Identifying and exploiting classes of nonconvex constraints whose feasible region is convex after branching can reduce the time to compute global solutions for nonlinear optimization problems. We develop techniques for identifying quadratic and nonlinear constraints whose feasible region can be represented as the union of a finite number of second-order cones, and we provide necessary … Read more
We define the inclusion certificate as a measure that expresses each point in the domain of a collection of functions as a convex combination of other points in the domain. Using inclusion certificates, we extend the convex extensions theory to enable simultaneous convexification of functions. We discuss conditions under which the domain of the functions … Read more
We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let F denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on F is dominated by an alternative … Read more
We provide Completely Positive and Copositive Programming formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which are used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, … Read more
In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of … Read more
Let P be a pointed, polyhedral cone in R_n. In this paper, we study the cone C = cone{xx^T: x \in P} of quadratic forms. Understanding the structure of C is important for globally solving NP-hard quadratic programs over P. We establish key characteristics of C and construct a separation algorithm for C provided one … Read more
The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then … Read more
Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_\y$) on $R^n$ whose description depends on the parameter $y\in Y$. We assume that one can compute all moments of some probability measure $\varphi$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and … Read more
We show that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or spectral norm; determining a best … Read more
Facial reduction algorithm reduces the size of the positive semidefinite cone in SDP. The elimination method for a sparse SOS polynomial ([3]) removes unnecessary monomials for an SOS representation. In this paper, we establish a relationship between a facial reduction algorithm and the elimination method for a sparse SOS polynomial. CitationTechnical Report CS-09-02, Department of … Read more