P. Jonker

Critical sets in parametric optimization

We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a ‘generalized critical point’ (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set∑ consisting of all g.c. points. Due to the parameter, the set∑ is pieced together from one-dimensional manifolds. The points of∑ can be divided into five (characteristic) types. The subset of ‘nondegenerate critical points’ (first type) is open and dense in∑ (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along∑ is presented. Finally, the Kuhn-Tucker subset of∑ is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.

One-parameter families of optimization problems: Equality constraints

In this paper, we introduce generalized critical points and discuss their relationship with other concepts of critical points [resp., stationary points]. Generalized critical points play an important role in parametric optimization. Under generic regularity conditions, we study the set of generalized critical points, in particular, the change of the Morse index. We focus our attention on problems with equality constraints only and provide an indication of how the present theory can be extended to problems with inequality constraints as well.

The continuous, desingularized Newton method for meromorphic functions

For any (nonconstant) meromorphic function, we present a real analytic dynamical system, which may be interpreted as an infinitesimal version of Newtons method for finding its zeros. A fairly complete description of the local and global features of the phase portrait of such a system is obtained (especially, if the function behaves not too bizarre at infinity). Moreover, in the case of rational functions, structural stability aspects are studied. For a generic class of rational functions, we give a complete graph-theoretical characterization, resp. classification, of these systems. Finally, we present some results on the asymptotic behaviour of meromorphic functions.

On the classification of plane graphs representing structurally stable rational Newton flows

We study certain plane graphs, called Newton graphs, representing a special class of dynamical systems which are closely related to Newtons iteration method for finding zeros of (rational) functions defined on the complex plane. These Newton graphs are defined in terms of nonvanishing angles between edges at the same vertex. We derive necessary and sufficient conditions -of purely combinatorial nature- for an arbitrary plane graph in order to be topologically equivalent with a Newton graph. Finally, we analyse the structure of Newton graphs and prove the existence of a polynomial algorithm to recognize such graphs.

One-Parametric Linear-Quadratic Optimization Problems

We consider families of optimization problems with quadratic object function and affine linear constraints, which depend smoothly on one real parameter. For a generic subclass of such problems only three different types of (generalized) critical points occur, whereas in the general case (of nonlinear one-parameter families of constrained optimization problems on Rn) five types are to be distinguished. We clarify the theoretical background of these phenomena and illustrate the underlying mechanism with simple examples.

Some Reflections on the Continuous Newton-Method for Rational Functions

We consider the desingularized continuous Newton-method for rational functions of one complex variable. Structural stability aspects are discussed. Special attention is given to rational functions with one (simple) pole. For these functions a full classification — up to topological equivalence — is obtained.

On the stratification of a class of specially structured matrices

We consider specially structured matrices representing optimization problems with quadratic objective functions and (finitely many) affine linear equality constraints in an n-dimensional Euclidean space. The class of all such matrices will be subdivided into subsets [‘strata’], reflecting the features of the underlying optimization problems. From a differential-topological point of view, this subdivision turns out to be very satisfactory: Our strata are smooth manifolds, constituting a so-called Whitney Regular Stratification, and their dimensions can be explicitly determined. We indicate how, due to Thoms Transversality Theory, this setting leads to some fundamental results on smooth one-parameter families of linear-quadratic optimization problems with (finitely many) equality and inequality constraints.

Morse theory (without constraints)

In this chapter we make more precise some of the intuitive ideas on Morse theory as exposured in Chapter 1.

Stability of optimization problems

In Section 4.1 we studied the dependence of nondegenerate critical points and their values on a finite number of parameters. This was done in order to describe the change of the extremal set (value) of the error function in Chebyshev approximation problems.

Homology, Morse relations

In Chapter 2, 3, 4 we considered changes in the structure of lower level sets of functions up to homotopy equivalence (cell-attaching etc.). Consequently, we may measure those changes by means of homotopy invariants. In Chapter 1, Fig. 1.6.3, we have given already an intuitive impression of what may happen if we attach a 2-cell, in terms of canceling, respectively creating a 2-, respectively 3-dimensional “hole”. Now we will make the notion of a “k-dimensional hole” mathematically precise. In fact, given a topological Hausdorff space X, for q = 0, 1, 2, ... , we associate with X a vector space (over ℝ): H q (X). In case H q (X) is finite dimensional, the dimension r q of H q (X) will represent the number of “(q + 1)-dimensional holes” in X. In this section we will give a brief introduction to the theory on the spaces H q (X). As far as proofs are concerned, we restrict ourselves to those proofs which are important for a good understanding. For those proofs which we delete, we refer to [A/B], [Gre], [Span].