We study some mathematical programming formulations for the origin-destination model in airline revenue management. In particular, we focus on the traditional probabilistic model proposed in the literature. The approach we study consists of solving a sequence of two-stage stochastic programs with simple recourse, which can be viewed as an approximation to a multi- stage stochastic … Read more
OOPS is an object oriented parallel solver using the primal dual interior point methods. Its main component is an object-oriented linear algebra library designed to exploit nested block structure that is often present is truly large-scale optimization problems. This is achieved by treating the building blocks of the structured matrices as objects, that can use … Read more
In power law graphs, cut quality varies inversely with cut balance. Using some million node social graphs as a test bed, we empirically investigate this property and its implications for graph partitioning. We use six algorithms, including Metis and MQI (state of the art methods for finding bisections and quotient cuts) and four relaxation/rounding methods. … Read more
A point x is an approximate solution of a generalized equation [b lies in F(x)] if the distance from the point b to the set F(x) is small. Metric regularity of the set-valued mapping F means that, locally, a constant multiple of this distance bounds the distance from x to an exact solution. The smallest … Read more
We introduce a new approach to minimizing a function defined as the pointwise maximum over finitely many convex real functions (next referred to as the “component functions”), with the aim of working on the basis of “incomplete knowledge” of the objective function. In fact, a descent algorithm is proposed which does not necessarily require at … Read more
The Netlib library of linear programming problems is a well known suite containing many real world applications. Recently it was shown by Ordonez and Freund that 71% of these problems are ill-conditioned. Hence, numerical difficulties may occur. Here, we present rigorous results for this library that are computed by a verification method using interval arithmetic. … Read more
We prove local and global convergence results for an interior-point method applied to mathematical programs with equilibrium constraints. The global result shows the algorithm minimizes infeasibility regardless of starting point, while one result proves local convergence when penalty functions are exact; another local result proves convergence when the solution is not even a KKT point. … Read more
This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. In contrast with classical approaches, the choice of the penalty parameter ceases to be … Read more
APPSPACK is software for solving unconstrained and bound constrained optimization problems. It implements an asynchronous parallel pattern search method that has been specifically designed for problems characterized by expensive function evaluations. Using APPSPACK to solve optimization problems has several advantages: No derivative information is needed; the procedure for evaluating the objective function can be executed … Read more
Let B_i be deterministic symmetric m\times m matrices, and \xi_i be independent random scalars with zero mean and “of order of one” (e.g., \xi_i are Gaussian with zero mean and unit standard deviation). We are interested in conditions for the “typical norm” of the random matrix S_N = \xi_1B_1+…+\xi_NB_N to be of order of 1. … Read more