F. Twilt

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.

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.

On Decomposition and Structural Stability in Non-Convex Optimization

We consider finite dimensional optimization problems. The basic idea of Morse theory is used to trace relevant combinatorial features of the structure of these problems. We use a concept of critical points which extends the concept of Kuhn-Tucker points. On one band this concept enables us to prove structural stability for a class of problems. On the other hand it turns out to be a natural tool for studying homotopies of the object function.

Newton flows for elliptic functions I Structural stability : characterization & genericity

Abstract Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e. doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed. With respect to an appropriate topology on the set of all elliptic functions f of fixed order we prove: For almost all functions f, the corresponding Newton flows are structurally stable i.e. topologically invariant under small perturbations of the zeros and poles for f [genericity]. They can be described in terms of nondegeneracy-properties of f similar to the rational case [characterization].

Newton flows for elliptic functions: a pilot study†

Elliptic Newton flows are generated by a continuous, desingularized Newton method for doubly periodic meromorphic functions on the complex plane. In the special case, where the functions underlying these elliptic Newton flows are of second-order, we introduce various, closely related, concepts of structural stability. In particular, within the class of all (second order) elliptic Newton flows, structural stability turns out to be a generic property. Moreover, the phase portraits of all such structurally stable flows are equal, up to conjugacy. As an illustration, we treat the elliptic Newton flows for the Jacobi functions sn, cn and dn. †Dedicated to H.Th. Jongen on the occasion of his 60th birthday.

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.

Newton flows for elliptic functions II

In our previous paper we associated to each non-constant elliptic function f on a torus T a dynamical system, the elliptic Newton flow corresponding to f. We characterized the functions for which these flows are structurally stable and showed a genericity result. In the present paper we focus on the classification and representation of these structurally stable flows. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded graph

Newton flows for elliptic functions III & IV: Pseudo Newton graphs: bifurcation and creation of flows

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