## PENNON – A Code for Convex Nonlinear and Semidefinite Programming

We introduce a computer program PENNON for the solution of problems of convex Nonlinear and Semidefinite Programming (NLP-SDP). The algorithm used in PENNON is a generalized version of the Augmented Lagrangian method, originally introduced by Ben-Tal and Zibulevsky for convex NLP problems. We present generalization of this algorithm to convex NLP-SDP problems, as implemented in … Read more

## A primal-dual symmetric relaxation for homogeneous conic systems

We address the feasibility of the pair of alternative conic systems of constraints Ax = 0, x \in C, and -A^T y \in C^*, defined by an m by n matrix A and a cone C in the n-dimensional Euclidean space. We reformulate this pair of conic systems as a primal-dual pair of conic programs. … Read more

## A new class of potential affine algorithms for linear convex programming

We obtain a new class of primal affine algorithms for the linearly constrained convex programming. It is constructed from a family of metrics generated the r power, r>=1, of the diagonal iterate vector matrix. We obtain the so called weak convergence. That class contains, as particular cases, the multiplicative Eggermont algorithm for the minimization of … Read more

We study the mesh topology design problem in overlay virtual private networks. Given a set of customer nodes and associated traffic matrix, tunnels that are connecting node pairs through a public service provider network subject to degree constraints are determined so as to minimize total multihopped traffic. Valid inequalities strengthening the LP relaxation and a … Read more

## Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S.~Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of objective quadratic functions and diagonal coefficient matrices … Read more

## Computing All Nonsingular Solutions of Cyclic-n Polynomial Using Polyhedral Homotopy Continuation Methods

All isolated solutions of the cyclic-n polynomial equations are not known for larger dimensions than 11. We exploit two types of symmetric structures in the cyclic-n polynomial to compute all isolated nonsingular solutions of the equations efficiently by the polyhedral homotopy continuation method and to verify the correctness of the generated approximate solutions. Numerical results … Read more

## Solving Stability Problems on a Superclass of Interval Graphs

We introduce a graph invariant, called thinness, and show that a maximum weighted stable set on a graph $G(V, E)$ with thinness $k$ may be found in $O(\frac{|V|}{k})^k$-time, if a certain representation is given. We show that a graph has thinness 1 if and only if it is an interval graph, while a graph with … Read more

## Relations between divergence of multipliers and convergence to infeasible points in primal-dual interior methods for nonconvex nonlinear programming

Recently, infeasibility issues in interior methods for nonconvex nonlinear programming have been studied. In particular, it has been shown how many line-search interior methods may converge to an infeasible point which is on the boundary of the feasible region with respect to the inequality constraints. The convergence is such that the search direction does not … Read more

## A characterization of the distance to infeasibility under block-structured perturbations

We discuss several generalizations of the classical Eckart and Young identity. We show that a natural extension of this identity holds for rectangular matrices defining conic systems of constraints, and for perturbations restricted to a particular block structure, such as those determined by a sparsity pattern. Our results extend and unify the classical Eckart and … Read more

## A binary LP model to the facility layout problem

In facility layout problems, a major concern is the optimal design or remodeling of the facilities of an organization. The decision-maker’s objective is to arrange the facility in an optimal way, so that the interaction among functions (i.e. machines, inventories, persons) and places (i.e. offices, work locations, depots) is efficient. A simple pure-binary LP model … Read more