Optimization problems involving eigenvalues arise in many applications. Let x be a vector of real parameters and let A(x) be a continuously differentiable symmetric matrix function of x. We consider a particular problem which occurs frequently: the minimization of the maximum eigenvalue of A(x), subject to linear constraints and bounds on x. The eigenvalues of A(x) are not differentiable at points x where they coalesce, so the optimization problem is said to be nonsmooth. Furthermore, it is typically the case that the optimization objective tends to make eigenvalues coalesce at a solution point. There are three main purposes of the paper. The first is to present a clear and self-contained derivation of the Clarke generalized gradient of the max eigenvalue function in terms of a "dual matrix". The second purpose is to describe a new algorithm, based on the ideas of a previous paper by the author (SIAM J. Matrix Anal. Appl. 9 (1988) 256-268), which is suitable for solving l...

In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [34] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQP-methods, as demonstrated by extensive numerical tests.

This paper describes Knitro 5.0, a C-package for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving large-scale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming. Various algorithmic options are available, including two interior methods and an active-set method. The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings. 1

this paper we investigate more closely the structures of two application examples which can be considered representative for the rather wide class of large-scale optimization problems in so far as they stem from discretizations:

In space based robotics, one of the most important problems is the disturbance to the satellite attitude and to the satellite microgravity environment caused by satellite mounted robot operation. This paper reports on computer-aided motion planning strategies to overcome this problem. Point to point motion designs are generated which not only connect prescribed start and end points of the robot motion, but also simultaneously return the satellite to its original attitude. Theoretical characterizations of some of those motion designs are presented, as well as numerical results. The computation time required to produce such motion designs is 1 or 2 minutes on a workstation. Thus, it can be practical to use these motion plans in space. Introduction Satellite mounted robots can aid in the automated assembly of large structures in space and sometimes eliminate the need for extra-vehicular activity or "space walks", which are time-consuming and dangerous for astronauts. Robots in space can ...

. The increasing demand for optimization in process control of reactive systems necessitates the development of fast and reliable software for the numerical computation of optimal controls taking into account the special structures of the chemical systems under investigation. This paper presents a new algorithm based on partially reduced SQP methods for the computation of optimal controls following the direct approach. It is coupled with a package for the simulation of homogeneous reactive systems, which provides the specific routines for the description of the problem at hand. The combined code takes advantage from the fact that the number of influence variables to be determined is rather low compared with the overall number of variables. Numerical results for two new and practically important application problems are presented. Indications for possible further developments are given as well. 1 Motivation and problem formulation Process optimization plays an important role for the ef...

This paper is concerned with the case of a robot mounted on a satellite with the attitude control system not in operation. This is realistic in the sense that the RMS is often operated with the attitude control system off, to prevent sudden corrections. The problem considered is the construction of point to point maneuvers of the robot, so that the final attitude of the satellite is identical to its initial attitude. The modelling assumptions are that the robot consists of rigid bodies with negligible mass and that the whole system is free floating and at rest in space. It has been shown in [5] that this path planning problem is theoretically solvable. There is no uniqueness to such solutions, and generally there is a continuum of such solutions. The one known solution generated in [5], although feasible, is complex and undesirable. There is a need to develop methods of generating robot paths that have certain desirable properties. In order to generate desirable paths the optimal control problem approach has been developed in [9]. The basic idea is to formulate an objective criterion characterizing the desirable paths, and to use the dynamic equations and initial and final configuration conditions as constraints. This defines an optimal control problem, the solution of which results in the desirable path. min