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
In a Standard Quadratic Optimization Problem (StQP), a possibly indefinite quadratic form (the simplest nonlinear function) is extremized over the standard simplex, the simplest polytope. Despite this simplicity, the nonconvex instances of this problem class allow for remarkably rich patterns of coexisting local solutions, which are closely related to practical difficulties in solving StQPs globally. … Read more
We propose to solve global optimal control problems with a new algorithm that is based on Bock’s direct multiple shooting method. We provide conditions and numerical evidence for a significant overall runtime reduction compared to the standard single shooting approach. CitationOptimal Control Applications and Methods, Vol. 39 (2), pp. 449–470, 2017 DOI 10.1002/oca.2324 Article online … Read more
In this paper we exploit a slight variant of a result previously proved in [11] to define a procedure which delivers the convex envelope of some bivariate functions over polytopes. The procedure is based on the solution of a KKT system and simplifies the derivation of the convex envelope with respect to previously proposed techniques. … Read more
For binary polynomial optimization problems (POPs) of degree $d$ with $n$ variables, we prove that the $\lceil(n+d-1)/2\rceil$th semidefinite (SDP) relaxation in Lasserre’s hierarchy of the SDP relaxations provides the exact optimal value. If binary POPs involve only even-degree monomials, we show that it can be further reduced to $\lceil(n+d-2)/2\rceil$. This bound on the relaxation order … Read more
Traditional decomposition based branch-and-bound algorithms, like branch-and-price, can be very efficient if the duality gap is not too large. However, if this is not the case, the branch-and-bound tree may grow rapidly, preventing the method to find a good solution. In this paper, we present a new decompositon algorithm, called ADGO (Alternating Direction Global Optimization … Read more
In this article we introduce a stability concept for the coercivity of multivariate polynomials $f \in \mathbb{R}[x]$. In particular, we consider perturbations of $f$ by polynomials up to the so-called degree of stable coercivity, and we analyze this stability concept in terms of the corresponding Newton polytopes at infinity. For coercive polynomials $f \in \mathbb{R}[x]$ … 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
When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the … Read more