A Lagrangian relaxation approach to the edge-weighted clique problem

The $b$-clique polytope $CP^n_b$ is the convex hull of the node and edge incidence vectors of all subcliques of size at most $b$ of a complete graph on $n$ nodes. Including the Boolean quadric polytope $QP^n$ as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over $QP^n_n$. The problem of optimizing linear functions over $CP^n_b$ has so far been approached via heuristic combinatorial algorithms and cutting-plane methods. We study the structure of $CP^n_b$ in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented. \u

The Steiner Linear Ordering Problem: Application to resource-constrained scheduling problems

When examined through polyhedral study, the resource-constrained scheduling problems have always dealt with processes which have the same priority. With the Steiner Linear Ordering problem, we can address systems where the elements involved have different levels of priority, either high or low. This allows us greater flexibility in modeling different resource-constrained scheduling problems. In this paper, we address both the linear ordering problem and its application to scheduling problems, and provide a polyhedral study of the associated polytopes

Tight Polyhedral Representations of Discrete Sets Using Projections, Simplices, and Base-2 Expansions

This research effort focuses on the acquisition of polyhedral outer-approximations to the convex hull of feasible solutions for mixed-integer linear and mixed-integer nonlinear programs. The goal is to produce desirable formulations that have superior size and/or relaxation strength. These two qualities often have great influence on the success of underlying solution strategies, and so it is with these qualities in mind that the work of this dissertation presents three distinct contributions. The first studies a family of relatively unknown polytopes that enable the linearization of polynomial expressions involving two discrete variables. Projections of higher-dimensional convex hulls are employed to reduce the dimensionality of the requisite linearizing polyhedra. For certain lower dimensions, a complete characterization of the convex hull is obtained; for others, a family of facets is acquired. Furthermore, a novel linearization for the product of a bounded continuous variable and a general discrete variable is obtained. The second contribution investigates the use of simplicial facets in the formation of novel convex hull representations for a class of mixed-discrete problems having a subset of their variables taking on discrete, affinely independent realizations. These simplicial facets provide new theoretical machinery necessary to extend the reformulation-linearization technique (RLT) for mixed-binary and mixed-discrete programs. In doing so, new insight is provided which allows for the subsumation of previous mixed-binary and mixed-discrete RLT results. The third contribution presents a novel approach for representing functions of discrete variables and their products using logarithmic numbers of 0-1 variables in order to economize on the number of these binary variables. Here, base-2 expansions are used within linear restrictions to enforce the appropriate behavior of functions of discrete variables. Products amongst functions are handled by scaling these linear restrictions. This approach provides insight into, improves upon, and subsumes recent related linearization methods from the literature