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October 29th, 2014, 05:13 PM  #1 
Newbie Joined: Oct 2014 From: Australia Posts: 1 Thanks: 0  Minimum for multivariable function using nonlinear Gauss Seidel
Let $X_1$...$X_p$ spaces without specific property (i.e., non convex and it's no necessary subset of real numbers $\mathbb{R}$). Let a continuous function $f:X_1\times...\times X_p\rightarrow \mathbb{R}$ such that $f$ has a minimum point $X^*=(X_1^*,...,X_p^*)$ such that $$X_j^*=g_j(X_1^*,...,X_{j1}^*,X_{j+1}^*,...,X_p^*),$$ where $g_j:X_1\times...\times X_{j1}\times X_{j+1}\times...\times X_p\rightarrow X_{j}$ is continuous for all $j=1,...,p$. Now, I define a sequence $$X_j^{(k+1)}=g_j(X_1^{(k+1)},...,X_{j1}^{(k+1)},X_{j+1}^k,...,X_p^k),$$ for all $k$. Let $X^k=(X_1^k,...,X_p^k)$. I could prove that sequence $\{f(X^{k})\}$ converge. Moreover, it is a decreasing sequence. But, Could I say $\{f(X^{k})\}$ converge to $\{f(X^*)\}$? Any answer will help me 

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function, gauss, gauss seidel, minimum, multivariable, nonlinear, seidel 
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