How to approach this minmax problem (nonconvex)

Question

This is my first time to solve the minmax problem, I also read the following discussion:

My problem is

\begin{equation} \begin{aligned} & {\underset{w_i,\ \ \ \ \ x_i}{\max\min}} & & w_1(x_2-x_1)^2+w_2(x_3-x_2)^2 \\ & \text{s.t.} & & w_1+w_2=1 \\ & & & x_1+x_2+x_3 = 0\\ & & & x_1^2+x_2^2+x_3^3 = 1 \end{aligned} \end{equation}

Assume I know how to solve the problem if $w_i$ for all $i$ are given. Then is there any suggested methods to solve this problem based on this assumption.

What sort of guarantees are you looking at? Convergence to globally optimal solutions? Convergence to stationary points? — madnessweasley, Jun 12 '19 at 10:09
Convergence to globally optimal solution. This problem admits a globally optimal solution. — sleeve chen, Jun 12 '19 at 20:01

score 1 · Accepted Answer · answered Jun 13 '19 at 01:19

One approach for solving the saddle point problem $$\underset{w \in W}{\max} \: \underset{x \in X}{\min} \: f(w,x)$$ to global optimality is to reformulate it as the semi-infinite program

\begin{align} \underset{w \in W, \: z \in \mathbb{R}}{\max} \:\: & z \\ \text{s.t.} \quad & z \leq f(w,x), \quad \forall x \in X. \end{align}

Since your problem is affine in the space $(w,z)$, there are several cutting plane algorithms for solving this problem assuming that, for a given $(\bar{w},\bar{z}) \in W \times \mathbb{R}$, you can solve the subproblem $\underset{x \in X}{\max} \left\lbrace \bar{z} - f(\bar{w},x) \right\rbrace$ to global optimality. One summary reference is here.

How did you say that my problem is affine in the space $(w,z)$? Thanks! — sleeve chen, Jun 13 '19 at 02:35
Well, I meant the function $z - f(w,x)$ is affine in the variables $w$ and $z$ for each fixed $x \in X$ plus $W \times \mathbb{R}$ is a polyhedral set — madnessweasley, Jun 13 '19 at 02:38

How to approach this minmax problem (nonconvex)

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