Jürgen Guddat

On stability and stationary points in nonlinear optimization

On propose trois theoremes concernant la stabilite et les points stationnaires du probleme de minimisation sous contrainte suivant: Min f(x) sur M[H,G]={x∈R n /H(x)=0, G(x)≥0}

Stability in convex quadratic parametric programming

In this paper we discuss convex quadratic programming problems with variable coefficients in the linear part of the objective function or/and in the right hand side of the constraints. Local and global stability statements are contained. An important global stability theorem is proved for a feneral non-linear programming problem arbitrary, where F is a continuous function over is a nonempty compact subset of E n . A possibility of calculating of a local stability set for the convex quadratic parametric programming problem is also given. This method is not based on an algorithm for quadratic programming problems.

A modified standard embedding with jumps in nonlinear optimization

The paper deals with a combination of pathfollowing methods (embedding approach) and feasible descent direction methods (so-called jumps) for solving a nonlinear optimization problem with equality and inequality constraints. Since the method that we propose here uses jumps from one connected component to another one, more than one connected component of the solution set of the corresponding one-parametric problem can be followed numerically. It is assumed that the problem under consideration belongs to a generic subset which was introduced by Jongen, Jonker and Twilt. There already exist methods of this type for which each starting point of a jump has to be an endpoint of a branch of local minimizers. In this paper the authors propose a new method by allowing a larger set of starting points for the jumps which can be constructed at bifurcation and turning points of the solution set. The topological properties of those cases where the method is not successful are analyzed and the role of constraint qualifications in this context is discussed. Furthermore, this new method is applied to a so-called modified standard embedding which is a particular construction without equality constraints. Finally, an algorithmic version of this new method as well as computational results are presented.

Eine modification der methode von theil und van de panne zur lösung einparametrischer quadratischer optimierungsprobleme

At first the authors give a conception of structural stability for a one-parametric optimization problem. I n the second part there are proved some interesting theorems on convexity intervals, which partially are of importance for the numerical computation of these intervals. At last the method of THEIL and VAN DE PANNE is modified in such a manner, that they get a decomposition of the given convexity interval into stability regions. I n the caae of non-degeneration this decomposition is unique.

Pathfollowing methods for nonlinear multiobjective optimization problems

Abstract The paper deals with multiobjective nonlinear optimization problems with equality and inequality con- straints. In a corresponding dialogue procedure the decision maker has to determine in each step the aspira- tion and reservation level expressing his goals. This leads to a nonlinear optimization problem which can be solved by using a combination of pathfollowing methods and feasible descent direction methods (so- called jumps). The authors propose the application of a recently published pathfollowing method with jumps to multiobjective optimization problems, discuss its topological properties and present several computational results.

New embeddings for nonlinear multiobjective optimization problems I

In a dialogue procedure the decision maker has to determine in each step the aspiration and reservation level expressing his wishes (goals). This leads to an optimization problem which is not easy to solve in the nonconvex case (the known starting point is not feasible). We propose a modified standard embedding (one parametric optimization). This problem will be discussed from the point of view of parametric optimization (non-degenerate critical points, singularities, pathfollowing methods to describe numerically a connected component in the set of stationary points and in the set of generalized critical points, respectively, and jumps (descent methods) to other connected components in these sets). This embedding is much better for computing a goal realizer or replaying that the goal was not realistic than the embeddings considered in the literature before, but in the worst case we have to find all connected components and this is an open problem.

A NOTE ON EMBEDDINGS FOR THE AUGMENTED LAGRANGE METHOD

Nonlinear programs (P) can be solved by embedding problem P into one parametric problem P(t), where P(1) and P are equivalent and P(0), has an evident solution. Some embeddings fulfill that the solutions of the corresponding problem P(t) can be interpreted as the points computed by the Augmented Lagrange Method on P. In this paper we study the Augmented Lagrangian embedding proposed in [6]. Roughly speaking, we investigated the properties of the solutions of P(t) for generic nonlinear programs P with equality constraints and the characterization of P(t) for almost every quadratic perturbation on the objective function of P and linear on the functions defining the equality constraints.

Parametric Multiobjective Optimization

The weighted sum approach for finding Pareto optimal solutions in multiobjective optimization has been presented depending on a parameter value. We show that the one-parametric optimization techniques can be applied to parametric multiobjective optimization.

On the calculation of a feasible point of a nonconvex set: pathfollowing with jumps

The article deals with an approach for the calculation of a feasible point of a nonconvex compact set, which is defined by finitely many equality and inequality constraints. This approach is a heuristic one (since the convergence to a feasible point cannot be ensured), and it combines pathfollowing methods with Nonlinear Programming (NLP) solvers (socalled jumps) for solving a sequence of auxiliary one-parametric problems. It is assumed that these one-parametric problems belong to a generic subset, which was introduced by Jongen, Jonker and Twilt [Jongen, H. Th., Jonker, P. and Twilt, F., 1986, Critical sets in parametric optimization. Mathematical Programming, 34, 333–353.] The crucial property of this approach is that the solution curves to be followed do not contain stationary points, which implies that certain singularities cannot appear. Furthermore, the new approach is applied to problems from global and multiobjective optimization and, finally, an algorithmic version as well as computational results are presented. §Dedicated to Prof. Dr. Diethard Pallaschke on the occasion of his 65th birthday.

Multiobjective Optimization: Singularities, Pathfollowing and Jumps

This investigation is a continuation of the considerations in Guddat et al. (1985), Guddat and Guerra Vasquez (1987), Guddat et al. (to appear) using pathfollowing methods with jumps (cf. e.g. Gfrerer et al. 1983, Gfrerer et al. 1985, Guddat 1987, Guddat et al. to appear, Guddat et al. 1988, Guddat and Nowack 1990, Jongen 1988) based on information on the singularities (cf. Jongen et al. 1986a, Jongen et al. 1986b).