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I'm trying to solve this problem, but I'm not sure if it is possible to check the optimality of this problem.

$$\min_{K,L} \quad Tr(L^\top L)\qquad\\ \text{s.t.} \quad K^\top L = A^\top Q\\ \qquad \qquad \qquad Q + A^\top Q A - K^\top K \succeq 0\\ \quad \quad Q - L^\top L \succeq 0$$

where A,Q are real $n\times n$ matrix, and Q is PD. Is it possible to generate optimal solution (K,L) as functions of (A,Q)? Additionally, is there conditions of solutions to guarantee the optimality of certain solution?

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  • $\begingroup$ Section 5.5 of web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf $\endgroup$ Nov 28, 2023 at 11:45
  • $\begingroup$ Thanks. But due to the first equality constraint, isn't this problem nonconvex and the KKT conditions do not guarantee the global optimality? $\endgroup$
    – Jisun Lee
    Nov 29, 2023 at 1:19
  • $\begingroup$ Yes, you are correct. The extended KKT conditions are necessary, but not sufficient in this case. $\endgroup$ Nov 29, 2023 at 2:11

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