Vrftjkry

Dear Optimization Experts,

Background:

I have a convex optimization problem on hand that can be shown in general form as given below
begin{equation}
begin{aligned}
& underset{x in mathbb{R}^N}{text{minimize}}
& & f(x) \
& text{subject to}
& & g_i(x) - alpha_i leq 0 forall i = 1,cdots,K ; ,
end{aligned}
end{equation}
where both functions $g_i: mathbb{R}^N rightarrowmathbb{R}$ and $f: mathbb{R}^N rightarrowmathbb{R}$ are convex, and $alpha_i in mathbb{R}$ is given.

The Lagrangian is:
begin{align}
Lleft(x, left{lambda_iright}right)
&= f(x) + sum limits_{i=1}^{K} lambda_i left(g_i(x) - alpha_i right) ; .
end{align}

Question:

If $K=1$ then I can obtain the closed-form solution $x$ (and analytical solution of the Lagrange multiplier $lambda_1$) by following the KKT conditions.

Now, the question arises when $K > 1$ then I can't obtain the closed-form solution, but I can compute $x$ analytically which is dependent on all the $lambda_i$. So, to compute the Lagrange multiplier say $lambda_i$, I resort to iterative solution where I fix other Lagrange multipliers ($lambda_j forall j = 1,cdots,K$ except $i$). Then repeat the above process for other Lagrange multipliers iteratively.

• do you have any standard name for such scheme to compute Lagrange multipliers cyclically?


• If not, can I say that this cyclic/iterative scheme is nothing but Coordinate Descent (or like)?

Thank you so much for your time in advance.