Juan C. Vera The maximum $k$-colorable subgraph (M$k$CS) problem is to find an induced $k$-colorable subgraph with maximum cardinality in a given graph. This paper is an in-depth analysis of the M$k$CS problem that considers various semidefinite programming relaxations including their theoretical and numerical comparisons. To simplify these relaxations we exploit the symmetry arising from permuting the colors, … Read more
We show the equivalence among the following three condition measures of a full column rank matrix $A$: the chi measure, the signed Hoffman constant, and the signed distance to ill-posedness. The latter two measures are constructed via suitable collections of matrices obtained by flipping the signs of some rows of $A$. Our results provide a … Read more
We give a characterization of the Hoffman constant of a system of linear constraints in $\R^n$ relative to a reference polyhedron $R\subseteq\R^n$. The reference polyhedron $R$ represents constraints that are easy to satisfy such as box constraints. In the special case $R = \R^n$, we obtain a novel characterization of the classical Hoffman constant. More … Read more
We propose a combinatorial algorithm to compute the Hoffman constant of a system of linear equations and inequalities. The algorithm is based on a characterization of the Hoffman constant as the largest of a finite canonical collection of easy-to-compute Hoffman constants. Our algorithm and characterization extend to the more general context where some of the … Read more
We consider the approximate minimization of a given polynomial on the standard simplex, obtained by taking the minimum value over all rational grid points with given denominator ${r} \in \mathbb{N}$. It was shown in [De Klerk, E., Laurent, M., Sun, Z.: An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric … Read more
Copositive programming is a relative young field which has evolved into a highly active research area in mathematical optimization. An important line of research is to use semidefinite programming to approximate conic programming over the copositive cone. Two major drawbacks of this approach are the rapid growth in size of the resulting semidefinite programs, and … Read more
Polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. There is a well stablished body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of … Read more
Certificates of non-negativity are fundamental tools in optimization. A “certificate” is generally understood as an expression that makes the non-negativity of the function in question evident. Some classical certificates of non-negativity are Farkas Lemma and the S-lemma. The lift-and-project procedure can be seen as a certificate of non-negativity for affine functions over the union of … 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