## A combined model for chain expansion including the possibility of locating a new facility and modification and/or closing of existing facilities

The problem of an expanding chain (it already has some facilities) in a given area is considered. It may locate a new facility, or vary (up or down) the quality of its existing facilities, or close some of them, or a combination of all those possibilities, whatever it is the best to maximize its profit, … Read more

## A Survey on Bilevel Optimization Under Uncertainty

Bilevel optimization is a very active field of applied mathematics. The main reason is that bilevel optimization problems can serve as a powerful tool for modeling hierarchical decision making processes. This ability, however, also makes the resulting problems challenging to solve—both in theory and practice. Fortunately, there have been significant algorithmic advances in the field … Read more

## Small polygons with large area

A polygon is {\em small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small $n$-gon for this case. The improvement affects the $1/n^3$ … Read more

## Generalization of Doubly Nonnegative Cone: Focusing on Inner-Approximation for Generalized Copositive Cone

We aim to provide better relaxation for generalized completely positive (copositive) programming. We first develop an inner-approximation hierarchy for the generalized copositive cone over a symmetric cone. Exploiting this hierarchy as well as the existing hierarchy proposed by Zuluaga et al. (SIAM J Optim 16(4):1076–1091, 2006), we then propose two (NN and ZVP) generalized doubly … Read more

## Escaping Spurious Local Minima of Low-Rank Matrix Factorization Through Convex Lifting

This work proposes a rapid global solver for nonconvex low-rank matrix factorization (MF) problems that we name MF-Global. Through convex lifting steps, our method efficiently escapes saddle points and spurious local minima ubiquitous in noisy real-world data, and is guaranteed to always converge to the global optima. Moreover, the proposed approach adaptively adjusts the rank … Read more

## Improved RIP-Based Bounds for Guaranteed Performance of Two Compressed Sensing Algorithms

Iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP) are two mainstream compressed sensing algorithms using the hard thresholding operator. The guaranteed performance of the two algorithms for signal recovery was mainly analyzed in terms of the restricted isometry property (RIP) of sensing matrices. At present, the best known bound using RIP of order … Read more

## A Hessian inversion-free exact second order method for distributed consensus optimization

We consider a standard distributed consensus optimization problem where a set of agents connected over an undirected network minimize the sum of their individual (local) strongly convex costs. Alternating Direction Method of Multipliers (ADMM) and Proximal Method of Multipliers (PMM) have been proved to be effective frameworks for design of exact distributed second order methods … Read more

## An SDP Relaxation for the Sparse Integer Least Square Problem

In this paper, we study the 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 provide sufficient conditions for our SDP relaxation to solve SILS. The class of data input which guarantee … Read more

## A deterministic solver for multiobjective mixed-integer convex and nonconvex optimization

This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are … 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