Hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small size. In this paper, we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs. When used iteratively, this scheme improves … Read more
This paper presents a backward stable preprocessing technique for (nearly) ill-posed semidefinite programming, SDP, problems, i.e.,~programs for which Slater’s constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on \emph{primal-dual interior-point, p-d i-p} methods. These algorithms require Slater’s constraint qualification for both the primal and dual problems. This … Read more
We present different robust frameworks using probabilistic ambiguity descriptions of the input data in the least squares problems. The three probability ambiguity descriptions are given by: (1) confidence interval over the first two moments; (2) bounds on the probability measure with moments constraints; (3) confidence interval over the probability measure by using the Kantorovich probability … Read more
Quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In … Read more
Continuous linear programs have attracted considerable interest due to their potential for modelling manufacturing, scheduling and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (non-separated) problem instances. In this paper we propose a more generic approximation scheme for … Read more
Matrix rank minimization problems are gaining a plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization … Read more
The convex hull relaxation (CHR) method (Albornoz 1998, Ahlatçıoğlu 2007, Ahlatçıoğlu and Guignard 2010) provides lower bounds and feasible solutions (thus upper bounds) on convex 0-1 nonlinear programming problems with linear constraints. In the quadratic case, these bounds may often be improved by a preprocessing step that adds to the quadratic objective function terms which … Read more
In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees … Read more
Channel-optimized index assignment of source codewords is arguably the simplest way of improving transmission error resilience, while keeping the source and/or channel codes intact. But optimal design of index assignment is an in- stance of quadratic assignment problem (QAP), one of the hardest optimization problems in the NP-complete class. In this paper we make a … Read more
In this paper, with the help of convex-like function, we discuss the duality theory for nonconvex semidefinite programming. Our contributions are: duality theory for the general nonconvex semidefinite programming when Slater’s condition holds; perfect duality for a special case of the nonconvex semidefinite programming for which Slater’s condition fails. We point out that the results … Read more