Quentin Louveaux

Inequalities from Two Rows of a Simplex Tableau

In this paper we explore the geometry of the integer points in a cone rooted at a rational point. This basic geometric object allows us to establish some links between lattice point free bodies and the derivation of inequalities for mixed integer linear programs by considering two rows of a simplex tableau simultaneously.

Lifting, superadditivity, mixed integer rounding and single node flow sets revisited

Abstract In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for mixed integer programs. The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0–1 single node flow set.

Split Rank of Triangle and Quadrilateral Inequalities

A simple relaxation consisting of two rows of a simplex tableau is a mixed-integer set with two equations, two free integer variables, and nonnegative continuous variables. Recently, Andersen et al. and Cornuejols and Margot showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from lattice-free triangles and quadrilaterals. From an example given by Cook et al. it is known that one particular class of facet-defining triangle inequality does not have finite split rank. In this paper we show that all other facet-defining triangle and quadrilateral inequalities have finite split rank.

An Analysis of Mixed Integer Linear Sets Based on Lattice Point Free Convex Sets

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Lifting, Superadditivity, Mixed Integer Rounding and Single Node Flow Sets Revisited

\max_{x \in L} \pi^T x - \min_{x \in L} \pi^T x \leq w

An efficient algorithm for the provision of a day-ahead modulation service by a load aggregator

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A quantitative Doignon-Bell-Scarf theorem

\pi^T x \leq \pi_0

Combining Problem Structure with Basis Reduction to Solve a Class of Hard Integer Programs

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An algorithm for the separation of two-row cuts

Abstract.In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for mixed integer programs. The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0-1 single node flow set.

Online Sparse Bandit for Card Games

This article studies a decision making problem faced by an aggregator willing to offer a load modulation service to a Transmission System Operator. This service is contracted one day ahead and consists in a load modulation option, which can be called once per day. The option specifies the range of a potential modification on the demand of the loads within a certain time interval. The specific case where the loads can be modeled by a generic tank model is considered. Under this assumption, the problem of maximizing the range of the load modulation service can be formulated as a mixed integer linear programming problem. A novel heuristic-method is proposed to solve this problem in a computationally efficient manner. This method is tested on a set of problems. The results show that this approach can be orders of magnitude faster than CPLEX without significantly degrading the solution accuracy.