We present a heuristic method for general 0-1 mixed integer programming, intended for eventual incorporation into parallel branch-and-bound methods for solving such problems exactly. The core of the heuristic is a rounding method based on simplex pivots, employing only gradient information, for a strictly concave, differentiable merit function measuring integer feasibility. When local minima of … Read more
This paper presents the results of the application of a non-differentiable optimization method in connection with the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW is an extension of the Vehicle Routing Problem. In the VRPTW the service at each customer must start within an associated time window. The Shortest Path decomposition of the … Read more
The $p$-Center problem consists in locating $p$ facilities among a set of $M$ possible locations and assigning $N$ clients to them in order to minimize the maximum distance between a client and the facility to which he is allocated. We present a new integer linear programming formulation for this Min-Max problem with a polynomial number … Read more
The classical non-linear mixed integer formulation of the transmission network expansion problem cannot guarantee finding the optimal solution due to its non-convex nature. We propose an alternative mixed integer linear disjunctive formulation, which has better conditioning properties than the standard disjunctive model. The mixed integer program is solved by a commercial Branch and Bound code, … Read more
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