The Lagrangian-doubly nonnegative (DNN) relaxation has recently been shown to provide effective lower bounds for a large class of nonconvex quadratic optimization problems (QOPs) using the bisection method combined with first-order methods by Kim, Kojima and Toh in 2016. While the bisection method has demonstrated the computational efficiency, determining the validity of a computed lower … Read more
We examine the convexity and tractability of the two-sided linear chance constraint model under Gaussian uncertainty. We show that these constraints can be applied directly to model a larger class of nonlinear chance constraints as well as provide a reasonable approximation for a challenging class of quadratic chance constraints of direct interest for applications in … Read more
Facial reduction heuristics are developed in the interest of added performance and reliability in methods for mixed-integer conic optimization. Specifically, the process of branch-and-bound is shown to spawn subproblems for which the conic relaxations are difficult to solve, and the objective bounds of linear relaxations are arbitrarily weak. While facial reduction algorithms already exist to … Read more
We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in … Read more
A “facial reduction”-like regularization algorithm is established for conic optimization problems by relaxing requirements on the reduction certificates. It requires only a linear number of reduction certificates from a constant time-solvable auxiliary problem, but is challenged by representational issues of the exposed reductions. A condition for representability is presented, analyzed for Cartesian product cones, and … 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
We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order $2n$ which … Read more
Let $\F$ be a quadratically constrained, possibly nonconvex, bounded set, and let $\E_1, \ldots, \E_l$ denote ellipsoids contained in $\F$ with non-intersecting interiors. We prove that minimizing an arbitrary quadratic $q(\cdot)$ over $\G := \F \setminus \cup_{k=1}^\ell \myint(\E_k)$ is no more difficult than minimizing $q(\cdot)$ over $\F$ in the following sense: if a given semidefinite-programming … Read more
The authors in a previous paper devised certain subcones of the semidefinite plus nonnegative cone and showed that satisfaction of the requirements for membership of those subcones can be detected by solving linear optimization problems (LPs) with $O(n)$ variables and $O(n^2)$ constraints. They also devised LP-based algorithms for testing copositivity using the subcones. In this … Read more
A framework is developed for applying accelerated methods to general hyperbolic programming, including linear, second-order cone, and semidefinite programming as special cases. The approach replaces a hyperbolic program with a convex optimization problem whose smooth objective function is explicit, and for which the only constraints are linear equations (one more linear equation than for the … Read more