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
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
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
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
This paper studies the possibilities of the Linear Matrix Inequality (LMI) characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real case analog, such studies were conducted in Sturm and Zhang in 2003. In this paper it is shown that stronger results can … Read more
In this paper we consider the following two types of complex quadratic maximization problems: (i) maximize $z^{\HH} Q z$, subject to $z_k^m=1$, $k=1,…,n$, where $Q$ is a Hermitian matrix with $\tr Q=0$ and $z\in \C^n$ is the decision vector; (ii) maximize $\re y^{\HH}Az$, subject to $y_k^m=1$, $k=1,…,p$, and $z_l^m=1$, $l=1,…,q$, where $A\in \C^{p\times q}$ and … Read more
In the paper a primal-infeasible interior point algorithm is proposed for linearly constrained convex programming. The starting point is any positive primal-infeasible dual-feasible point in a large region. The method maintains positivity of the iterates which point satisfies primal-infeasible dual-feasible point. At each iterates it requires to solve approximately a nonlinear system. It is shown … Read more