# Sourour ElloumiENSTA Paris · Unité de Mathématiques Appliquées

Sourour Elloumi

## About

100

Publications

5,364

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

895

Citations

## Publications

Publications (100)

The p-median problem is a classic discrete location problem with several applications. It aims to open p sites while minimizing the sum of the distances of each client to its nearest open site. We study a Benders decomposition of the most efficient formulation in the literature. We prove that the Benders cuts can be separated by a polynomial time a...

Solution robustness focuses on structural similarities between the nominal solution and the scenario solutions. Most other robust optimization approaches focus on the quality robustness and only evaluate the relevance of their solutions through the objective function value. However, it can be more important to optimize the solution robustness and,...

We propose a method called Polynomial Quadratic Convex Reformulation (PQCR) to solve exactly unconstrained binary polynomial problems (UBP) through quadratic convex reformulation. First, we quadratize the problem by adding new binary variables and reformulating (UBP) into a non-convex quadratic program with linear constraints (MIQP). We then consid...

Wireless sensor networks have been widely deployed in the last decades to provide various services, like environmental monitoring or object tracking. Such a network is composed of a set of sensor nodes which are used to sense and transmit collected information to a base station. To achieve this goal, two properties have to be guaranteed: (i) the se...

We analyze a product pricing problem with single-minded customers, each interested in buying a bundle of products. The objective is to maximize the total revenue and we assume that supply is unlimited for all products. We contribute to a missing piece of literature by giving some mathematical formulations for this single-minded bundle pricing probl...

Optimal Power Flow (OPF) can be modeled as a non-convex Quadratically Constrained Quadratic Program (QCQP). Our purpose is to solve OPF to global optimality. To this end, we specialize the Mixed-Integer Quadratic Convex Reformulation method (MIQCR) to (OPF). This is a method in two steps. First, a Semi-Definite Programming (SDP) relaxation of (OPF)...

We propose a solution approach for the problem (P) of minimizing an unconstrained binary polynomial optimization problem. We call this method PQCR (Polynomial Quadratic Convex Reformulation). The resolution is based on a 3-phase method. The first phase consists in reformulating (P) into a quadratic program (QP). For this, we recursively reduce the...

The p-center problem consists in selecting p centers among M to cover N clients, such that the maximal distance between a client and its closest selected center is minimized. For this problem we propose two new and compact integer formulations.

The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QCQPs with the following restriction: all quadratic sub-functions of purely con...

In this paper, we focus on the problem of robust rolling-stock planning for French passenger trains. First, we characterize robustness and define some indicators for the evaluation of rolling-stock rosters. We take a particular interest in homogenizing turning-times in a roster in order to absorb potential delays. Then, we propose a new approach to...

We consider the (QAP) that consists in minimizing a quadratic function subject to assignment constraints where the variables are binary. In this paper, we build two families of equivalent quadratic convex formulations of (QAP). The continuous relaxation of each equivalent formulation is then a convex problem and can be used within a B&B. In this wo...

We present algorithm MIQCR-CB that is an advancement of method MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem $(SDP)$, and, in the s...

Nous proposons une formalisation en Coq des graphes orientés et non orientés sans arête multiple. La bibliothèque développée offre non seulement l'expressivité requise pour exprimer et démontrer des propriétés sur les graphes mais aussi une implantation purement fonctionnelle permettant de mettre en oeuvre efficacement les algorithmes de graphes. N...

Frequent upgrades of equipment in the telecommunications industry occur due to the emergence of new services or technological breakthroughs. In this work, we consider a network where each client is linked to a site and handled by a card located on that site. A technological migration has to be undertaken within a short horizon of a few years and it...

To ensure the robust delivery of video streams, a network must emit additional data that will replace the missing ones in case of a failure of some server. One key issue is then to keep this so-called redundancy as low as possible while still ensuring some quality of service. In this paper, we provide protectionoriented models to compute the minimu...

We consider problem (QP) of minimizing a quadratic function subject to linear or quadratic constraints. Variables are integer and bounded. This very general problem can model many classical problems in Combinatorial Optimization.
A major difference between (QP) and integer linear programs lies in the fact that, in general, its continuous relaxation...

