Random projections for quadratic programs

Random projections map a set of points in a high dimensional space to a lower dimen- sional one while approximately preserving all pairwise Euclidean distances. While random projections are usually applied to numerical data, we show they can be successfully applied to quadratic programming formulations over a set of linear inequality constraints. Instead of solving … Read more

Random projections for trust region subproblems

The trust region method is an algorithm traditionally used in the field of derivative free optimization. The method works by iteratively constructing surrogate models (often linear or quadratic functions) to approximate the true objective function inside some neighborhood of a current iterate. The neighborhood is called “trust region” in the sense that the model is … Read more

Random projections for linear programming

Random projections are random linear maps, sampled from appropriate distributions, that approximately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well-known Johnson-Lindenstrauss lemma states that there are \LL{random matrices with surprisingly few rows} that approximately preserve pairwise Euclidean distances among a set of points. This is … Read more

Fast approximate solution of large dense linear programs

We show how random projections can be used to solve large-scale dense linear programs approximately. This is a new application of techniques which are now fairly well known in probabilistic algorithms, but have never yet been systematically applied to the fundamental class of Linear Programming. We develop the necessary theoretical framework, and show that this … Read more

Bilevel mixed-integer linear programs and the zero forcing set

We study a class of bilevel binary linear programs with lower level variables in the upper-level constraints. Under certain assumptions, we prove that the problem can be reformulated as a single-level binary linear program, and propose a finitely terminating cut generation algorithm to solve it. We then relax the assumptions by means of a general … Read more

The Power Edge Set problem

The automated real time control of an electrical network is achieved through the estimation of its state using Phasor Measurement Units (PMUs). Given an undirected graph representing the network, we study the problem of finding the minimum number of PMUs to place on the edges such that the graph is fully observed. This problem is … Read more

Using the Johnson-Lindenstrauss lemma in linear and integer programming

The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as $k$-means or $k$ nearest neighbours since they only use Euclidean distances, and has sometimes been used in optimization algorithms involving the minimization of Euclidean distances. In this paper we … Read more

Robust optimal sizing of an hybrid energy stand-alone system

This paper deals with the optimal design of a stand-alone hybrid system composed of wind turbines, solar photovoltaic panels and batteries. To compensate for a possible lack of energy from these sources, an auxiliary fuel generator uarantees to meet the demand in every case but its use induces important costs. We have chosen a two-stage … Read more

2-Stage Robust MILP with continuous recourse variables

We solve a linear robust problem with mixed-integer first-stage variables and continuous second stage variables. We consider column wise uncertainty. We first focus on a problem with right hand-side uncertainty which satisfies a “full recourse property” and a specific definition of the uncertainty. We propose a solution based on a generation constraint algorithm. Then we … Read more