Book ChapterDOI
Strongly Stable Stationary Solutions in Nonlinear Programs.
Masakazu Kojima
- pp 93-138
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TLDR
In this paper, the authors focus on strongly stable stationary solutions in nonlinear programs and present several lemmas and theorems for the stability and sensitivity of local minimum solutions to parametric programs.Abstract:
Publisher Summary This chapter focuses on strongly stable stationary solutions in nonlinear programs. Many studies have been made on the stability or the sensitivity of local minimum solutions to parametric programs. Those studies mainly discussed the continuity of the minimum value of the objective function, the continuity of the set of minimum solutions and/or the continuity of an isolated local minimum or stationary solution with respect to a small change of the parameter vector. In the case where the strict complementarity does not hold, the approach based on the standard implicit function theorem for continuously differentiable maps cannot be used. Various lemmas and theorems are discussed in the chapter. Stationary index, s-stable local minimum solutions, and degenerate s-stable stationary solutions are discussed. An application to a class of continuous deformation methods is reviewed.read more
Citations
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Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications
Patrick T. Harker,Jong-Shi Pang +1 more
TL;DR: The field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory as mentioned in this paper.
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Introduction to Numerical Continuation Methods
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TL;DR: The Numerical Continuation Methods for Nonlinear Systems of Equations (NCME) as discussed by the authors is an excellent introduction to numerical continuuation methods for solving nonlinear systems of equations.
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Implicit Functions and Solution Mappings
TL;DR: In this article, the authors provide a reference on the topic and a unified collection of a number of results which are currently scattered throughout the literature, including implicit mappings defined by relations other than equations.
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Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity
Holger Scheel,Stefan Scholtes +1 more
TL;DR: Several stationarity concepts, based on a piecewise smooth formulation, are presented and compared and Fiacco-McCormick type second order optimality conditions and an extension of the stability results of Robinson and Kojima are presented.
References
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Strongly Regular Generalized Equations
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Bimatrix Equilibrium Points and Mathematical Programming
TL;DR: In this paper, simple constructive proofs are given of solutions to the matric matric system Mz − ω = q; z ≧ 0; ω ≧ 1; zT = 0, for various kinds of data M, q, which embrace quadratic programming and the problem of finding equilibrium points of bimatrix games.
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Complementary pivot theory of mathematical programming
TL;DR: The role of problems of the form w and z satisfying w = q + Mz, w = or 0, z = or0, zw = 0 play a fundamental role in mathematical programming.
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The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints
Olvi L. Mangasarian,S Fromovitz +1 more
TL;DR: In this article, the Kuhn-Tucker criterion was extended to the case of equalities and inequalities, and a new generalization of the Fritz-John criterion was proposed.
Journal ArticleDOI
Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations
Eugene L. Allgower,Kurt Georg +1 more
TL;DR: In this paper, a digest of simplicial and continuation methods for approximating fixed-points or zero-points of nonlinear finite-dimensional mappings is presented, where the following curves are implicitly defined, as for example, in the case of homotopies.