## Nonexpansive Markov Operators and Random Function Iterations for Stochastic Fixed Point Problems

We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant mea- sure the stochastic fixed point problem. This generalizes earlier work studying the stochastic feasibility problem, namely, to find points that are, with probability 1, fixed points of … Read more

## A Radial Basis Function Method for Noisy Global Optimisation

We present a novel response surface method for global optimisation of an expensive and noisy (black-box) objective function, where error bounds on the deviation of the observed noisy function values from their true counterparts are available. The method is based on the well-established RBF method by Gutmann (2001a,c) for minimising an expensive and deterministic objective … Read more

## A Slightly Lifted Convex Relaxation for Nonconvex Quadratic Programming with Ball Constraints

 Globally optimizing a nonconvex quadratic over the intersection of $$m$$ balls in $$\mathbb{R}^n$$ is known to be polynomial-time solvable for fixed $$m$$. Moreover, when $$m=1$$, the standard semidefinite relaxation is exact, and when $$m=2$$, it has recently been shown that an exact relaxation can be constructed via a disjunctive semidefinite formulation based on essentially two copies of the $$m=1$$ case. … Read more

## The multilinear polytope of beta-acyclic hypergraphs has polynomial extension complexity

Article Download View The multilinear polytope of beta-acyclic hypergraphs has polynomial extension complexity

## 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, … 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

Our objective was to generalize the well-known doubly nonnegative (DNN) cone, and compare it with a generalized cone proposed by Burer and Dong in 2012. Several inner-approximation hierarchies for the generalized copositive cone over a closed cone have been proposed as generalizations of that proposed by Parrilo in 2000. By exploiting these hierarchies, we generalized … 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