This paper presents a new method for solving global optimization problems. We use a local technique based on the notion of discrete gradients for finding a cone of descent directions and then we use a global cutting angle algorithm for finding global minimum within the intersection of the cone and the feasible region. We present … Read more
In this paper, we consider the nonconvex minimization problem of the value-at-risk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined … Read more
The natural approach for describing network flow problems is to introduce side constraints that capture restrictions of a logical or technological nature, e.g., capacity constraints. We study the traffic equilibria arising from selfish routing of individual users in networks with side constraints. We examine first the case of linear latency functions. Under very general assumptions … Read more
The prescribed goals of radiation treatment planning are often expressed in terms of dose-volume constraints. We present a novel formulation of a dose-volume constraint satisfaction search for the discretized radiation therapy model. This approach does not rely on any explicit cost function. The inverse treatment planning uses the aperture based approach with predefined, according to … Read more
We describe a complete solution of the linear-quaratic control problem with the linear term in the objective function on a semiinfinite interval. This problem has important applications to calculation of Nesterov-Todd and other primal-dual directions in infinite-dimensional setting. CitationTechnical report, University of Notre Dame, December, 2003ArticleDownload View PDF
Three-dimensional electron microscopy (3D-EM) is a powerful tool for visualizing complex biological systems. As any other imaging device, the electron microscope introduces a transfer function (called in this field the Contrast Transfer Function, CTF) into the image acquisition process that modulates the various frequencies of the signal. Thus, 3D reconstructions performed with these CTF-affected projections … Read more
In the context of SQP methods or, more recently, of sequential semidefinite programming methods, it is common practice to construct a positive semidefinite approximation of the Hessian of the Lagrangian. The Hessian of the augmented Lagrangian is a suitable approximation as it maintains local superlinear convergence under appropriate assumptions. In this note we give a … Read more
In the context of integer programming, we develop a polyhedral method for linearizing a product of a pair of real linear functions in 0/1 variables. As an example, by writing a pair of integer variables in binary expansion, we have a technique for linearizing their product. We give a complete linear description for the resulting … Read more
Starting from known necessary extremality conditions in terms of strict subdifferentials and normals the notion of weak stationarity is introduced. It is defined in terms of initial space elements. The necessary conditions become necessary and sufficient (for stationarity). CitationSchool of Information Technology and Mathematical Sciences, Centre of Information and Applied Optimization, University of Ballarat, POB … Read more
Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by … Read more