... more As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).
As an active participant of a competitive energy market, the generator (the energy supplier) chal... more As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).
As an active participant of a competitive energy market, the generator (the energy supplier) chal... more As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).
An early motivation for this study was the problem of relocation of scarce or endangered species ... more An early motivation for this study was the problem of relocation of scarce or endangered species of animals for breeding and/or reintroduction to establish new populations in the wild. In this paper, we introduce single and multiple objective optimization models which are designed to comprehend a wide variety of objectives which are of interest to conservation and wildlife managers. We present the models in a general way and point out special features relative to ecology as they arise. Thus, the models may be used for relocation decisions analysis in diverse fields, not only in conservation and ecology. After presentation of the models in such a general way, we reformulate the models to make use of the special structure present. Such reformulation reduces the number of decision variables and constraints and, in general, makes solutions easy to obtain. By easy to obtain, we mean that tools from linear and mixed-integer programming together with elementary sorting procedures provide the basis for solving the models.
Scope and Purpose---The classical interactive procedures for multiple criteria decision analysis ... more Scope and Purpose---The classical interactive procedures for multiple criteria decision analysis assume the so-called rational behavior of decision makers: they know the decision problem, and they are consistent and coherent in the decision process. However, as has been stressed by many researchers and practitioners, decision makers usually learn the decision problem during the interactive session with the decision support system, and there are numerous examples in which people systematically violate the consistency and coherence of their preferences. Therefore, the so-called aspiration-based interactive decision support schemes seem to be much more interesting for practical implementations. This interactive process explicitly depends on the aspiration levels stated and modified by the decision maker, and thereby makes operational the concept of the adaptive dependence of the decision process on learning and context. The paper presents an implementation of an aspiration-based interactive procedure to solve various multiobjective transshipment problems with facility location.
2006 2nd Conference on Next Generation Internet Design and Engineering, 2006
Expanding demand on the Internet services leads to an increased role of the network dimensioning ... more Expanding demand on the Internet services leads to an increased role of the network dimensioning problem for elastic traffic where one needs to allocate bandwidth to maximize service flows and simultaneously to reach a fair treatment of all the elastic services. Thus, both the overall efficiency (throughput) and the fairness (equity) among various services are important. The Max-Min Fairness (MMF) approach, widely used to this problem, guarantees fairness but may lead to significant losses in the overall throughput of the network. In this paper we show how the concepts of multiple criteria equitable optimization can be effectively used to generate various fair resource allocation schemes. We introduce a multiple target model equivalent to equitable optimization and we develop a corresponding procedure to generate fair efficient bandwidth allocations. The procedure is tested on a sample network dimensioning problem and its abilities to model various preferences are demonstrated.
This paper discusses connections between the multi-criteria techniques of goal programming (GP) a... more This paper discusses connections between the multi-criteria techniques of goal programming (GP) and the reference point method (RPM). Both approaches use a certain target point in the criterion (outcome) space as a key element to model decision maker preferences. Therefore, RPM can be expressed, similarly to GP, in the modelling framework of deviational variables. The paper gives a systematic survey and analysis of the lexicographic GP models of RPM. The corresponding preference models are formalised and analysed with respect to target values interpretations as well as the Pareto-efficiency of their solutions. The properties of equity among the individual achievements of solutions are also analysed with respect to the Rawlsian principle of justice.
