Maarten H. van der Vlerk

Stochastic integer programming:General models and algorithms

We survey structural properties of and algorithms for stochastic integer programmingmodels, mainly considering linear two‐stage models with mixed‐integer recourse (and theirmulti‐stage extensions).

Integrated chance constraints: Reduced forms and an algorithm

We consider integrated chance constraints (ICCs), which provide quantitative alternatives for traditional chance constraints. We derive explicit polyhedral descriptions for the convex feasible sets induced by ICCs, for the case that the underlying distribution is discrete. Based on these reduced forms, we propose an efficient algorithm for this problem class.The relation to conditional value-at-risk models and (simple) recourse models is discussed, leading to a special purpose algorithm for simple recourse models with discretely distributed technology matrix. For both algorithms, numerical results are presented.

Solving stochastic programs with integer recourse by enumeration: a framework using Gro¨bner basis reductions

In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Gröbner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the expected integer recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed.

On the convex hull of the simple integer recourse objective function

We consider the objective function of a simple integer recourse problem with fixed technology matrix.Using properties of the expected value function, we prove a relation between the convex hull of this function and the expected value function of a continuous simple recourse program.We present an algorithm to compute the convex hull of the expected value function in case of discrete right-hand side random variables. Allowing for restrictions on the first stage decision variables, this result is then extended to the convex hull of the objective function.

Stochastic programming with simple integer recourse

Stochastic integer programs are notoriously difficult. Very few properties are known and solution algorithms are very scarce. In this paper, we introduce the class of stochastic programs with simple integer recourse, a natural extension of the simple recourse case extensively studied in stochastic continuous programs.Analytical as well as computational properties of the expected recourse function of simple integer recourse problems are studied. This includes sharp bounds on this function and the study of the convex hull. Finally, a finite termination algorithm is obtained that solves two classes of stochastic simple integer recourse problems.

Modelling aspects of distributed processingin telecommunication networks

The purpose of this paper is to formally describe new optimization models for telecommunicationnetworks with distributed processing. Modern distributed networks put morefocus on the processing of information and less on the actual transportation of data than weare traditionally used to in telecommunications. This paper introduces new approaches formodelling decision support at operational, tactical and strategic levels. One of the mainadvantages of the technological framework we are working within is its inherent flexibility,which enables us to dynamically plan and consider uncertainty when decisions are made.When we present the models, emphasis is placed on the modelling discussions around theshift of focus towards processing, the new technological aspects, and how to utilize flexibilityto cope with uncertainty.

An ALM model for pension funds using integrated chance constraints

We discuss integrated chance constraints in their role of short-term risk constraints in a strategic ALM model for Dutch pension funds. The problem is set up as a multistage recourse model, with special attention for modeling short-term risk prompted by the development of new guidelines by the regulating authority for Dutch pension funds. The paper concludes with a numerical illustration of the importance of such short-term risk constraints.

Convex approximations for complete integer recourse models

Abstract.We consider convex approximations of the expected value function of a two-stage integer recourse problem. The convex approximations are obtained by perturbing the distribution of the random right-hand side vector. It is shown that the approximation is optimal for the class of problems with totally unimodular recourse matrices. For problems not in this class, the result is a convex lower bound that is strictly better than the one obtained from the LP relaxation.

Integrated chance constraints in an ALM model for pension funds

We discuss integrated chance constraints in their role of short-term risk constraints in a strategic ALM model for Dutch pension funds. The problem is set up as a multistage recourse model, with special attention for modeling the guidelines proposed by the regulating authority for Dutch pension funds. The paper concludes with an outline of a special-purpose heuristic, which is used to approximately solve the resulting model which contains many binary decision variables.

Convex approximations for a class of mixed-integer recourse models

We consider mixed-integer recourse (MIR) models with a single recourse constraint. We relate the second-stage value function of such problems to the expected simple integer recourse (SIR) shortage function. This allows to construct convex approximations for MIR problems by the same approach used for SIR models.