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I have functions defined as follows:

$f_1(A) = \sum\|x_i-x_j\|_A = \sum\sqrt{(x_i-x_j)^TA(x_i-x_j)}$ and $f_2(A) = \sum\|x_k-x_l\|^2_A$ where $A$ is a positive semi-definite (PSD) matrix, $x$ are number vectors.

Task is to minimize function $f(A) = f_2(A)$ subject to: $f_1(A) \geq 1$ and $A \succeq 0$. I can compute its gradient and hessian to use in Newton's method for minimization (which I want to use).

My question is: how can I incorporate above constraints (to be satisfied) into the algorithm?

Can the first constraint be solved by rewriting the main function $f(A)$ to contain slack variable -> something like: $f(A) = f_2(A) + 1 - f_1(A)$ ?

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Hi, I recommend you could post your question at Computational Science . –  Shuhao Cao May 7 '12 at 3:58
    
Hi, thanks for advice ... i supposed this to be rather mathematical issue. –  mko May 7 '12 at 10:48
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