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This question is about solving the problem on CVX using MATLAB:

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My variable is W which is a N X N matrix. a is n X 1 vector. Sigma is a PSD matrix.

a) Can I solve this problem in Disciplined manner? b) How to reformulate this in cvx since CVX says the Quadratic forms need to be with scalars. (equalities are matrix equalities).

Ashutosh Gupta
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  • No, cvx will not address this as it is nonconvex. – Johan Löfberg Aug 27 '15 at 10:14
  • Why is it non-convex? I am sorry but having a matrix variable confuses me The objective is quadratic in entries of W, and so are the constraints..then why is this non convex? And how should I try solving this problem? – Ashutosh Gupta Aug 27 '15 at 10:23
  • Take the trivial scalar case, and think of the feasible points of the scalar quadratic constraint w^2=1 – Johan Löfberg Aug 27 '15 at 10:59
  • Yeah Got it. w^2 = 1, w = +/- 1 -> feasible region is non convex. same is true if sigma is I. Thnks. – Ashutosh Gupta Aug 27 '15 at 19:21
  • How do I then solve this problem? Newton's method? – Ashutosh Gupta Sep 01 '15 at 08:47
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    Local method (using Newton etc), or a global approach using spatial branch-and-bound or semidefinite relaxations. However, the problem is strange, as the second equation yields $W^{-1}=W^T$ which when you plug into the first $\Sigma W^T = W^T$ which only can be solved when $\Sigma = I$ – Johan Löfberg Sep 01 '15 at 12:53

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