In the past decade, increasing interest in equity issues resulted in new methodologies in the are... more In the past decade, increasing interest in equity issues resulted in new methodologies in the area of operations re-10 search. This paper deals with the concept of equitably efficient solutions to multiple criteria optimization problems. 11 Multiple criteria optimization usually starts with an assumption that the criteria are incomparable. However, many 12 applications arise from situations which present equitable criteria. Moreover, some aggregations of criteria are often 13 applied to select efficient solutions in multiple criteria analysis. The latter enforces comparability of criteria (possibly 14 rescaled). This paper presents aggregations which can be used to derive equitably efficient solutions to both linear and 15 nonlinear multiple optimization problems. An example with equitable solutions to a capital budgeting problem is 16 analyzed in detail. An equitable form of the reference point method is introduced and analyzed. 17
This note discusses the properties of solutions generated by the minmax models of goal programmin... more This note discusses the properties of solutions generated by the minmax models of goal programming (GP) and compromise programming (CP). GP approaches use a certain target point in the criterion (attribute) space to model decision maker's preferences. When the ideal (utopia) point is used as the target, the minmax GP model coincides with the minmax (Chebyshev) CP model. In a recent review of the current GP state-of-the-art, there have been included suggestions that the two equivalent models ensure Pareto eciency of solutions and they guarantee a perfectly balanced allocation among the achievement of the individual targets. In this note, it is shown that the models, in general, do not ensure the eciency of solutions and they do not guarantee the perfect equity among the individual achievements. Moreover, there are given sucient and necessary conditions clarifying when the discussed properties of minmax solutions do occur. Ó
Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic domi... more 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 tradeoff 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.
We analyze relations between two methods frequently used for modeling the choice among uncertain ... more 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.
International Transactions in Operational Research, 2002
Following the seminal work by Markowitz, the portfolio selection problem is usually modeled as a ... more Following the seminal work by Markowitz, the portfolio selection problem is usually modeled 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. Several other risk measures have been later considered thus creating the entire family of mean-risk (Markowitz type) models. In this paper, we analyze mean-risk models using quantiles and tail characteristics of the distribution. Value at risk (VAR), defined as the maximum loss at a specified confidence level, is a widely used quantile risk measure. The corresponding second order quantile measure, called the worst conditional expectation or Tail VAR, represents the mean shortfall at a specified confidence level. It has more attractive theoretical properties and it leads to LP solvable portfolio optimization models in the case of discrete random variables, i.e., in the case of returns defined by their realizations under the specified scenarios. We show that the mean-risk models using the worst conditional expectation or some of its extensions are in harmony with the stochastic dominance order. For this purpose, we exploit duality relations of convex analysis to develop the quantile model of stochastic dominance for general distributions.
The Markowitz model for single period portfolio optimization quantifies the problem by means of o... more The Markowitz model for single period portfolio optimization quantifies the problem by means of only two criteria: the mean, representing the expected outcome, and the risk, a scalar measure of the variability of outcomes. The classical Markowitz model uses the variance as the risk measure, thus resulting in a quadratic optimization problem. Following Sharpe's work on linear approximation to the mean-variance model, many attempts have been made to linearize the portfolio optimization problem. There were introduced several alternative risk measures which are computationally attractive as (for discrete random variables) they result in solving Linear Programming (LP) problems. The LP solvability is very important for applications to real-life financial decisions where the constructed portfolios have to meet numerous side constraints and take into account transaction costs. This paper provides a systematic overview of the LP solvable models with a wide discussion of their properties.
The Reference Point Method (RPM) is based on the so-called augmented max-min aggregation. The reg... more The Reference Point Method (RPM) is based on the so-called augmented max-min aggregation. The regularization by the average achievement is easily implementable but it may disturb the basic max-min model. The only consequent regularization of the max-min aggregation is the lexicographic max-min (nucleolar) solution concept where in addition to the worst achievement, the second worst achievement is also optimized (provided that the worst remains on the optimal level), the third worst is optimized (provided that the two worst remain optimal), and so on. Such a nucleolar regularization satisfies the addition/deleting principle thus making the corresponding nucleolar RPM not affected by any passive criteria. The nucleolar regularization is more complicated in implementation but the recent progress made in optimization methods for ordered averages allows one to implement the nucleolar RPM quite effectively. Both the theoretical and implementation issues of the nucleolar RPM are analyzed.
