Abstract
In this chapter we set Lagrange functions and a related general duality principle at the pinnacle of duality theory. We treat important examples of Lagrange functions in:
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(α)
Section 49.3 (linear optimization).
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(β)
Section 50.1 (Kuhn-Tucker theory).
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(γ)
Section 51.6 (Trefftz duality for linear elliptic partial differential equations).
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(δ)
Section 51.7 (quasilinear elliptic partial differential equations).
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
Godfrey Harold Hardy (1877“1947)
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References to the Literature
Classical work: Farkas (1902) (Farkas’ Lemma on inequalities); J. von Neumann (1928) (existence of saddle points and game theory).
Saddle points and optimization in ℝN: Rockafellar (1970, M,B,H); Stoer and Witzgall (1970, M); Elster (1977, M).
Introduction to linear optimization and duality: Franklin (1980, M).
General minimax theorems: J. von Neumann (1928); Ky Fan (1952); Sion (1958); Browder (1968a); Brézis, Nirenberg, and Stampacchia (1972); Holmes ( 1975, M); Aubin ( 1979, M).
Saddle points, multivalued mappings, and variational inequalities: Browder (1968a); Mosco ( 1976, S); Gwinner ( 1981, S).
Saddle points and convex analysis: Ekeland and Temam (1974, M); Barbu and Precupanu (1978, M); Aubin (1979, M).
Saddle points and geometric functional analysis: Holmes (1975, M).
Saddle points and duality theory: Ekeland and Temam (1974, M) (recommended as an introduction, numerous applications); Pšeničnyi (1972, M); Göpfert (1973, M); Sander (1973, M); Ioffe and Tihomirov (1974, M); Golštein (1975, M) (generalized concept of solution); Krabs (1975, M); Barbu and Precupanu ( 1978, M).
Saddle functions and maximal monotone operators: Krauss (1984).
Critical points and nonlinear differential equations: Nirenberg ( 1981, S); Berkeley ( 1983, P).
Critical points and semilinear elliptic differential equations: Clark (1972); Ambrosetti and Rabinowitz (1973); Rabinowitz (1974, S), (1978); Ahmad, Lazer, and Paul (1976); Berger (1977, M); Amann (1979); Amann and Zehnder (1980); Hess (1980); Struwe (1980), (1982).
Critical points and semilinear wave equations: Rabinowitz (1978b); Amann (1979); Amann and Zehnder (1980); Brézis, Coron, and Nirenberg (1980); Brézis (1983) (duality principle).
Critical points and periodic solutions of Hamiltonian systems: Berger (1977, M); Rabinowitz (1978a), (1980); Benci and Rabinowitz (1979); Amann (1979); Amann and Zehnder (1980); Nirenberg (1981, S); Clarke and Ekeland (1980), (1981).
Saddle points and applications to economics: Aubin (1979, M) (comprehensive presentation).
Saddle points and game theory: Compare the references to the literature in Chapter 9 and in Section 37.8.
Critical points and Ljusternik-Schnirelman theory as well as Morse theory: Compare the references to the literature in Chapter 4 4.
Approximation methods for determining saddle points: Auslender (1972, L), (1976, M); Ekeland and Temam (1974, M) (Uzawa’s algorithm); Demjanov and Malozemov (1975, M); Belenkii and Volkonskii (1976, M) (handbook); Glowinski, Lions, and Trémolières ( 1976, M).
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© 1985 Springer Science+Business Media New York
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Zeidler, E. (1985). General Duality Principle by Means of Lagrange Functions and Their Saddle Points. In: Nonlinear Functional Analysis and its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5020-3_14
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DOI: https://doi.org/10.1007/978-1-4612-5020-3_14
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