We give an explicit characterization of all extreme rays of the cone of 5×5 copositive matrices. The results are based on the work of Baumert [L. D. Baumert, “Extreme copositive quadratic forms”, PhD thesis, 1965], where an implicit characterization was given. We show that the class of extreme rays found by Baumert forms a 10-dimensional … Read more
There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called … Read more
The classical trust-region subproblem (TRS) minimizes a nonconvex quadratic objective over the unit ball. In this paper, we consider extensions of TRS having extra constraints. When two parallel cuts are added to TRS, we show that the resulting nonconvex problem has an exact representation as a semidefinite program with additional linear and second-order-cone constraints. For … Read more
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
We consider an extension of ordinary linear programming (LP) that adds weighted logarithmic barrier terms for some variables. The resulting problem generalizes both LP and the problem of finding the weighted analytic center of a polytope. We show that the problem has a dual of the same form and give complexity results for several different … 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
The paper proposes a polynomial algorithm for solving systems of linear inequalities. The algorithm uses a polynomial relaxation-type procedure which either finds a solution for Ax = b, 0
In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the dual simplex method with the most negative pivoting rule for LP. The bound is comparable with the bound given by Kitahara and Mizuno (2010) for the primal simplex method. We apply the result to the … Read more