Copositive optimization is a special case of convex conic programming, and it optimizes a linear function over the cone of all completely positive matrices under linear constraints. Copositive optimization provides powerful relaxations of NP-hard quadratic problems or combinatorial problems, but there are still many open problems regarding copositive or completely positive matrices. In this paper, … Read more
In the High-Rank Matrix Completion (HRMC) problem, we are given a collection of n data points, arranged into columns of a matrix X, and each of the data points is observed only on a subset of its coordinates. The data points are assumed to be concentrated near a union of low-dimensional subspaces. The goal of … Read more
The Home Service Assignment, Routing, and Appointment scheduling (H-SARA) problem integrates the strategical fleet-sizing, tactical assignment, operational vehicle routing and scheduling subproblems at different decision levels, with a single period planning horizon and uncertainty (stochasticity) from the service duration, travel time, and customer cancellation rate. We propose a two-stage stochastic mixed-integer linear programming model for … Read more
In this paper, we consider an integrated vehicle routing and service scheduling problem for serving customers in distributed locations who need pick-up, drop-off or delivery services. We take into account the random trip time, non-negligible service time and possible customer cancellations, under which an ill-designed schedule may lead to undesirable vehicle idleness and customer waiting. … 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
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
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
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
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