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I am a noob in mathematic, so I would need your help in solving the optimization problem below

\begin{array}{l} \max\limits_{\bf l} \min \left( \left| {\bf g}_1 {\bf Ml} \right|^2, \left| {\bf g}_2 \bf Ml \right|^2 \right) \\ \text{ s.t. } \left\| {\bf Ml} \right\|^2 = \text{positive const}. \end{array}

where ${\bf g}_1, {\bf g}_2 \in {^{1 \times M}}$ and ${\bf M} \in {^{M \times 2}}$ are given vectors and matrix, $\bf{l}$ has the size $2\times1$.

Many thanks for your help, Anh

anhledc
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1 Answers1

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Transform the problem by introducing a new decision variable $z$ which captures $\min (|\mathbf{g}_1\mathbf{Ml}|^2,|\mathbf{g}_2 \mathbf{Ml}|^2)$. Your problem now becomes $$ \max_{\mathbf{l},z} z $$ subject to: $$ \begin{aligned} \|\mathbf{Ml}\|^2 &=k \\ z &\le |\mathbf{g}_1 \mathbf{Ml}|^2 \\ z &\le |\mathbf{g}_2 \mathbf{Ml}|^2 \end{aligned} $$

baudolino
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  • Thank you Michael, I think the first constraint must be $|\mathbf{Ml}|^2 &=k$. And what should I do next? – anhledc Aug 22 '13 at 23:05
  • If you need the actual numerical answer, plug this model into a convex solver like BARON or LOQO. If you'd like an analytical solution, try to form the Lagrangian and write the KKT conditions. Don't forget to upvote and mark the question as answered, if the proposed solution makes sense for you. – baudolino Aug 23 '13 at 01:35
  • Thank you very much! I will try to follow your instruction. – anhledc Aug 23 '13 at 02:31