I am trying to minimize the largest component of a vector $x = [x_1, x_2, x_3, x_4]$, where $x_1 \ge x_2 ... \ge x_4$, such that it satisfies a set of linear inequalities $A, b$ in the following way:

$$ Ax \le b. $$

Furthermore, I want that, the Shannon entropy of the vector $x$ satisfies the following:

$$ -\sum_i x_i \log_2(x_i) = q, $$ for some constant $q$.

I can write the following for the first constraint:

cvx_begin sdp

variable x(4, 1) 

minimize x(1)
subject to 
    A * x <= b


However, when I try to include the second constraint, like: quantum_entr(diag(x)) == q.

I get the following error:

Invalid constraint: {concave} == {real constant}

Is there a way to mix these two types of constraints in a semi-definite program? Thanks!

  • 2
    $\begingroup$ Semidefinite programs have only linear and semidefinite constraints. I don't see any semidefinite constraints so I'm not quite sure why this is tagged as an SDP. Also cvx has a function for the shannon entropy entr(). $\endgroup$
    – Rammus
    Commented Oct 26, 2021 at 21:12
  • $\begingroup$ I see, thanks Rammus. Is there a way around this? $\endgroup$ Commented Oct 26, 2021 at 21:18
  • 1
    $\begingroup$ Secondly the error you are getting from CVX is telling you that a constraint $f(x) = c$ where $f$ is a concave function is not a valid constraint. It violates the DCP ruleset. Basically your problem is not guaranteed to be a convex optimization problem (and it probably isn't), so cvx can't help you. You can't have in general a convex/concave function equality constraint. For example, consider a constraint $x^2 = 1$, then the feasible set is $x = \{-1, +1\}$ which is not a convex set. Note however that $x^2 \leq 1$ does lead a convex set. $\endgroup$
    – Rammus
    Commented Oct 26, 2021 at 21:22
  • 1
    $\begingroup$ You can relax your problem to $- \sum_i x_i \log x_i \geq q$ if you want to make it convex. $\endgroup$
    – Rammus
    Commented Oct 26, 2021 at 21:27
  • $\begingroup$ Thanks a lot, Rammus. Let me try this. $\endgroup$ Commented Oct 26, 2021 at 21:28


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