zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On generalized semi-infinite optimization and bilevel optimization. (English) Zbl 1081.90063
Summary: The paper studies the connections and differences between bilevel problems (BL) and generalized semi-infinite problems (GSIP). Under natural assumptions (GSIP) can be seen as a special case of a (BL). We consider the so-called reduction approach for (BL) and (GSIP) leading to optimality conditions and Newton-type methods for solving the problems. We show by a structural analysis that for (GSIP)-problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general (BL)-problems not. The genericity behavior of (BL) and (GSIP) is in particular studied for linear problems.

90C34Semi-infinite programming
90C46Optimality conditions, duality
Full Text: DOI
[1] Bard, J. F.; Moore, J. T.: A branch and bound algorithm for the bilevel programming problem. SIAM journal of science and statistical computation 11, No. 2, 281-292 (1990) · Zbl 0702.65060
[2] Falk, E. J.; Liu, J.: On bilevel programming, part I: General nonlinear cases. Mathematical programming 70, 47-72 (1995) · Zbl 0841.90112
[3] Gibson, C. G.; Wirthmüller, K.; Du Plessis, A. A.; Looijenga, E. J. N.: Topological stability of smooth mappings. Lecture notes in mathematics 552 (1976) · Zbl 0377.58006
[4] R. Hamming, On the bilevel programming problem, Thesis in the Faculty of Mathematics, University of Twente, 1998 · Zbl 1019.00507
[5] Hettich, R.; Jongen, H. Th.: Semi-infinite programming: conditions of optimality and applications. Optimization techniques 2, 1-11 (1978) · Zbl 0381.90085
[6] Hettich, R.; Still, G.: Semi-infinite programming models in robotics. Parametric optimization and related topics II (1991) · Zbl 0737.90068
[7] Hettich, R.; Kortanek, K.: Semi-infinite programming: theory methods and applications. SIAM review 35, No. 3, 380-429 (1993) · Zbl 0784.90090
[8] Hettich, R.; Still, G.: Second order optimality conditions for generalized semi-infinite programming problems. Optimization 34, 195-211 (1995) · Zbl 0855.90129
[9] A. Hoffmann, R. Reinhardt, On reverse Chebyshev approximation problems, Technical University of Illmenau, Preprint No. M08/94, 1994
[10] Jongen, H. Th.; Rückmann, J. -J.; Stein, O.: Generalized semi-infinite optimization: A first order optimality condition and examples. Mathematical programming 83, 145-158 (1998) · Zbl 0949.90090
[11] Jongen, H. Th.; Zwier, G.: On the local structure of the feasible set in semi-infinite optimization. International series of numerical mathematics 72, 185-202 (1984)
[12] Kaplan, A.; Tichatschke, R.: On a class of terminal variational problems. Parametric optimization and related topics IV (1997) · Zbl 0885.49018
[13] Rockafellar, R. T.: Directional differentiability of the optimal value function in nonlinear programming problem. Mathematical programming study 21, 213-226 (1994) · Zbl 0546.90088
[14] Rückmann, J. -J.; Shapiro, A.: First-order optimality conditions in generalized semi-infinite programming. Journal of optimization theory and applications 101, 677-691 (1999) · Zbl 0956.90055
[15] Rückmann, J. -J.; Stein, O.: On linear and linearized generalized semi-infinite problems. Annals of operations research 101, 191-208 (2001) · Zbl 1055.90080
[16] Shimizu, K.; Ishizuka, Y.; Bard, J.: Nondifferentiable and two-level mathematical programming. (1997) · Zbl 0878.90088
[17] Stein, O.: On level sets of marginal functions. Optimization 48, 43-67 (2000) · Zbl 0962.90048
[18] O. Stein, First order optimality conditions for degenerate index sets in generalized semi-infinite programming, Preprint No. 91, Lehrstuhl C für Mathematik, RWTH Aachen, 2000
[19] Stein, O.; Still, G.: On optimality conditions for generalized semi-infinite programming problems. Journal of optimization theory and applications 104, 443-458 (2000) · Zbl 0964.90047
[20] Still, G.: Generalized semi-infinite programming: theory and methods. European journal of operational research 119, 301-313 (1999) · Zbl 0933.90063
[21] Still, G.: Generalized semi-infinite programming: numerical aspects. Optimization 49, 223-242 (2001) · Zbl 1039.90083
[22] G. Still, Linear bilevel problems: Genericity results and an efficient method for computing local minima, Mathematical Methods of Operations Research, to appear
[23] G.-W. Weber, Generalized semi-infinite optimization and related topics, Habilitation Thesis, Darmstadt University of Technology, 1999