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
Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak$^*$ sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, … Read more
In this paper, we study the class of linear and discrete optimization problems in which the objective coefficients are chosen randomly from a distribution, and the goal is to evaluate robust bounds on the expected optimal value as well as the marginal distribution of the optimal solution. The set of joint distributions is assumed to … Read more
We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the … Read more
We consider generalizations of parity polytopes whose variables, in addition to a parity constraint, satisfy certain ordering constraints. More precisely, the variable domain is partitioned into k contiguous groups, and within each group, we require the variables to be sorted nonincreasingly. Such constraints are used to break symmetry after replacing an integer variable by a … Read more
We consider the maximum $k$-cut problem that involves partitioning the vertex set of a graph into $k$ subsets such that the sum of the weights of the edges joining vertices in different subsets is maximized. The associated semidefinite programming (SDP) relaxation is known to provide strong bounds, but it has a high computational cost. We … Read more
We present a heuristic approach for convex optimization problems containing sparsity constraints. The latter can be cardinality constraints, but our approach also covers more complex constraints on the support of the solution. For the special case that the support is required to belong to a matroid, we propose an exchange heuristic adapting the support in … Read more
The application of decision diagrams in combinatorial optimization has proliferated in the last decade. In recent years, authors have begun to investigate how to utilize not one, but a set of diagrams, to model constraints and objective function terms. Optimizing over a collection of decision diagrams, the problem we refer to as the consistent path … Read more
We investigate the NP-hard problem of computing the spark of a matrix (i.e., the smallest number of linearly dependent columns), a key parameter in compressed sensing and sparse signal recovery. To that end, we identify polynomially solvable special cases, gather upper and lower bounding procedures, and propose several exact (mixed-)integer programming models and linear programming … Read more
Experimental design is a classical statistics problem and its aim is to estimate an unknown m-dimensional vector from linear measurements where a Gaussian noise is introduced in each measurement. For the combinatorial experimental design problem, the goal is to pick k out of the given n experiments so as to make the most accurate estimate … Read more