Consider the optimisation problem expressed in a crude form
$\max_{\mathbf{Q}}\sum w_ir_i$
where $w_i$ are constants, $r_i$ are concave functions of positive semidefinite matrix $\mathbf{Q}$ satisfying $\text{trace}[\mathbf{QA}]\leq P$ for some other positive semidefinite $\mathbf{A}$.
Given the objective function and the feasible region, the problem is obviously a convex problem. I studied about the concept of Lagrange and KKT multiplier applied to constraints expressed in terms of real valued functions. But for the positive definite constraint on $\mathbf{Q}$, is it possible to attach a KKT multiplier with it, as if $\mathbf{Q}$ is a real number? According to some articles, it's possible. But any explanation on this concept of treating positive definite matrices as positive numbers and why is this justified, which, I assume is part of a more generalised KKT condition?
P. S. The problem is part of my research problem and the exact function isn't important here. All I need is an explanation of using KKT condition on $\mathbf{Q}$.
Best Answer
Have a look for SDP (semidefinite programming).
Your problem is \begin{align*} \text{Minimize} \quad &f(Q)\\ \text{such that} \quad & Q \succeq 0 \\ \text{and} \quad &\mathrm{trace}(Q \, A ) \le P\end{align*} You can associate the Lagrangian $$ L(Q, R, \lambda) = f(Q) + \mathrm{trace}(Q \, R) + \lambda \, \mathrm{trace}(Q \, A ), $$ for $R \preceq 0$ and $\lambda \ge 0$. If a constraint qualification (CQ) is satisfied, the first order optimality conditions are \begin{align*}L_Q = f'(Q) + R + \lambda \, A &= 0\\ \mathrm{trace}(Q \, R) &= 0\\ \lambda \, \mathrm{trace}(Q \, A ) &= 0 \end{align*} That is, the positive-semidefinite constraint gets a negative-semidefinite multiplier $R$, satisfying the complementary slackness condition $\mathrm{trace}(Q \, R) = 0$. This is similar to the $\le$ constraint, which gets a positive multiplier.