Reformulation and Sampling to Solve a Stochastic Network Interdiction Problem

The Network Interdiction Problem involves interrupting an adversary’s ability to maximize flow through a capacitated network by destroying portions of the network. A budget constraint limits the amount of the network that can be destroyed. In this paper, we study a stochastic version of the network interdiction problem in which the successful destruction of an … Read more

Stochastic Programming Approach to Optimization under Uncertainty

In this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems. We argue that two stage (say linear) stochastic programming problems can be solved with a reasonable accuracy by Monte Carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the … Read more

Clique-based facets for the precedence constrained knapsack problem

We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in management and machine scheduling, and also appears as a subproblem in decomposition techniques for network design and other related problems. We present a new approach for determining facets of … Read more

Continuous Optimization Methods for Structure Alignments

Structural Alignment is an important tool for fold identification of proteins, structural screening on ligand databases, pharmacophore identification and other applications. In the general case, the optimization problem of superimposing two structures is nonsmooth and nonconvex, so that most popular methods are heuristic and do not employ derivative information. Usually, these methods do not admit … Read more

Single-Product Pricing via Robust Optimization

We present a robust optimization approach to the problem of pricing a capacitated product over a finite time horizon in the presence of demand uncertainty. This technique does not require the knowledge of the underlying probability distributions, which in practice are difficult to estimate accurately, and instead models random variables as uncertain parameters belonging to … Read more

A pricing problem under Monge property

We study a pricing problem where buyers with non-uniform demand purchase one of many items. Each buyer has a known benefit for each item and purchases the item that gives the largest utility, which is defined to be the difference between the benefit and the price of the item. The optimization problem is to decide … Read more

Dynamic Enumeration of All Mixed Cells

The polyhedral homotopy method, which has been known as a powerful numerical method for computing all isolated zeros of a polynomial system, requires all mixed cells of the support of the system to construct a family of homotopy functions. Finding the mixed cells is formulated in terms of a linear inequality system with an additional … Read more

On the complexity of optimization over the standard simplex

We review complexity results for minimizing polynomials over the standard simplex and unit hypercube. In addition, we show that there exists a polynomial time approximation scheme (PTAS) for minimizing some classes of functions (including Lipschitz continuous functions) over the standard simplex. The main tools used in the analysis are Bernstein approximation and Lagrange interpolation on … Read more

The Rate of Convergence of the Augmented Lagrangian Method for Nonlinear Semidefinite Programming

We analyze the rate of local convergence of the augmented Lagrangian method for nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and certain variational analysis on the projection operator in the symmetric-matrix space. Without … Read more