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Let $X, Y \in \mathbb{R}^{n\times n}$. How to deal with the constraint $X^T X - Y^T Y \geq 0$ in CVX? Thanks very much!

Ryan
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1 Answers1

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Whether you mean that inequality element-wise or in the semi-definite sense, it is ono-convex either way. So CVX can't be used for this problem.

Please read the CVX FAQ

Mark L. Stone
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  • Thanks very much for your answer, and I did understand sometimes the problem can be non-convex. But I am not sure if this case can be converted to a convex one. – Ryan Nov 10 '22 at 09:28
  • I both $X$ and $Y$ are variables, that constraint is inherently non-convex and can't be converted to be convex. You could consider using YALMIP, https://yalmip.github.io/tutorial/basics/ which with the right choice of solver, allows non-convex objective function and constraints. – Mark L. Stone Nov 10 '22 at 10:55
  • Thanks Mark, are there any iteration-based methods to solve that (like cone complementarity linearization)? – Ryan Nov 10 '22 at 11:02
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    You need to first clarify whether you mean hat inequality element-wise or in the semi-definite sense. If the former, it is non-convex quadratic nonlinear) inequality constraint,. If the latter, it is a BMI (Bilinear Matrix Inequality) constraint, which is non-convex, and is a special case of nonlinear SDP, which is much more difficult than Linear SDP; and is not in scope of Mosek or other convex conic solvers. In either case, you can try YALMIP's BMIBNB global optimization solver. If the former, Gurobi and BARON could be used, depending on what's in the test of the problem formulation. – Mark L. Stone Nov 10 '22 at 11:11
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    FMINCON or KNITRO under YALMIIP are possibilities if you are content with a local, not necessarily global optimum. Whether you can get a useful solution within a timeframe you are prepared to accept, depends on the problem size and other aspects of the problem difficulty. – Mark L. Stone Nov 10 '22 at 11:13
  • Thanks so much for your detailed reply. I will have a try :-) – Ryan Nov 10 '22 at 11:45
  • Typo in my previous comment: Meant " If the former, it is non-convex quadratic (bilinear) inequality constrain" – Mark L. Stone Nov 10 '22 at 11:53