Let J be the zero set of the gradient fx of a function f:Rn→R. Under fairly general conditions the stochastic approximation algorithm ensures d(f(xk),f(J))→0, as k→∞. First of all, the paper considers this proble...Let J be the zero set of the gradient fx of a function f:Rn→R. Under fairly general conditions the stochastic approximation algorithm ensures d(f(xk),f(J))→0, as k→∞. First of all, the paper considers this problem: Under what conditions the convergence d(f(xk),f(J)) → 0 implies k →∞ d(xk,J)→O. It is shown that such implication takes place if fx is continuous and f(J) is nowhere dense. Secondly, an intensified version of Sard's theorem has been proved, which itself is interesting. As a particular case, it provides two independent sufficient conditions as answers to the previous question: If f is a C1 function and either i) J is a compact set or ii) for any bounded set B, f-1(B)is bounded, then f(J) is nowhere dense. Finally, some tools in algebraic geometry are used to prove that j(J) is a finite set if f is a polynomial. Hence f(J) is nowhere dense in the polynomial case.展开更多
基金Supported by the National Natural Science Foundation of China (G69774008, G59837270, G1998020308)and National Key Project.
文摘Let J be the zero set of the gradient fx of a function f:Rn→R. Under fairly general conditions the stochastic approximation algorithm ensures d(f(xk),f(J))→0, as k→∞. First of all, the paper considers this problem: Under what conditions the convergence d(f(xk),f(J)) → 0 implies k →∞ d(xk,J)→O. It is shown that such implication takes place if fx is continuous and f(J) is nowhere dense. Secondly, an intensified version of Sard's theorem has been proved, which itself is interesting. As a particular case, it provides two independent sufficient conditions as answers to the previous question: If f is a C1 function and either i) J is a compact set or ii) for any bounded set B, f-1(B)is bounded, then f(J) is nowhere dense. Finally, some tools in algebraic geometry are used to prove that j(J) is a finite set if f is a polynomial. Hence f(J) is nowhere dense in the polynomial case.