Let
$(MQP)$
be a general mixed-integer quadratic program that consists of minimizing a quadratic function
$f(x) = x^TQx +c^Tx$
subject to linear constraints. Our approach to solve
$(MQP)$
is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables
$y_{ij}$
, additional quadrat...

We review Quadratic Convex Reformulation (QCR) for quadratic pro-
grams with general integer variables. This solution 2-phase approach
consist in first reformulating the quadratic program into an equivalent
other problem having a convex ob jective function. The second phase
relies on MIP solvers that solve the reformulated problem by standard
branc...

We address survivability considerations for telecommunication networks where a part of the network may fail. We focus on single arc failures in multicast networks, with or without network coding. The problem is to compute a routing such that, if any single arc failure occurs, the remaining throughput is as large as possible. In the case of multicas...

Nous considérons le problème (QP) de la minimisation d'une fonction quadratique sous des contraintes linéaires ou quadratiques. Les variables sont entières et bornées. Ce problème très général permet de modéliser de nombreux problèmes classiques en Optimisation Combinatoire et constitue une première généralisation de la programmation linéaire en no...

Many graph partitionning problems can be formulated by quadratic programs (QP) with binary variables and linear and quadratic constraints. We apply the general approach that consists in first reformulating the initial (QP) into an equivalent program (QP ). Problem (QP ) has the additional property that its continuous relaxation is a convex quadrati...

We review the quadratic convex reformulation approach for quadratic programs with integer variables. We also show the recent extensions to quadratically constrained programs and to the case of mixed-integer variables. In all these extensions, the global framework is the same: in a preprocessing step, we compute a tight equivalent reformulation of t...

We present a solution approach for the general problem (QP) of minimizing a quadratic function of integer variables subject to a set of quadratic constraints. The resolution is divided into two phases. The ?rst phase is to reformulate the initial problem as an equivalent quadratic problem which continuous relaxation is convex; the second phase is t...

Given a telecommunication network, modelled by a capacitated digraph,
we are interested in comparing the behaviour and usefulness of two
information propagation schemes, namely multicast and network coding,
when the aforementioned network is subject to simple arc failure.
We consider the case with a single source node and a set of terminal
nodes. T...

We address the exact solution of general integer quadratic programs with linear constraints. These programs constitute a particular case of mixed-integer quadratic programs for which we introduce in Billionnet et al. (Math. Program., 2010) a general solution method based on quadratic convex reformulation, that we called MIQCR. This reformulation co...

We consider two problems that arise in designing two-level star networks taking into account service quality considerations. Given a set of nodes with pairwise traffic demand and a central hub, we select p hubs and connect them to the central hub with direct links and then we connect each nonhub node to a hub. This results in a star/star network. I...

A robust-planning methodology for
S. Tréfond1, H. Djellab1, E. Escobar1, A. Billionnet2 & S. Elloumi2
1SNCF-Innovation & Research department, France
2CEDRIC-ENSIIE, France
Abstract
This paper deals with an investigation of combinatorial and robust optimization
models to solve rolling-stock planning problems for passenger trains. Here
robustness mea...

We consider binary quadratic programs (QP) having a quadratic objective
function, linear constraints, and binary variables. Many classical solution methods of
these problems are based on exact reformulation of QP into an
equivalent mixed integer linear program. Several linearization methods were
studied in the literature. More recent solution metho...

Let (MQP) be a MIQP that consists in minimizing a quadratic function subject to linear constraints. Our approach to solve (MQP) is first to consider (MQP'), an equivalent MIQP that has a convex objective function, additional variables and constraints, and additionnal quadratic constraints. Then, we propose a new Branch and Bound based on the relaxa...

Let (MQP) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints.
In this paper, we present a convex reformulation of (MQP), i.e. we reformulate (MQP) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that
uses a...

