Akaike’s information criterion (AIC) is a measure of the quality of a statistical model for a given set of data. We can determine the best statistical model for a particular data set by the minimization of the AIC. Since we need to evaluate exponentially many candidates of the model by the minimization of the AIC, … Read more
Representability results for mixed-integer linear systems play a fundamental role in optimization since they give geometric characterizations of the feasible sets that can be formulated by mixed-integer linear programming. We consider a natural extension of mixed-integer linear systems obtained by adding just one ellipsoidal inequality. The set of points that can be described, possibly using … Read more
In this paper, we consider the Multilinear set defined as the set of binary points satisfying a collection of multilinear equations. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is … Read more
Concave Mixed-Integer Quadratic Programming is the problem of minimizing a concave quadratic polynomial over the mixed-integer points in a polyhedral region. In this work we describe an algorithm that finds an ε-approximate solution to a Concave Mixed-Integer Quadratic Programming problem. The running time of the proposed algorithm is polynomial in the size of the problem … Read more
We introduce a new technique to generate good feasible points of mixed-integer nonlinear optimization problems. It makes use of the so-called inner parallel set of the relaxed feasible set, which was employed in O. Stein, Error bounds for mixed integer linear optimization problems, Mathematical Programming, Vol. 156 (2016), 101-123, as well as O. Stein, Error … Read more
This paper describes the extensions that were added to the constraint integer programming framework SCIP in order to enable it to solve convex and nonconvex mixed-integer nonlinear programs (MINLPs) to global optimality. SCIP implements a spatial branch-and-bound algorithm based on a linear outer-approximation, which is computed by convex over- and underestimation of nonconvex functions. An … Read more
We consider a challenging resource allocation problem arising in mobile wireless communications. The goal is to allocate the available channels and power in a so-called OFDMA system, in order to maximise the transmission rate, subject to quality of service (QoS) constraints. Standard MINLP software struggled to solve even small instances of this problem. Using outer … Read more
In this paper we analyze general two-term disjunctions on a regular cone $\K$ and derive a general form for a family of convex inequalities which are valid for the resulting nonconvex sets. Under mild technical assumptions, these inequalities collectively describe the closed convex hulls of these disjunctions, and if additional conditions are satisfied, a single … Read more
Feasibility pumps are highly effective primal heuristics for mixed-integer linear and nonlinear optimization. However, despite their success in practice there are only few works considering their theoretical properties. We show that feasibility pumps can be seen as alternating direction methods applied to special reformulations of the original problem, inheriting the convergence theory of these methods. … Read more
One of the fundamental tasks in compressed sensing is finding the sparsest solution to an underdetermined system of linear equations. It is well known that although this problem is NP-hard, under certain conditions it can be solved by using a linear program which minimizes the 1-norm. The restricted isometry property has been one of the … Read more