Abdel Lisser

Graph partitioning using linear and semidefinite programming

Abstract.u2002Graph partition is used in the telecommunication industry to subdivide a transmission network into small clusters. We consider both linear and semidefinite relaxations for the equipartition problem and present numerical results on real data from France Telecom networks with up 900 nodes, and also on randomly generated problems.

Upper bounds for the 0-1 stochastic knapsack problem and a B&B algorithm

In this paper we study and solve two different variants of static knapsack problems with random weights: The stochastic knapsack problem with simple recourse as well as the stochastic knapsack problem with probabilistic constraint. Special interest is given to the corresponding continuous problems and three different problem solving methods are presented. The resolution of the continuous problems allows to provide upper bounds in a branch-and-bound framework in order to solve the original problems. Numerical results on a dataset from the literature as well as a set of randomly generated instances are given.

A second-order cone programming approach for linear programs with joint probabilistic constraints

Abstract This paper deals with a special case of Linear Programs with joint Probabilistic Constraints (LPPC) with normally distributed coefficients and independent matrix vector rows. Through the piecewise linear approximation and the piecewise tangent approximation, we approximate the stochastic linear programs with two second-order cone programming (SOCP for short) problems. Furthermore, the optimal values of the two SOCP problems are a lower and upper bound of the original problem respectively. Finally, numerical experiments are given on randomly generated data.

Knapsack problem with probability constraints

This paper is dedicated to a study of different extensions of the classical knapsack problem to the case when different elements of the problem formulation are subject to a degree of uncertainty described by random variables. This brings the knapsack problem into the realm of stochastic programming. Two different model formulations are proposed, based on the introduction of probability constraints. The first one is a static quadratic knapsack with a probability constraint on the capacity of the knapsack. The second one is a two-stage quadratic knapsack model, with recourse, where we introduce a probability constraint on the capacity of the knapsack in the second stage. As far as we know, this is the first time such a constraint has been used in a two-stage model. The solution techniques are based on the semidefinite relaxations. This allows for solving large instances, for which exact methods cannot be used. Numerical experiments on a set of randomly generated instances are discussed below.

On two-stage stochastic knapsack problems

Abstract In this paper we study a particular version of the stochastic knapsack problem with normally distributed weights: the two-stage stochastic knapsack problem. Contrary to the single-stage knapsack problem, items can be added to or removed from the knapsack at the moment the actual weights become known (second stage). In addition, a chance-constraint is introduced in the first stage in order to restrict the percentage of cases where the items chosen lead to an overload in the second stage. To the best of our knowledge, there is no method known to exactly evaluate the objective function for a given first-stage solution. Therefore, we propose methods to calculate the upper and lower bounds. These bounds are used in a branch-and-bound framework in order to search the first-stage solution space. Special interest is given to the case where the items have similar weight means. Numerical results are presented and analyzed.

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

We present integer linear models with a polynomial number of variables and constraints for combinatorial optimization problems in graphs: optimum elementary cycles, optimum elementary paths and optimum tree problems.

Telecommunication Network Capacity Design for Uncertain Demand

The expansion of telecommunication services has increased the number of users sharing network resources. When a given service is highly demanded, some demands may be unmet due to the limited capacity of the network links. Moreover, for such demands, telecommunication operators should pay penalty costs. To avoid rejecting demands, we can install more capacities in the existing network. In this paper we report experiments on the network capacity design for uncertain demand in telecommunication networks with integer link capacities. We use Poisson demands with bandwidths given by normal or log-normal distribution functions. The expectation function is evaluated using a predetermined set of realizations of the random parameter. We model this problem as a two-stage mixed integer program, which is solved using a stochastic subgradient procedure, the Barahonas volume approach and the Benders decomposition.

On a stochastic bilevel programming problem

In this article, a mixed integer bilevel problem having a probabilistic knapsack constraint in the first level is proposed. The problem formulation is mainly motivated by practical pricing and service provision problems as it can be interpreted as a model for the interaction between a service provider and customers. A discrete probability space is assumed which allows a reformulation of the problem as an equivalent deterministic bilevel problem. The problem is further transformed into a linear bilevel problem, which in turn yields a quadratic optimization problem, namely the global linear complementarity problem. Based on this quadratic problem, a procedure to compute upper bounds on the initial problem by using a Lagrangian relaxation and an iterative linear minmax scheme is proposed. Numerical experiments confirm that the scheme practically converges.© 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Enhancing a Branch-and-Bound Algorithm for Two-Stage Stochastic Integer Network Design-Based Models

In this paper we present branch-and-bound (B&B) strategies for two-stage stochastic integer network design-based models with integrality constraints in the first-stage variables. These strategies are used within L-shaped decomposition-based B&B framework. We propose a valid inequality in order to improve B&B performance. We use this inequality to implement a multirooted B&B tree. A selective use of optimality cuts is explored in the B&B approach and we also propose a subgradient-based technique for branching on 0-1 feasible solutions. Finally, we present computational results for a fixed-charge network design problem with random demands.

Stochastic Shortest Path Problem with Delay Excess Penalty

Abstract We study and solve a particular stochastic version of the Restricted Shortest Path Problem, the Stochastic Shortest Path Problem with Delay Excess Penalty. While arc costs are kept deterministic, arc delays are assumed to be normally distributed and a penalty per time unit occurs whenever the given delay constraint is not satisfied. The objective is to minimize the sum of path cost and total delay penalty.