In this paper, we consider problem (P ) of minimizing a quadratic function q(x)
=
x
t
Qx
+
c
t
x of
binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers.
But, for this, we have to first convexify the objective function q(x). A classical trick is to raise up the diagonal
entries of Q by a vector u u...

We consider an integer program (QQP) where both the objective function and the constraints contain quadratic terms. We show that the quadratic convex reformulation approach can be extended to that case. We start by solving a semidefinite programming problem (SDP). From the dual solution of SDP, we deduce reformulation of QQP as an equivalent proble...

Given a set of clients and a set of potential sites for facilities, the p-median problem consists of opening a set of p sites and assigning each client to the closest open facility to it. In [Elloumi, S., A tighter formulation of the p-median problem, J. Comb. Optim., 19 (2004), 69–83], a new formulation of this problem was proposed that takes bene...

Given a set of clients and a set of potential sites for facilities, the p-median problem consists of opening a set of p sites and assigning each client to the closest open facility to it. It can be viewed as a variation of the uncapacitated
facility location problem. We propose a new formulation of this problem by a mixed integer linear problem. We...

Let (QP) be a mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of (QP), i.e. we reformulate (QP) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and...

In this paper, we consider the problem of designing a two level telecommunications network with service quality considerations. We are given a set of users or demand nodes and each of these nodes wants to communicate with all others. A fixed central hub is given and $p$ additional hubs should be chosen among the user nodes. Then each hub is connect...

-Let (QP) be an integer quadratic program that consists in minimizing a quadratic function
subject to linear constraints. To solve (QP), we reformulate it into an equivalent program with a convex objective function, and we use a Mixed Integer Quadratic Programming solver. This reformulation, called IQCR, is optimal in a certain sense from the conti...

-Let (QP) be a binary quadratic program that consists in minimizing a quadratic function subject to linear constraints. To solve (QP) we reformulate it into an equivalent program with a convex objective function. Our reformulation, that we call EQCR (Extended Quadratic Convex Reformulation), is optimal from the continuous relaxation bound point of...

Frequent upgrades of equipments in the telecommunications industry occur due to the emergence of new services or technologic breakthroughs. In this work, we consider a network where each client is linked to a site and handled by a card located on that site. A technological migration has to be operated and it consists in replacing all the existing c...

Let be a 0-1 quadratic program which consists in minimizing a quadratic function subject to linear equality constraints. In this paper, we present QCR, a general method to reformulate into an equivalent 0-1 program with a convex quadratic objective function. The reformulated problem can then be efficiently solved by a classical branch-and-bound alg...

Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalitie...

Let (QP) be an integer quadratic program that consists in minimizing a quadratic function subject to linear constraints. In this
paper, we present several linearizations of (QP). Many linearization methods for the quadratic 0-1 programs are known. A natural approach when considering (QP) is to reformulate it into a quadratic 0-1 program. However, t...

Given a set of clients and a set of potential sites for facilities, several location problems consist of opening a set of sites and assigning each client to the closest open facility to it. It can be viewed as a variation of the uncapacitated facility location problem. We propose a new formulation of this problem by a mixed integer linear problem....

Many combinatorial optimization problems can be formu- lated as the minimization of a 0-1 quadratic function subject to linear constraints. In this paper, we are interested in the exact solution of this problem through a two-phase general scheme. The first phase con- sists in reformulating the initial problem either into a compact mixed integer lin...

In this paper, we consider problem (P) of minimizing a quadratic function q(x)=x
t
Qx+c
t
x of binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers. But, for this, we have to first convexify the objective function q(x). A classical trick is to raise up the diagonal entries of Q by a vector u until...

Given an undirected graph G = (V, E), we consider the graph bisection problem, which consists in partitioning the nodes of G in two disjoined sets with p and n − p nodes respectively such that the total weight of edges crossing between subsets is minimal. We apply QCR to it, a general method, presented in [4], which combines semidefinite programmin...

Let (QP) be a 0-1 quadratic program which consists in minimizing a quadratic function subject to linear equality constraints. In this paper, we present QCR, a general method to reformulate (QP) into an equivalent 0-1 program with a convex quadratic objective function. The reformulated problem can then be efficiently solved by a classical branch-and...

