J Glob Optim (2008) 42:533–547 DOI 10.1007/s10898-007-9275-5 A geometric characterization of “optimality-equivalent” relaxations Walid Ben Ameur · José Neto Received: 17 July 2007 / Accepted: 21 December 2007 / Published online: 15 January 2008 © Springer Science+Business Media, LLC. 2008 Abstract An optimization problem is defined by an objective function to be maximized with respect to a set of constraints. To overcome some theoretical and practical difficulties, the constraint-set is sometimes relaxed and “easier” problems are solved. This led us to study relaxations providing exactly the same set of optimal solutions. We give a complete charac- terization of these relaxations and present several examples. While the relaxations introduced in this paper are not always easy to solve, they may help to prove that some mathematical programs are equivalent in terms of optimal solutions. An example is given where some of the constraints of a linear program can be relaxed within a certain limit. Keywords Convex relaxation · Convex geometry · Sensitivity analysis 1 Introduction An approach that is commonly taken when dealing with optimization problems consists in relaxing some constraints and solving easier problems. Lagrangean relaxations [17], linear relaxations for integer [18, 20, 22] and convex problems [3, 11, 14], semidefinite relaxations [24], convex relaxations [6, 7, 13, 19] are often used to get either optimal or approximate solutions of the original problem. Given an optimization problem, a relaxation will be said to be optimality-equivalent if it has the same set of optimal solutions as the original problem. We will assume that the set of feasible solutions S of the original problem is convex and the objective function is linear. Notice that S is not necessarily given in an explicit way. It can be, for example, the convex hull of the integer solutions of a linear system. Then we are looking for sets T containing the W. Ben Ameur (B ) · J. Neto Institut National des Télécommunications, GET/INT CNRS/SAMOVAR, 9 rue Charles Fourier, 91011 Evry, France e-mail: walid.benameur@int-edu.eu J. Neto e-mail: neto@isima.fr 123