Download Computing Equilibria and Fixed Points: The Solution of by Zaifu Yang PDF
By Zaifu Yang
Computing Equilibria and stuck Points is dedicated to the computation of equilibria, fastened issues and desk bound issues. This quantity is written with 3 targets in brain: (i) to provide a finished advent to fastened element tools and to the definition and building of Gröbner bases; (ii) to debate numerous attention-grabbing functions of those equipment within the fields of common equilibrium concept, online game idea, mathematical programming, algebra and symbolic computation; (iii) To introduce numerous complicated fastened element and desk bound element theorems. those tools and issues will be of curiosity not just to economists and video game theorists fascinated with the computation and lifestyles of equilibrium results in financial types and cooperative and non-cooperative video games, but in addition to utilized mathematicians, machine scientists and engineers facing types of hugely nonlinear structures of equations (or polynomial equations).
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Extra info for Computing Equilibria and Fixed Points: The Solution of Nonlinear Inequalities
Sample text
The problem of finding a stationary point is called stationary point problem. Before giving an existence theorem of stationary points, we introduce the following result. 9 For any nonempty, convex and compact subset C ofR,n, define a function r : ]Rn f---7 C by r(x) = argmin{lIy - xl1 2lYE C}. t. Y E C. Since C is a nonempty compact set, the problem has an optimal solution, say, yi. Suppose it is not unique. Then there exists y2 E C such that Ilyi - xW = IIy2 - xii = 0: and yi i= y2. Take yO = (yi + y2)/2.
Suppose that u is a point in R n such that g(u) ~ x~~ng(x) + f.. d(v, w) for any w '# v. d(x, un· It is easy to verify that X is nonempty and closed. d(u, x) ~ g(u) - g(x) ~ g(u) - wifJng(w) ~ f.. Hence d(u, x) ~ 1. It also holds that g(x) ~ g(u). 7 THEOREMS OF TARSKI, CARISTI AND EKELAND 35 Suppose to the contrary that for any x E X, there exists w E R n such that w =f:. x and g(w) S g(x) - Ed(x,w). Then we have Ed(w,u) S Ed(w, x) + Ed(x,u) S g(x) - g(w) + g(u) - g(x) = g(u) - g(w). This implies that w E X.
Another important problem related to fixed point problems is the socalled complementarity problem. It is a problem of finding a point x* E R+. such that j(x*) E R+. and (x*) T j(x*) = 0, where f : R n 1-7 R n is a function. The problems above can easily be extended to the simplotope and the existence theorems can also be easily obtained. , (pj)T Ji(p*) > (pj)T Ji(p*), j = 1, ... , N, for all pES. Define a function gj : S 1-7 snj for j = 1, ... , N by iljEIN Rnj gj(x) = argmin{ IIYj - (Xj + Ji (x)) II IYj E snj }.