Let (QP) be a 0-1 quadratic program which consists in minimizing a quadratic function subject to linear constraints. In this paper, we present a general method to solve (QP) by reformulation of the problem into an equivalent 0-1 program with a convex quadratic objective function, followed by the use of a standard mixed integer quadratic programming...

In this paper, we address the SDH network design problem (SDHNDP) which arises while designing the fixed part of global system for mobile communications access networks using synchronous digital hierarchy (SDH) rings.An SDH ring is a simple cycle that physically links a subset of antennae to a single concentrator. Inside a ring, a concentrator hand...

We consider the Module Allocation Problem with Non-Uniform communica- tion costs (MAPNU), where a set of program modules must be assigned to a set of processors. The optimal assignment minimizes the sum of execution costs and communication costs between modules. This problem is naturally formulated as a quadratic 0-1 problem with linear constraints...

The p-Center problem consists in locating p facilities among a set of M possible locations and assigning N clients to them in order to minimize the maximum distance between a client and the facility to which it is allocated. We present a new integer linear programming formulation for this Min-Max problem with a polynomial number of variables and co...

We address the Quadratic Assignment Problem following a polyhedral method. We consider the Quadratic Assignment Polytopc defined as the convex hull of the solutions of the linearized problem. Its dimension and a minimal description of its affine hull have been given by Padberg and Rijal (1996). Here we propose a large family of valid inequalities i...

We address the Quadratic Assignment Problem following a polyhedral method. We consider the Quadratic Assignment Polytope defined as the convex hull of the solutions of the linearized problem. Its dimension and a minimal description of its affine hull have been given by Padberg and Rijal (1996). Here we propose a large family of valid inequalities i...

We address the Quadratic Assignment Problem following a polyhedral method. We consider the Quadratic Assignment Polytope defined as the convex hull of the solutions of the linearized problem. Its dimension and a minimal description of its affine bull have been given by Padberg and Rijal (1996). Here we propose a large family of valid inequalities i...

We consider the Constrained Module Allocation Problem (CMAP), where a set of program modules must be assigned to a set of processors having a limited capacity. The optimal assignment minimizes the sum of execution costs and communication costs between modules. This problem is naturally formulated as a quadratic 0-1 problem with linear constraints....

The p-Center problem consists in locating p facilities among a set of M possible locations and assigning N clients to them in order to minimize the maximum distance between a client and the facility to which it is allocated. We present a new integer linear programming formulation for this Min-Max problem with a polynomial number of variables and co...

We consider the quadratic semi-assignment problem in which we minimize a quadratic pseudo-Boolean function F subject to the semi-assignment constraints. We propose in this paper a linear programming method to obtain the best reduction of this problem, i.e. to compute the greatest constant c such that F is equal to c plus F′ for all feasible solutio...

Québec Canada 9-12 Mai 2001

This paper presents a general decomposition method to compute bounds for constrained 0-1 quadratic programming. The best decomposition is found by using a Lagrangian decomposition of the problem. Moreover, in its simplest version this method is proved to give at least the bound obtained by the LP-relaxation of a non-trivial linearization. To illust...

We consider the problem of allocating n tasks of a distributed program to m processors of a distributed system in order to minimize total communication and processing costs. If the intertask communication can be represented by a tree and if the communication costs are uniform, it is known that an optimal allocation can be determined in O(nm) time....

We consider a set of n processors p1, P2,…,pn which communicate via a network and a modular program consisting of m tasks t1, t2,…, tm. Tasks can be assigned to any processor and some of them exchange data.The network configuration can be represented by a graph Gp = (Vp, Ep); the set of vertices Vp corresponds to the set of processors and two verti...

On considere un ensemble de tâches devant d'executer sur un reseau totalement maille de processeurs heterogenes disposant de ressources limitees. Un placement des tâches sur les processeurs doit tenir compte des contraintes de ressources et engendre un cout egal a la somme des couts d'execution des tâches sur les processeurs auxquels elles sont aff...