— The rapid growth of traffic induced by Internet services makes the simple over-provisioning of ... more — The rapid growth of traffic induced by Internet services makes the simple over-provisioning of resources not economical and hence imposes new requirements on the di-mensioning methods. Therefore, the problem of network design with the objective of minimizing the cost and at the same time solving the tradeoff between maximizing the service data flows and providing fair treatment of all demands becomes more and more important. In this context, the so-called Max-Min Fair (MMF) principle is widely considered to help finding reasonable bandwidth allocation schemes for competing demands. Roughly speaking, MMF assumes that the worst service performance is maximized, and then is the second worst performance, the third one, and so on, leading to a lexicographically maximized vector of sorted demand bandwidth allocations. It turns out that the MMF optimal solution cannot be approached in a standard way (i.e., as a mathematical programming problem) due to the necessity of lexicographic maximization of ordered quantities (bandwidth allocated to demands). Still, for convex models, it is possible to formulate effective sequential procedures for such lexicographic optimization. The purpose of the presented paper is threefold. First, it discusses resolution algorithms for a generic MMF problem related to telecommunications network design. Second, it gives a survey of network design instances of the generic formulation, and illustrates the efficiency of the general algorithms in these particular cases. Finally, the paper discusses extensions of the formulated problems into more practical (unfortunately non-convex) cases, where the general for convex MMF problems approach fails.
As an active participant of a competitive energy market, the generator (the energy supplier) chal... more As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).
As an active participant of a competitive energy market, the generator (the energy supplier) chal... more As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).
As an active participant of a competitive energy market, the generator (the energy supplier) chal... more As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).
As an active participant of a competitive energy market, the generator (the energy supplier) chal... more As an active participant of a competitive energy market, the generator (the energy supplier) challenges new management decisions being exposed to the financial risk environment. There is a strong need for the decision support models and tools for energy market participants. This paper shows that the stochastic short-term planning model can be effectively used as a key analytical tool within the decision support process for relatively small energy suppliers (price-takers). A self-scheduling method for the thermal units on the energy market is addressed. A schedule acquired for given preferences can be used as a desired pattern for bidding process. The uncertainty of the market prices is modeled by a set of possible scenarios with assigned probabilities. Several risk criteria are introduced leading to a multiple criteria optimization problem. The risk criteria are well appealing and easily computable (by means of linear programming) but they meet the formal risk aversion standards. The aspiration/reservation based interactive analysis applied to the multiple criteria problem allows us to find an efficient solution (generation scheme) well adjusted to the generator preferences (risk attitude).
An early motivation for this study was the problem of relocation of scarce or endangered species ... more An early motivation for this study was the problem of relocation of scarce or endangered species of animals for breeding and/or reintroduction to establish new populations in the wild. In this paper, we introduce single and multiple objective optimization models which are designed to comprehend a wide variety of objectives which are of interest to conservation and wildlife managers. We present the models in a general way and point out special features relative to ecology as they arise. Thus, the models may be used for relocation decisions analysis in diverse fields, not only in conservation and ecology. After presentation of the models in such a general way, we reformulate the models to make use of the special structure present. Such reformulation reduces the number of decision variables and constraints and, in general, makes solutions easy to obtain. By easy to obtain, we mean that tools from linear and mixed-integer programming together with elementary sorting procedures provide the basis for solving the models.
Scope and Purpose---The classical interactive procedures for multiple criteria decision analysis ... more Scope and Purpose---The classical interactive procedures for multiple criteria decision analysis assume the so-called rational behavior of decision makers: they know the decision problem, and they are consistent and coherent in the decision process. However, as has been stressed by many researchers and practitioners, decision makers usually learn the decision problem during the interactive session with the decision support system, and there are numerous examples in which people systematically violate the consistency and coherence of their preferences. Therefore, the so-called aspiration-based interactive decision support schemes seem to be much more interesting for practical implementations. This interactive process explicitly depends on the aspiration levels stated and modified by the decision maker, and thereby makes operational the concept of the adaptive dependence of the decision process on learning and context. The paper presents an implementation of an aspiration-based interactive procedure to solve various multiobjective transshipment problems with facility location.
