We study the classical 0-1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be … Read more
We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1 polynomial programming). For polyhedral relaxations of such problems it is generally not true that variables integer in the relaxed solution will retain the same values in … Read more
In this paper we address the Adjacent Quadratic Assignment Problem (AQAP) which is a variant of the QAP where the cost coefficient matrix has a particular structure. Motivated by strong lower bounds obtained by applying Reformulation Linearization Technique (RLT) to the classical QAP, we propose two special RLT representations for the problem. The first is … Read more
Motivated by concave cost combinatorial optimization problems, we study the following mixed integer nonlinear set: P = {(w,x) : w >= f(a’x), e’x
When a weighted graph is an instance of the Distance Geometry Problem (DGP), certain types of vertex orders (called discretization orders) allow the use of a very efficient, precise and robust discrete search algorithm (called Branch-and-Prune). Accordingly, finding such orders is critically important in order to solve DGPs in practice. We discuss three types of … Read more
This paper provides a study of integer linear programming formulations for the diameter constrained spanning tree problem (DMSTP) in the natural space of edge design variables. After presenting a straightforward model based on the well known jump inequalities a new stronger family of circular-jump inequalities is introduced. These inequalities are further generalized in two ways: … Read more
We study the discrete optimization problem under the distributionally robust framework. We optimize the Entropic Value-at-Risk, which is a coherent risk measure and is also known as Bernstein approximation for the chance constraint. We propose an efficient approximation algorithm to resolve the problem via solving a sequence of nominal problems. The computational results show that … Read more
A reformulation of quadratically constrained binary programs as duals of set-copositive linear optimization problems is derived using either \(\{0,1\}\)-formulations or \(\{-1,1\}\)-formulations. The latter representation allows an extension of the randomization technique by Goemans and Williamson. An application to the max-clique problem shows that the max-clique problem is equivalent to a linear program over the max-cut … Read more
In the \emph{incremental knapsack problem} ($\IK$), we are given a knapsack whose capacity grows weakly as a function of time. There is a time horizon of $T$ periods and the capacity of the knapsack is $B_t$ in period $t$ for $t = 1, \ldots, T$. We are also given a set $S$ of $N$ items … Read more
Typical output from an optimization solver is a single optimal solution. At the same time, a set of high-quality and diverse solutions could be beneficial in a variety of contexts, for example problems involving imperfect information, or those for which the structure of high-quality solution vectors can reveal meaningful insights. In view of this, we … Read more