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Semi-infinite Optimization

Authors: Friedrich Juhnke
Staff Members: Jörn Grey
Cooperations: Olaf Sarges (WLB Düsseldorf)




Semi-infinite Optimization deals with the problem of minimizing (maximizing) a real-valued objective function of a finite number of variables with respect to an (possibly and generally) infinite number of constraints.
There is a great variety of (classical) applications of semi-infinite optimization, including problems in approximation theory (with respect to polyhedral norms), operation research, optimal control, boundary value problems and others. These applications and appealing theoretical properties of semi-infinite problems gave rise to intensive (and up to now undiminished) research activities in this field since its inceptive appearing in the 1960s.
Recent applications of semi-infinite optimization techniques to geometric extremal problems are opened up in the last years, first of all in convex geometry.
Describing an n-dimensional convex body by its Minkowski support function, there occur in a very natural way systems of (infinitely many) linear inequalities with a finite number of variables. Additionally, any inclusion $C \subseteq K$ of two convex bodies $C,K \subseteq {\bf R}^n $ can equivalently be formulated by the inequality $ h(u) \le k(u)$ for all directions $u \in {\bf R}^n$, where h,k are the support functions of C,K, respectively. So the feasible regions of extremum problems corresponding to coverings or embeddings in convex geometry can be described by semi-infinite systems and semi-infinite optimization techniques turn out to be an appropriate tool for handling them.
Our research activities are adressed to applications of semi-infinite optimization techniques to general covering and embedding problems in convex geometry and deal with the following more specific questions:




Selected Publications
1
Juhnke, F.: Inradius und Dicke konvexer Koerper aus optimierungstheoretischer Sicht, Beitraege zur Algebra u.Geometrie 27 (1988), 13 - 20
2
Juhnke, F.: Das Umkugelproblem und lineare semi-infinite Optimierung, Beitraege zur Algebra u. Geometrie 28 (1989), 147-156.
3
Juhnke, F.: Volumenminimale Ellipsoidueberdeckungen, Beitraege zur Algebra u. Geometrie 30 (1990), 143 - 153
4
Juhnke, F.: Extremal spheres and semi-infinite duality theory, in: Extended abstracts of the 16th Symposium on Operations Research Trier 1991, Physica-Verlag Heidelberg 1992, 43-47
5
Juhnke, F.: Extremal circumscribed ellipsoids and generalized convexity, Approximation $\&$ Optimization, ed. Guddat, Jongen, Kummer, Nozicka, Peter Lang Verlag Frankfurt (Main), 1993, 323-340
6
Juhnke, F.: Circumscribed spheres via semi-infinite optimization, in: 17th Symposium on Operations Research, Hamburg 1992, Physica-Verlag Heidelberg 1993, 197-200
7
Juhnke, F.: Embedded maximal ellipsoids and semi-infinite optimization,
Beitr„ge zur Algebra und Geometrie / Contributions to algebra and geometrie, 35 (1994), No.2, 163-171
8
Juhnke, F.: Polarity of embedded and circumscribed ellipsoids, Beitraege zur Algebra und Geometrie / Contributions to Algebra and Geometry, 36 (1995), No.1, 17-24
9
Juhnke, F.: Embedded ellipsoids and generalized convexity, in: Parametric Optimization and Related Topics, eds. J. Guddat, H.Th. Jongen, F. Nozicka, G. Still and F. Twilt, Peter Lang Verlag Frankfurt (Main), 1996, 177 - 184
10
Sarges, O: Redundance of Vertices of the Cube relatively to its Minimal Ellipsoid , Beitraege zur Algebra und Geometrie / Contributions to Algebra and Geometry, 37 (1996), No.1, 41-49
11
Juhnke, F., O. Sarges: Minimal Spherical Shells and Linear Semi-infinite Optimization, Beitraege zur Algebra und Geometrie / Contributions to Algebra and Geometry, (to appear)


 
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Hugo
1999-07-19