Optimize problem without objective function

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I can find the optimal x by fixing B, u, v. However, after that, I try to fix x to get B, u, v. I do not have an objective function. What should I do in this case ? Can I freely create an objective functions ?
Thank you very much.

CVX does not require an objective function. In such case, the solver finds whatever feasible solution “it feels like”, and CVX returns that to the user (or the problem is declared infeasible, or the optimization fails due to numerical issues).

Whether your alternating variables scheme converges to anything, let alone a global, or even local minimum of the original problem, is another matter, and is out of scope of the forum,.