2006 2nd Conference on Next Generation Internet Design and Engineering, 2006
Expanding demand on the Internet services leads to an increased role of the network dimensioning ... more Expanding demand on the Internet services leads to an increased role of the network dimensioning problem for elastic traffic where one needs to allocate bandwidth to maximize service flows and simultaneously to reach a fair treatment of all the elastic services. Thus, both the overall efficiency (throughput) and the fairness (equity) among various services are important. The Max-Min Fairness (MMF) approach, widely used to this problem, guarantees fairness but may lead to significant losses in the overall throughput of the network. In this paper we show how the concepts of multiple criteria equitable optimization can be effectively used to generate various fair resource allocation schemes. We introduce a multiple target model equivalent to equitable optimization and we develop a corresponding procedure to generate fair efficient bandwidth allocations. The procedure is tested on a sample network dimensioning problem and its abilities to model various preferences are demonstrated.
This paper discusses connections between the multi-criteria techniques of goal programming (GP) a... more This paper discusses connections between the multi-criteria techniques of goal programming (GP) and the reference point method (RPM). Both approaches use a certain target point in the criterion (outcome) space as a key element to model decision maker preferences. Therefore, RPM can be expressed, similarly to GP, in the modelling framework of deviational variables. The paper gives a systematic survey and analysis of the lexicographic GP models of RPM. The corresponding preference models are formalised and analysed with respect to target values interpretations as well as the Pareto-efficiency of their solutions. The properties of equity among the individual achievements of solutions are also analysed with respect to the Rawlsian principle of justice.
In the past decade, increasing interest in equity issues resulted in new methodologies in the are... more In the past decade, increasing interest in equity issues resulted in new methodologies in the area of operations re-10 search. This paper deals with the concept of equitably efficient solutions to multiple criteria optimization problems. 11 Multiple criteria optimization usually starts with an assumption that the criteria are incomparable. However, many 12 applications arise from situations which present equitable criteria. Moreover, some aggregations of criteria are often 13 applied to select efficient solutions in multiple criteria analysis. The latter enforces comparability of criteria (possibly 14 rescaled). This paper presents aggregations which can be used to derive equitably efficient solutions to both linear and 15 nonlinear multiple optimization problems. An example with equitable solutions to a capital budgeting problem is 16 analyzed in detail. An equitable form of the reference point method is introduced and analyzed. 17
This note discusses the properties of solutions generated by the minmax models of goal programmin... more This note discusses the properties of solutions generated by the minmax models of goal programming (GP) and compromise programming (CP). GP approaches use a certain target point in the criterion (attribute) space to model decision maker's preferences. When the ideal (utopia) point is used as the target, the minmax GP model coincides with the minmax (Chebyshev) CP model. In a recent review of the current GP state-of-the-art, there have been included suggestions that the two equivalent models ensure Pareto eciency of solutions and they guarantee a perfectly balanced allocation among the achievement of the individual targets. In this note, it is shown that the models, in general, do not ensure the eciency of solutions and they do not guarantee the perfect equity among the individual achievements. Moreover, there are given sucient and necessary conditions clarifying when the discussed properties of minmax solutions do occur. Ó
Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic domi... more 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 tradeoff 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.
We analyze relations between two methods frequently used for modeling the choice among uncertain ... more 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.
International Transactions in Operational Research, 2002
Following the seminal work by Markowitz, the portfolio selection problem is usually modeled as a ... more Following the seminal work by Markowitz, the portfolio selection problem is usually modeled 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. Several other risk measures have been later considered thus creating the entire family of mean-risk (Markowitz type) models. In this paper, we analyze mean-risk models using quantiles and tail characteristics of the distribution. Value at risk (VAR), defined as the maximum loss at a specified confidence level, is a widely used quantile risk measure. The corresponding second order quantile measure, called the worst conditional expectation or Tail VAR, represents the mean shortfall at a specified confidence level. It has more attractive theoretical properties and it leads to LP solvable portfolio optimization models in the case of discrete random variables, i.e., in the case of returns defined by their realizations under the specified scenarios. We show that the mean-risk models using the worst conditional expectation or some of its extensions are in harmony with the stochastic dominance order. For this purpose, we exploit duality relations of convex analysis to develop the quantile model of stochastic dominance for general distributions.
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Papers by W. Ogryczak