Deterministic annealing and relaxation labeling algorithms for classification and matching are presented and discussed. A new approach—self annealing—is introduced to bring deterministic annealing and relaxation labeling into accord. Self annealing results in an emergent linear schedule for winner-take-all and linear assignment problems. Self annihilation, a generalization of self annealing is ...

We study non-convex quadratic minimization problems under (possibly non-convex) quadratic and linear constraints, and characterize both Lagrangian and Semi-Lagrangian dual bounds in terms of conic optimization. While the Lagrangian dual is equivalent to the SDP relaxation (which has been known for quite a while, although the presented form, incorporating explicitly linear constraints, seems to ...

The choice network revenue management model incorporates customer purchase behavior as a function of the offered products, and is the appropriate model for airline and hotel network revenue management, dynamic sales of bundles, and dynamic assortment optimization. The optimization problem is a stochastic dynamic program and is intractable. A certainty-equivalence relaxation of the dynamic progr...

We present methods for constructing approximation algorithms to NP-hard optimization problems using linear programming (LP). We typically obtain an LP relaxation of the problem by formulating it as an integer linear program and dropping the integrality constraints. We then solve the LP relaxation exactly, and apply various rounding strategies (including randomized rounding and iterative roundin...

We consider two linear relaxations of the asymmetric traveling salesman problem (TSP), the Held-Karp relaxation of the TSP’s arc-based formulation, and a particular approximate linear programming (ALP) relaxation obtained by restricting the dual of the TSP’s shortest path formulation. We show that the two formulations produce equal lower bounds for the TSP’s optimal cost regardless of cost stru...

Branch and cut methods for integer programming problems solve a sequence of linear programming problems. Traditionally, these linear programming relaxations have been solved using the simplex method. The reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one. If the solution to a relaxation is fractional, additional constraints can b...

In this article we study the effectiveness of alternative integer programming formulations for area constrained harvest scheduling, known as the area restriction model (ARM). Empirical assessment of the effect of area constraints on the optimal objective value of these alternative approaches is carried out, focusing on the root Linear Programming relaxation. We also examine how this relates to ...

Metric facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. The capacity-constrained generalizations, such as capacitated facility location (Cfl) and lower-bounded facility location (Lbfl), have proved notorious as far as LP-based approximation is concerned: while there are local-search-based con...

Many inequalities for Mixed-Integer Linear Programs (MILPs) or pure Integer Linear Programs (ILPs) are derived from the Gomory corner relaxation, where all the nonbinding constraints at an optimal LP vertex are relaxed. Computational results show that the corner relaxation gives a good approximation of the integer hull for problems with general-integer variables, but the approximation is less s...

The choice network revenue management model incorporates customer purchase probabilities as a function of the offered products, and is the appropriate model for airline and hotel network revenue management, dynamic sales of bundles, and dynamic assortment optimization. The optimization problem is a stochastic dynamic program and is intractable; a linear programming approximation called choice d...