On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems

We consider the semi-infinite system of polynomial inequalities of the form \[ \mathbf{K}:=\{x\in\mathbb{R}^m\mid p(x,y)\ge 0,\ \ \forall y\in S\subseteq\mathbb{R}^n\}, \] where $p(X,Y)$ is a real polynomial in the variables $X$ and the parameters $Y$, the index set $S$ is a basic semialgebraic set in $\mathbb{R}^n$, $-p(X,y)$ is convex in $X$ for every $y\in S$. We … Read more

The convex hull of a quadratic constraint over a polytope

A quadratically constrained quadratic program (QCQP) is an optimization problem in which the objective function is a quadratic function and the feasible region is defined by quadratic constraints. Solving non-convex QCQP to global optimality is a well-known NP-hard problem and a traditional approach is to use convex relaxations and branch-and-bound algorithms. This paper makes a … Read more

Generating irreducible copositive matrices using the stable set problem

In this paper it is considered how graphs can be used to generate copositive matrices, and necessary and sufficient conditions are given for these generated matrices to then be irreducible with respect to the set of positive semidefinite plus nonnegative matrices. This is done through combining the well known copositive formulation of the stable set … Read more

Volumetric barrier decomposition algorithms for two-stage stochastic linear semi-infinite programming

In this paper, we study the two-stage stochastic linear semi-infinite programming with recourse to handle uncertainty in data defining (deterministic) linear semi-infinite programming. We develop and analyze volumetric barrier decomposition-based interior point methods for solving this class of optimization problems, and present a complexity analysis of the proposed algorithms. We establish our convergence analysis by … Read more

A Linear Programming Based Approach to the Steiner Tree Problem with a Fixed Number of Terminals

We present a set of integer programs (IPs) for the Steiner tree problem with the property that the best solution obtained by solving all, provides an optimal Steiner tree. Each IP is polynomial in the size of the underlying graph and our main result is that the linear programming (LP) relaxation of each IP is … Read more

A New Face Method for Linear Programming

An attractive feature of the face method \cite{pan14} for solving LP problems is that it uses the orthogonal projection of the negative objective gradient on the related null space as the search direction. However, the method would not be amenable for solving large sparse problems, since it handles the involved normal system by orthogonal transformations. … Read more

A dual spectral projected gradient method for log-determinant semidefinite problems

We extend the result on the spectral projected gradient method by Birgin et al in 2000 to a log-determinant semidefinite problem (SDP) with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the … Read more

A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks

The computation of the Newton direction is the most time consuming step of interior-point methods. This direction was efficiently computed by a combination of Cholesky factorizations and conjugate gradients in a specialized interior-point method for block-angular structured problems. In this work we apply this algorithmic approach to solve very large instances of minimum cost flows … Read more

Compact Disjunctive Approximations to Nonconvex Quadratically Constrained Programs

Decades of advances in mixed-integer linear programming (MILP) and recent development in mixed-integer second-order-cone programming (MISOCP) have translated very mildly to progresses in global solving nonconvex mixed-integer quadratically constrained programs (MIQCP). In this paper we propose a new approach, namely Compact Disjunctive Approximation (CDA), to approximate nonconvex MIQCP to arbitrary precision by convex MIQCPs, which … Read more

A branch and price algorithm for the resource constrained home health care vehicle routing problem

We consider the vehicle routing problem with resource constraints motivated by a home health care application. We propose a branch and price algorithm to solve the problem. In our problem, we consider different types of patients that require a nurse or a health aid or both. The patients can be serviced by the appropriate vehicles … Read more