We recall the use of squared slacks used to transform inequality constraints into equalities and several reasons why their introduction may be harmful in many algorithmic frameworks routinely used in nonlinear programming. Numerical examples performed with the sequential quadratic programming method illustrate those reasons. Citation Cahier du GERAD G-2007-62, Aug. 2007 Article Download View The … Read more
The transportation paradox is related to the classical transportation problem. For certain instances of this problem an increase in the amount of goods to be transported may lead to a decrease in the optimal total transportation cost. Even though the paradox has been known since the early days of linear programming, it has got very … Read more
Traditionally, optimality and uniqueness of an (n,N,t) spherical code is proved using linear programming bounds. However, this approach does not apply to the parameter (4,10,1/6). We use semidefinite programming bounds instead to show that the Petersen code (which are the vertices of the 4-dimensional second hypersimplex or the midpoints of the edges of the regular … Read more
We study cooperative games with supermodular costs. We show that supermodular costs arise in a variety of situations: in particular, we show that the problem of minimizing a linear function over a supermodular polyhedron–a problem that often arises in combinatorial optimization–has supermodular optimal costs. In addition, we examine the computational complexity of the least core … Read more
We consider the classical problem of estimating a density on $[0,1]$ via some maximum entropy criterion. For solving this convex optimization problem with algorithms using first-order or second-order methods, at each iteration one has to compute (or at least approximate) moments of some measure with a density on $[0,1]$, to obtain gradient and Hessian data. … Read more
In the paper, we focus primarily on the problem of recovering a linear form g’*x of unknown “signal” x known to belong to a given convex compact set X in R^n from N independent realizations of a random variable taking values in a finite set, the distribution p of the variable being affinely parameterized by … Read more
In engineering design, an optimized solution often turns out to be suboptimal, when implementation errors are encountered. While the theory of robust convex optimization has taken significant strides over the past decade, all approaches fail if the underlying cost function is not explicitly given; it is even worse if the cost function is nonconvex. In … Read more
We present a general metaheuristic for joint chance-constrained stochastic programs with discretely distributed parameters. We give a reformulation of the problem that allows us to define a finite solution space. We then formulate a novel neighborhood for the problem and give methods for efficiently searching this neighborhood for solutions that are likely to be improving. … Read more
In this paper, we propose lifting techniques for generating strong cuts for nonlinear programs that are globally-valid. The theory is geometric and provides intuition into lifting-based cut generation procedures. As a special case, we find short proofs of earlier results on lifting techniques for mixed-integer programs. Using convex extensions, we obtain conditions that allow sequence-independent … Read more
We present a strong duality theory for optimization problems over symmetric cones without assuming any constraint qualification. We show important complexity implications of the result to semidefinite and second order conic optimization. The result is an application of Borwein and Wolkowicz’s facial reduction procedure to express the minimal cone. We use Pataki’s simplified analysis and … Read more