Given a non-convex optimization problem, we study conditions under which every Karush-Kuhn-Tucker (KKT) point is a global optimizer. This property is known as KT-invexity and allows to identify the subset of problems where an interior point method always converges to a global optimizer. In this work, we provide necessary conditions for KT-invexity in n-dimensions and … Read more
Cutting planes are derived from specific problem structures, such as a single linear constraint from an integer program. This paper introduces cuts that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set $S\cap P$, where $S$ is a closed set, … Read more
Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of $[0,1]^n$. This … Read more
We consider the problem of packing congruent circles with the maximum radius in a unit square. As a mathematical program, this problem is a notoriously difficult nonconvex quadratically constrained optimization problem which possesses a large number of local optima. We study several convexification techniques for the circle packing problem, including polyhedral and semi-definite relaxations and … Read more
We study the simultaneous convexification of graphs of bilinear functions that contain bilinear products between variables x and y, where x belongs to a general polytope and y belongs to a simplex. We propose a constructive procedure to obtain a linear description of the convex hull of the resulting set. This procedure can be applied … Read more
Representability results play a fundamental role in optimization since they provide characterizations of the feasible sets that arise from optimization problems. In this paper we study the sets that appear in the feasibility version of mixed binary convex quadratic optimization problems. We provide a complete characterization of the sets that can be obtained as the … Read more
The Alternating Direction Method of Multipliers (ADMM) has been proved to be effective for solving separable convex optimization subject to linear constraints. In this paper, we propose a Generalized Symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two … Read more
Consider a multi-user multi-carrier communication system where multiple users share multiple discrete subcarriers. To achieve high spectrum efficiency, the users in the system must choose their transmit power dynamically in response to fast channel fluctuations. Assuming perfect channel state information, two formulations for the spectrum management (power control) problem are considered in this paper: the … Read more
We consider the Multilinear polytope defined as the convex hull of the set of binary points satisfying a collection of multilinear equations. Such sets are of fundamental importance in many types of mixed-integer nonlinear optimization problems, such as binary polynomial optimization. Utilizing an equivalent hypergraph representation, we study the facial structure of the Multilinear polytope … Read more
This paper proposes a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. … Read more