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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
of two convex bodies
can equivalently be formulated by the inequality
for all directions
,
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:
- Investigation of the minimal circumscribed ellipsoid and
the maximal inscribed ellipsoid of convex bodies in
by means
of semi-infinite optimization techniques. This approach presents necessary
and sufficient optimality conditions
and (finite) characterizations of the extremal ellipsoids.
Furtheron the uniqueness of the optimal ellipsoids can be proved to be a
consequence of these optimality conditions.
The optimization models of these two covering and embedding problems can
be constructed in a very similar way, such that simultaneous investigations
are possible what allows to obtain common properties
and connections between extremal covering and embedding.
Particularily, polarity relations between the circumscribed ellipsoid
of minimal volume and the inscribed ellipsoid of maximal volume of a
convex body are established.
-
In the special cases of inscribed/circumscribed spheres the arising
optimization models are linear ones. The classical results (characterizations
of inspheres and circumspheres of convex bodies, inequalities of Jung and
Steinhagen between inradius, circumradius, width and diameter) occur in a new
light as consequences of the linear semi-infinite duality theory.
- The investigations are extended to spherical shells
of a convex body K with
,
where
denote concentric balls with center x and radii r,R,
respectively.
A minimal shell (minimal with respect to the width R-r ) can be obtained
as a solution of a linear semi-infinite optimization problem.
The linear semi-infinite duality theory provides characterizing properties of
the minimal shell.
Further research activities are intended to similar questions e.g.
ellipsoidal and more general convex shells.
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