# Why can we rewrite the optimization problem (modify “max-min” to “min-max” )

I saw this paper,and it said to make the first problem "tractable" ,so we have to modify the first problem to the second problem

## First problem:

$$\max\limits_{\{\mathbf v\},\rho_k}\min\limits_{\mathbf h_k \in H_k} SINR_k(\{\mathbf v_k\},\rho_k)$$

s.t.

$$\zeta(1-\rho_k)(\sum\limits_{j=1}^{K}|\mathbf h_k^H \mathbf v_j|^2+\sigma^2_k) \ge e_k,\forall k$$

$$\sum\limits_{k=1}^{K}tr(\mathbf D_m \mathbf v_k \mathbf v_k^H) \le P_m,\forall m$$

$$0 \le \rho_k \le 1,\forall k$$

the paper said is non-convex due to coupled variables {ρk} and {vk} in both the objective function and the EH constraint, and thus, is difﬁcult to solve efﬁciently.

## Second problem

$$\min\limits_{\{\mathbf v\},\rho_k}\max\limits_{1 \le m \le M} \frac{\sum\limits_{k=1}^{K}tr(\mathbf D_m \mathbf v_k \mathbf v_k^H )}{P_m}$$

s.t.

$$\zeta(1-\rho_k)(\sum\limits_{j=1}^{K}|\mathbf h_k^H \mathbf v_j|^2+\sigma^2_k) \ge e_k,\forall k$$

$$SINR_k(\{\mathbf v_k\},\rho_k) \ge \Gamma,$$

$$0 \le \rho_k \le 1,\forall k$$

the paper said To make problem (3) tractable, we decompose the problem into a set of the min-max per-DA port power balancing problems, one for each given SINR target Γ > 0 . Using bisection search over Γ

I want to ask that what does that "tractable" mean?nonconvex?hard to calculate?can't be calculate?why can we just want to make the first problem "trackble" ,then modify the objective function from max-min to min-max?

I don't know why will we rewrite the first formula to the second formula,because it seems that the objected functions of two problem are inverse proportion,so the max-min will rewrite to min-max,but i don't know why will they be inverse proportion

• The relationship between the two problems is not clear. Sometimes such statements mean "we don't know how to solve the original problem, so let's solve this other one instead". – copper.hat Jun 27 at 5:29
• @copper.hat yes i know,but i don't know why will we rewrite the first formula to the second formula,because it seems that the objected functions of two problem are inverse proportion,so the max-min will rewrite to min-max,but i don't know why will they be inverse proportion – XM551 Jun 27 at 5:41