Applications

Abstract

In this chapter we consider three applications of our approach: globally convergent algorithms for nonlinear optimization problems (Section 6.2), global optimization (Section 6.3) and multi-objective optimization (Section 6.4). It will be shown that pathfollowing methods with jumps can be useful for the considered applications but not successful in every case. This is no surprise, since we do not have jumps in all situations that can appear in the set ∑_gc under the assumption (A1) (cf. Chapter 5). The following problems remain open:

• (i) Find an actually globally convergent algorithm for a non-convex optimization problem with an arbitrarily chosen non-feasible starting point without any additional information.

• (ii) Find a deterministic approach to solve surely the problem of global optimization.

• (iii) Find surely a goal realizer (cf. Definition 6.4.1) for the multi-objective optimization problem if the goal (which expresses the current wishes of the decision-maker in a dialogue procedure) has been a realistic one and give an answer to the question whether the goal has been a realistic one or not.