Inverse functions and existence principles. (English. Russian original) Zbl 1352.49019
J. Math. Sci., New York 218, No. 5, 572-580 (2016); translation from Fundam. Prikl. Mat. 19, No. 5, 35-47 (2014).
Summary: Each one of six general existence principles of compactness (the extreme value theorem), completeness (the Newton method or the modified Newton method), topology (Brouwer’s fixed-point theorem), homotopy (on contractions of a sphere to its center), variational analysis (Ekeland’s principle), and monotonicity (the Minty-Browder theorem) is shown to lead to the inverse function theorem, each one giving some novel insight. There are differences in assumptions and algorithmic properties; some of the propositions have been constructed specially for this paper. Simple proofs of the last two principles are included. The proof by compactness is shorter and simpler than the shortest and simplest known proof, that by completion. This gives a very short self-contained proof of the Lagrange multiplier rule, which depends only on optimization methods. The proofs are of independent interest and are intended as well to be useful in the context of the ongoing efforts to obtain new variants of methods that are based on the inverse function theorem, such as comparative statics methods.
MSC:
49J99 | Existence theories in calculus of variations and optimal control |
49J45 | Methods involving semicontinuity and convergence; relaxation |
49K21 | Optimality conditions for problems involving relations other than differential equations |
26B10 | Implicit function theorems, Jacobians, transformations with several variables |
Keywords:
inverse function theorem; extreme value theorem; Ekeland’s variational principle; Newton method; Brouwer’s fixed point theorem; homotopy; monotonicity; optimization; Lagrange multiplier ruleReferences:
[1] | D. Acemoglu and M. K. Jensen, Robust Comparative Statics in Large Dynamic Economies, http://economics.mit.edu/files/9619 (2014). · Zbl 0286.49015 |
[2] | J. M. Borwein and A. S. Lewis, Convex Calculus and Nonlinear Optimization: Theory and Examples, Springer, New York (2006). · Zbl 1116.90001 · doi:10.1007/978-0-387-31256-9 |
[3] | I. Ekeland, “On the variational principle,” J. Math. Anal. Appl., 47, 324-353 (1974). · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0 |
[4] | S. G. Krantz and H. R. Park, The Implicit Function Theorem: History, Theory and Applications, Birkhäuser, Basel (2012). |
[5] | P. Milgrom and J. Roberts, “Computing Equilibria,” Am. Econ. Rev., 84, 441-459 (1994). |
[6] | M. Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, Westview Press (1971). · Zbl 0141.05403 |
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