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October 29th, 2014, 05:13 PM   #1
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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
$$X_j^*=g_j(X_1^*,...,X_{j-1}^*,X_{j+1}^*,...,X_p^*),$$ where $g_j:X_1\times...\times X_{j-1}\times X_{j+1}\times...\times X_p\rightarrow X_{j}$ is continuous for all $j=1,...,p$.

Now, I define a sequence
$$X_j^{(k+1)}=g_j(X_1^{(k+1)},...,X_{j-1}^{(k+1)},X_{j+1}^k,...,X_p^k),$$ 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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