This article introduces a new retraction on the symplectic Stiefel manifold. The operation that requires the highest computational cost to compute the novel retraction is a matrix inversion of size $2p$–by–$2p$, which is much less expensive than those required for the available retractions in the literature. Later, with the new retraction, we design a constraint … Read more
Euclidean Jordan algebras are the abstract foundation for symmetriccone optimization. Every element in a Euclidean Jordan algebra has a complete spectral decomposition analogous to the spectral decomposition of a real symmetric matrix into rank-one projections. The spectral decomposition in a Euclidean Jordan algebra stems from the likewise-analogous characteristic polynomial of its elements, whose degree is … Read more
We present a fixed point approach to find the whole solution set of a set-valued optimization problem though a parametric problem, in which the height of the level set of the objective function is regarded as the parameter. First, the solution concept based on the vector approach is considered in this method. Then, we propose … Read more
The purpose of this technical report is to review the main properties of an accelerated composite gradient (ACG) method commonly referred to as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). In addition, we state a version of FISTA for solving both convex and strongly convex composite minimization problems and derive its iteration complexities to generate iterates … Read more
Linear complementarity problems (LCPs) are an important modeling tool for many practically relevant situations but also have many important applications in mathematics itself. Although the continuous version of the problem is extremely well studied, much less is known about mixed-integer LCPs (MILCPs) in which some variables have to be integer-valued in a solution. In particular, … Read more
A surprising result is presented in this paper with possible far reaching consequences for any optimization technique which relies on Krylov subspace methods employed to solve the underlying linear equation systems. In this paper the advantages of the new technique are illustrated in the context of Interior Point Methods (IPMs). When an iterative method is … Read more
A conic optimization problem is a problem involving a constraint that the optimization variable be in some closed convex cone. Prominent examples are second order cone programs (SOCP), semidefinite problems (SDP), and copositive problems. We survey recent progress made in this area. In particular, we highlight the connections between nonconvex quadratic problems, binary quadratic problems, … Read more
The share-of-choice product design (SOCPD) problem is to find the product, as defined by its attributes, that maximizes market share arising from a collection of customer types or segments. When customers follow a logit model of choice, the market share is given by a weighted sum of logistic probabilities, leading to the logit-based share-of-choice product … Read more
The COVID-19 pandemic has had an unprecedented impact on global health and the economy since its inception in December, 2019 in Wuhan, China. Non-pharmaceutical interventions (NPI) like lockdowns and curfews have been deployed by affected countries for controlling the spread of infections. In this paper, we develop a Mixed Integer Non-Linear Programming (MINLP) epidemic model … Read more
We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of … Read more