# Standard name for the computation of Lagrange multipliers iteratively by fixing other multipliers?

Dear Optimization Experts,

Background:

I have a convex optimization problem on hand that can be shown in general form as given below \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} where both functions $$g_i: \mathbb{R}^N \rightarrow\mathbb{R}$$ and $$f: \mathbb{R}^N \rightarrow\mathbb{R}$$ are convex, and $$\alpha_i \in \mathbb{R}$$ is given.

The Lagrangian is: \begin{align} L\left(x, \left\{\lambda_i\right\}\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.

• Depending on how you select $\lambda$, this may be coordinate descent in the dual problem. – LinAlg Nov 18 '18 at 13:53
• thank you LinAlg! – user550103 Nov 18 '18 at 15:51