Hubertus Th. Jongen

P. Jonker

F. Twilt

In this second volume we pay attention to three important subjects in nonlinear optimization. The first one concerns the dependence of an optimization problem on the problem data: stability, resp. sensitivity aspects are considered from both a local and global point of view. Since stability results are intimately related with some concepts of transversality, we treat transversality theory in detail. A second subject consists on one hand of gradients, the gradient differential equation and its use in the search for several local minima. On the other hand we treat the underlying differential equation for Newton's method for finding critical points of a function. Again, transversality plays a crucial role. Finally, we consider optimization problems depending on parameters (critical sets, bifurcation of the feasible set, etc.) The present volume is preceded by a first one in which we study manifolds with boundary and critical point theory (Morse theory) within the framework of nonlinear optimization.

Nonlinear Optimization in IRN: Transversality, flows, parametric aspects

The topology of global optimization is treated in detail (Morse Theory, Karush-Kuhn-Tucker points, Chebyshev Approximation). Moreover, three further basic subjects in nonlinear optimization are treated: Stability and sensitivity, gradient- and Newton's differential equations and, finally, parametric aspects. Audience: Researchers and graduate students working in optimization and operations research.

Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects

Nonlinear Optimization in Finite Dimensions

Peter Jonker

Frank Twilt

In this first volume manifolds with boundary and critical point theory play an important role. We focus our attention to topological aspects in nonlinear optimization. In this way, we do not treat numerical and algorithmic features; however, we believe that the present work will provide a good insight into nonlinear phenomenae and that it will be helpful in order to device computational strategies towards global optimization. In order to underline the geometry of our analysis, many figures are included. In a second volume, we treat structural stability, transversality, (gradient-, Newton-)flows and parametric aspects.

Hubertus Th. Jongen

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