A polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs
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Global Optimization Theory
Approximation hierarchies for copositive cone over symmetric cone and their comparison
We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a … Read more
Explicit convex hull description of bivariate quadratic sets with indicator variables
We consider the nonconvex set \(S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}\), which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we … Read more
Recursive McCormick Linearization of Multilinear Programs
Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLP). One example is the family of Recursive McCormick Linearization (RML) strategies, where bilinear products are substituted for artificial variables, which deliver a relaxation of the original problem when introduced together with concave and convex envelopes. In this article, we introduce … Read more
Generalizations of doubly nonnegative cones and their comparison
In this study, we examine the various extensions of the doubly nonnegative (DNN) cone, frequently used in completely positive programming (CPP) to achieve a tighter relaxation than the positive semidefinite cone. To provide tighter relaxation for generalized CPP (GCPP) than the positive semidefinite cone, inner-approximation hierarchies of the generalized copositive cone are exploited to obtain … Read more
An SDP Relaxation for the Sparse Integer Least Square Problem
In this paper, we study the polynomial approximability or solvability of sparse integer least square problem (SILS), which is the NP-hard variant of the least square problem, where we only consider sparse {0, ±1}-vectors. We propose an l1-based SDP relaxation to SILS, and introduce a randomized algorithm for SILS based on the SDP relaxation. In … Read more
Solving Two-Trust-Region Subproblems using Semidefinite Optimization with Eigenvector Branching
Semidefinite programming (SDP) problems typically utilize the constraint that X-xx’ is PSD to obtain a convex relaxation of the condition X=xx’, where x is an n-vector. In this paper we consider a new hyperplane branching method for SDP based on using an eigenvector of X-xx’. This branching technique is related to previous work of Saxeena, … Read more
Strong duality of a conic optimization problem with a single hyperplane and two cone constraints
Strong (Lagrangian) duality of general conic optimization problems (COPs) has long been studied and its profound and complicated results appear in different forms in a wide range of literatures. As a result, characterizing the known and unknown results can sometimes be difficult. The aim of this article is to provide a unified and geometric view … Read more
On Piecewise Linear Approximations of Bilinear Terms: Structural Comparison of Univariate and Bivariate Mixed-Integer Programming Formulations
Bilinear terms naturally appear in many optimization problems. Their inherent nonconvexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of … Read more
A Reformulation Technique to Solve Polynomial Optimization Problems with Separable Objective Functions of Bounded Integer Variables
Real-world problems are often nonconvex and involve integer variables, representing vexing challenges to be tackled using state-of-the-art solvers. We introduce a mathematical identity-based reformulation of a class of polynomial integer nonlinear optimization (PINLO) problems using a technique that linearizes polynomial functions of separable and bounded integer variables of any degree. We also introduce an alternative … Read more