This paper presents and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance. Roughly, a filter method accepts a step that either improves the objective function value or the value of some function that measures the constraint violation. The new algorithm does not compute or approximate any derivatives, … Read more
Recently, general definitions of pattern search methods for both unconstrained and linearly constrained optimization were presented. It was shown under mild conditions, that there exists a subsequence of iterates converging to a stationary point. In the unconstrained case, stronger results are derived under additional assumptions. In this paper, we present three small dimensioned examples showing … Read more
Many engineering optimization problems involve a special kind of discrete variable that {\em can} be represented by a number, but this representation has no significance. Such variables arise when a decision involves some situation like a choice from an unordered list of categories. This has two implications: The standard approach of solving problems with continuous … Read more
In the literature, thermal insulation systems with a fixed number of heat intercepts have been optimized with respect to intercept locations and temperatures. The number of intercepts and the types of insulators that surround them were chosen by parametric studies. This was because the optimization methods used could not treat such categorical variables. Discrete optimization … Read more
A series of numerical experiments with interior point (LOQO, KNITRO) and active-set SQP codes (SNOPT, filterSQP) are reported and analyzed. The tests were performed with small, medium-size and moderately large problems, and are examined by problem classes. Detailed observations on the performance of the codes, and several suggestions on how to improve them are presented. … Read more
We consider numerical methods for finding (weak) second-order critical points for large-scale non-convex quadratic programming problems. We describe two new methods. The first is of the active-set variety. Although convergent from any starting point, it is intended primarily for the case where a good estimate of the optimal active set can be predicted. The second … Read more
The following is a list of all of the published and unpublished works on quadratic programming that we are aware of. Some are general references to background material, while others are central to the development of the quadratic programming methods and to the applications we intend to cover in our evolving book on the subject. … Read more
A slack-based feasible interior point method is described which can be derived as a modification of infeasible methods. The modification is minor for most line search methods, but trust region methods require special attention. It is shown how the Cauchy point, which is often computed in trust region methods, must be modified so that the … Read more
This paper introduces and analyses a new algorithm for minimizing a convex function subject to a finite number of convex inequality constraints. It is assumed that the Lagrangian of the problem is strongly convex. The algorithm combines interior point methods for dealing with the inequality constraints and quasi-Newton techniques for accelerating the convergence. Feasibility of … Read more
A reduction in the computational work is possible if we do not require that the nonzeros of a Jacobian matrix be determined directly. If a column or row partition is available, the proposed substitution technique can be used to reduce the number of groups in the partition further. In this chapter, we present a substitution … Read more