Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization. Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton’s method for a point satisfying the perturbed optimality conditions. These equations involve both the primal and dual … Read more
We propose a new version of potentially optimal intervals for the DIRECT algorithm. A two-points based sampling method is presented. The method starts from a distingished point (the peak point) by forming an initial triangle. The idea is to sample the midpoint of a specific interval: the basis of the resulting triangle. This specific interval … Read more
This paper is concerned with the optimization of continuum structures under dynamic loading using methods from topology design. The constraint functions are non-linear and implicit, their evaluation requires the resolution of a computation-intensive finite-element analysis performed by a black-box commercial structural mechanics software such as MSC/Nastran. We first present a brief overview of topology optimization … Read more
A new quasi-Newton scheme for updating a low rank positive semi-definite Hessian approximation is described, primarily for use in sequential quadratic programming methods for nonlinear programming. Where possible the symmetric rank one update formula is used, but when this is not possible a new rank two update is used, which is not in the Broyden … Read more
We consider 3-partitioning the vertices of a graph into sets $S_1, S_2$ and $S_3$ of specified cardinalities, such that the total weight of all edges joining $S_1$ and $S_2$ is minimized. This problem is closely related to several NP-hard problems like determining the bandwidth or finding a vertex separator in a graph. We show that … Read more
We present a simple, scaleable, distributed simplex implementation for large linear programs. It is designed for coarse grained computation, particularly, readily available networks of workstations. Scalability is achieved by using the standard form of the simplex rather than the revised method. Virtually all serious implementations are based on the revised method because it is much … Read more
In the recent years, there has been a considerable amount of work in the development of numerical methods for derivative free optimization problems. Some of this work relies on the management of the geometry of sets of sampling points for function evaluation and model building. In this paper, we continue the work developed in [Conn, … Read more
In this paper we propose a new class of test functions for unconstrained global optimization problems for which, however, it is a priori known that the global minimum lies in the interior of a sphere centered at the origin. The class depends on some parameters through which the difficulty of the test problems can be … Read more
We propose a unified theory that links uncertainty sets in robust optimization to risk measures in portfolio optimization. We illustrate the correspondence between uncertainty sets and some popular risk measures in finance, and show how robust optimization can be used to generalize the concepts of these measures. We also show that by using properly defined … Read more
The purpose of this paper is to derive computational checks for the simplex method of Linear Programming which can be applied at any iteration. The paper begins with a quick review of the simplex algorithm. It then goes through a theoretical development of the simplex method and its dual all the time focusing on the … Read more