The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges … Read more
In 1961 Leon Steinberg formulated a “backboard wiring” problem that has resisted solution for 40 years. Steinberg’s wiring problem is to determine the locations of 34 computer components on a 4 by 9 grid so as to minimize the total length of the wiring required to interconnect them. The problem is an example of a … Read more
We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov\’asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem. This relaxation $Q_t(G)$ can be defined as the projection on … Read more
A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous … Read more
Inspired by an algorithm due to Minieka, we develop a simple and yet very efficient exact algorithm for the problem of locating p facilities and assigning clients to them in order to minimize the maximum distance between a client and the facility it is assigned to. After a lower bounding phase, the algorithm iteratively sets … Read more
The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. … Read more
Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the … Read more
The Cardinality Constrained Circuit Problem (CCCP) is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most $k$ edges. The CCCP is NP-Hard. We present classes of facet-inducing inequalities for the convex hull of feasible circuits. ArticleDownload View PDF
Sherali and Adams \cite{SA90}, Lov\’asz and Schrijver \cite{LS91} and, recently, Lasserre \cite{Las01b} have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a $0-1$ polytope $P\subseteq \oR^n$ converging to $P$ in $n$ steps. Lasserre’s approach uses results about representations of positive polynomials as sums of squares and the dual … Read more
In this paper we study special barrier functions for the convex cones, which are the sum of a self-concordant barrier for the cone and a positive-semidefinite quadratic form. We show that the central path of these augmented barrier functions can be traced with linear speed. We also study the complexity of finding the analytic center … Read more