A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs

In this paper we generalize N-fold integer programs and two-stage integer programs with N scenarios to N-fold 4-block decomposable integer programs. We show that for fixed blocks but variable N, these integer programs are polynomial-time solvable for any linear objective. Moreover, we present a polynomial-time computable optimality certificate for the case of fixed blocks, variable N and any convex separable objective function. We conclude with two sample applications, stochastic integer programs with second-order dominance constraints and stochastic integer multi-commodity flows, which (for fixed blocks) can be solved in polynomial time in the number of scenarios and commodities and in the binary encoding length of the input data. In the proof of our main theorem we combine several non-trivial constructions from the theory of Graver bases. We are confident that our approach paves the way for further extensions

Stochastic programming models and methods for portfolio optimization and risk management

This project is focused on stochastic models and methods and their application in portfolio optimization and risk management. In particular it involves development and analysis of novel numerical methods for solving these types of problem. First, we study new numerical methods for a general second order stochastic dominance model where the underlying functions are not necessarily linear.Specifically, we penalize the second order stochastic dominance constraints to the objective under Slater’s constraint qualification and then apply the well known stochastic approximation method and the level function methods to solve the penalized problem and present the corresponding convergence analysis. All methods are applied to some portfolio optimization problems, where the underlying functions are not necessarily linear all results suggests that the portfolio strategy generated by the second order stochastic dominance model outperform the strategy generated by the Markowitz model in a sense of having higher return and lower risk. Furthermore a nonlinear supply chain problem is considered, where the performance of the level function method is compared to the cutting plane method. The results suggests that the level function method is more efficient in a sense of having lower CPU time as well as being less sensitive to the problem size. This is followed by study of multivariate stochastic dominance constraints. We propose a penalization scheme for the multivariate stochastic dominance constraint and present the analysis regarding the Slater constraint qualification. The penalized problem is solved by the level function methods and a modified cutting plane method and compared to the cutting surface method proposed in [70] and the linearized method proposed in [4]. The convergence analysis regarding the proposed algorithms are presented. The proposed numerical schemes are applied to a generic budget allocation problem where it is shown that the proposed methods outperform the linearized method when the problem size is big. Moreover, a portfolio optimization problem is considered where it is shown that the a portfolio strategy generated by the multivariate second order stochastic dominance model outperform the portfolio strategy generated by the Markowitz model in sense of having higher return and lower risk. Also the performance of the algorithms is investigated with respect to the computation time and the problem size. It is shown that the level function method and the cutting plane method outperform the cutting surface method in a sense of both having lower CPU time as well as being less sensitive to the problem size. Finally, reward-risk analysis is studied as an alternative to stochastic dominance. Specifically, we study robust reward-risk ratio optimization. We propose two robust formulations, one based on mixture distribution, and the other based on the first order moment approach. We propose a sample average approximation formulation as well as a penalty scheme for the two robust formulations respectively and solve the latter with the level function method. The convergence analysis are presented and the proposed models are applied to Sortino ratio and some numerical test results are presented. The numerical results suggests that the robust formulation based on the first order moment results in the most conservative portfolio strategy compared to the mixture distribution model and the nominal model

Optimization with multivariate conditional value-at-risk constraints

For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods

Optimization with multivariate conditional value-at-risk constraints

For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods

Minimizing value-at-risk in the single-machine total weighted tardiness problem

The vast majority of the machine scheduling literature focuses on deterministic problems, in which all data is known with certainty a priori. This may be a reasonable assumption when the variability in the problem parameters is low. However, as variability in the parameters increases incorporating this uncertainty explicitly into a scheduling model is essential to mitigate the resulting adverse effects. In this paper, we consider the celebrated single-machine total weighted tardiness (TWT) problem in the presence of uncertain problem parameters. We impose a probabilistic constraint on the random TWT and introduce a risk-averse stochastic programming model. In particular, the objective of the proposed model is to find a non-preemptive static job processing sequence that minimizes the value-at-risk (VaR) measure on the random TWT at a specified confidence level. Furthermore, we develop a lower bound on the optimal VaR that may also benefit alternate solution approaches in the future. In this study, we implement a tabu-search heuristic to obtain reasonably good feasible solutions and present results to demonstrate the effect of the risk parameter and the value of the proposed model with respect to a corresponding risk-neutral approach

Processing second-order stochastic dominance models using cutting-plane representations

This is the post-print version of the Article. The official published version can be accessed from the links below. Copyright @ 2011 Springer-VerlagSecond-order stochastic dominance (SSD) is widely recognised as an important decision criterion in portfolio selection. Unfortunately, stochastic dominance models are known to be very demanding from a computational point of view. In this paper we consider two classes of models which use SSD as a choice criterion. The first, proposed by Dentcheva and Ruszczyński (J Bank Finance 30:433–451, 2006), uses a SSD constraint, which can be expressed as integrated chance constraints (ICCs). The second, proposed by Roman et al. (Math Program, Ser B 108:541–569, 2006) uses SSD through a multi-objective formulation with CVaR objectives. Cutting plane representations and algorithms were proposed by Klein Haneveld and Van der Vlerk (Comput Manage Sci 3:245–269, 2006) for ICCs, and by Künzi-Bay and Mayer (Comput Manage Sci 3:3–27, 2006) for CVaR minimization. These concepts are taken into consideration to propose representations and solution methods for the above class of SSD based models. We describe a cutting plane based solution algorithm and outline implementation details. A computational study is presented, which demonstrates the effectiveness and the scale-up properties of the solution algorithm, as applied to the SSD model of Roman et al. (Math Program, Ser B 108:541–569, 2006).This study was funded by OTKA, Hungarian National Fund for Scientific Research, project 47340; by Mobile Innovation Centre, Budapest University of Technology, project 2.2; Optirisk Systems, Uxbridge, UK and by BRIEF (Brunel University Research Innovation and Enterprise Fund)