Interior point methods (IPM) are the most popular approaches to solve Second Order Cone Optimization (SOCO) problems, due to their theoretical polynomial complexity and practical performance. In this paper, we present a warm-start method for primal-dual IPMs to reduce the number of IPM steps needed to solve SOCO problems that appear in a Branch and … Read more
Triggered by Burer’s seminal characterization from 2009, many copositive (CP) reformulations of mixed-binary QPs have been discussed by now. Most of them can be used as proper relaxations, if the intractable co(mpletely )positive cones are replaced by tractable approximations. While the widely used approximation hierarchies have the disadvantage to use positive-semidefinite (psd) matrices of orders … 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
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
Let K be a closed convex cone with dual K^* in a finite-dimensional real inner-product space V. The complementarity set of K is C(K) = { (x, s) in K × K^* | = 0 }. We say that a linear transformation L : V -> V is Lyapunov-like on K if = 0 for all (x, … Read more
We establish connections between the facial reduction algorithm of Borwein and Wolkowicz and the self-dual homogeneous model of Goldman and Tucker when applied to conic optimization problems. Specifically, we show the self-dual homogeneous model returns facial reduction certificates when it fails to return a primal-dual optimal solution or a certificate of infeasibility. Using this observation, … Read more
In this work we study the sets of two-party correlations generated from a Bell scenario involving two spatially separated systems with respect to various physical models. We show that the sets of classical, quantum, no-signaling and unrestricted correlations can be expressed as projections of affine sections of appropriate convex cones. As a by-product, we identify … Read more
Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a quadratic optimization problem with an $\ell_0$-norm penalty. Exactly enforcing the $\ell_0$-norm penalty is computationally intractable for larger scale problems, so different sparsity-inducing penalty … Read more
A framework is presented whereby a general convex conic optimization problem is transformed into an equivalent convex optimization problem whose only constraints are linear equations and whose objective function is Lipschitz continuous. Virtually any subgradient method can be applied to solve the equivalent problem. Two methods are analyzed. (In version 2, the development of algorithms … Read more
This paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization—a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way. We intend the paper for readers new to the area, and hence the exposition is largely self-contained. We focus on examples having just a few variables … Read more