I have the following maximization problem:

$$\max_{\mathbf{w}, t, \Theta} t$$

$$\text{s.t. } ||\mathbf{w}||^{2} \leq P$$

$$\mathbf{S} \succ 0$$

$$||\mathbf{u}^{(n)}||+2\sum_{i=1}^{2} \mathbf{u}^{(n)} [\mathbf{u}(i) - \mathbf{u}^{(n)}(i)] \geq t$$

$$|\theta_{n}| = 1 \text{ } \forall n = 1,...,M$$

I want to reformulate it into a minimization problem so that I can be able to use the Manifold optimization algorithm to tackle the unit modulus constraints. I am aware that we could employ the exact penalty method as defined in Boyd's book. However, I am struggling to understand how to formulate the penalty parameter. May somebody kindly help me with understanding the exact penalty method. Alternatively, I am also open to other methods that can be used for the reformulation of the problem.

  • 6
    $\begingroup$ Change it to min -t subject to the same constraints. Now it's a minimization problem, very much a non-convex minimization problem. Whether a manifold optimization algorithm, such as available through Manopt, will work out well, or at all, is another matter. $\endgroup$ – Mark L. Stone Feb 20 at 18:26

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