Maximin program to solve in matlab

Nonconvex
1

$$\eqalignno{ &\mathop{\max}{{X,y}}\ \mathop{\min}{i=1,2,…M}\ \varphi_{i}({ X}, y_{i}):={\langle{ l}{i}{ l}{i}^{H}{ Xh}{i}{ h}{i}^{H}{ X}^{H}\rangle\over y_{i}+\sigma_{{\rm de}}^{2}/\sigma_{s}^{2}}:&\hbox{(1a)}\cr&\qquad \qquad \qquad { P}{n}({ X})\leq\rho{n},\ n=1,2, \ldots, N,&\hbox{(1b)}\cr&\sum_{j\neq i}\langle { l}{i}{ l}{i}^{H}{ Xh}{j}{ h}{j}^{H}{ X}^{H}\rangle +{\sigma_{{\rm re}}^{2}\over\sigma_{s}^{2}} \langle { l}{i}{ l}{i}^{H}{ XX}^{H}\rangle\leq y_{i},\cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad i=1,2, \ldots, M,&\hbox{(1c)}}$$

where y=(y_1,y_2…y_M)^T and \varphi_{i}(X,y_i) is smooth and convex while all constraints (1b) and (1c) are convex. please help me for solving it using cvx.equation (1c) in LMI is written as:$$\eqalignno{& \left[\matrix{ {\sum_{j\neq i}^{M}h_{j}h_{j}^{H}+{\sigma_{{\rm re}}^{2}\over \sigma_{s}^{2}}I_{N}} & { \Bigg(\sum_{\neq i}^{M}h_{j}h_{j}^{H}+{\sigma_{{\rm re}}^{2}\over \sigma_{s}^{2}}I_{N})x^{H}{l}{i}}\cr {l{{\rm i}}^{H}X\Bigg(\sum_{j \neq {\rm i}}^{M}h_{j}h_{j}^{H}+{\sigma_{{\rm re}}^{2}\over \sigma_{s}^{2}}{I}{N})\ y{i}}}\right]\succeq 0, \cr &\qquad \qquad i=1,2, \ldots, M,&\hbox{(2)}}$$

2

The individual functions \varphi_i are indeed convex when y_i+\sigma^2_{de}/\sigma_s^2>0. But the minimum of convex functions is, in general, not convex. CVX cannot solve this problem for M>1.

3

If we decompose the objective function in terms of a minimization function. i.e min_{1,2...,M}\varphi_{i}(X,y)=f(X,y)-g(X,y) where f(X,y)= max_{1,2,....M}\sum_{j\neq I}^M \varphi_{j}(X,y_{i}) and g(X,y)=\sum_{I=1}^M\varphi_{i}(X,y_i)

4

then the optimization becomes min_{X,y=(y_1,y_2,....,y_M)^T}f(X,y) - g(X,y).subject to (1b) and (2). now it becomes the convex function.

5

g is not a concave function either. This simply is not convex. But regardless, it must adhere to the rules of the DCP ruleset or CVX cannot solve it even if it is convex. And you haven’t even shared what P(X) is.

6

P_{n}(X)= trace((\sigma_{s}^2HH^H+\sigma_{re}^2I_{N})X^H(n,.)X(n,.))

7

That can be made convex: if you let Q be a symmetric square root of \sigma_s^2HH^H+\sigma_{re}^2I, then P_n(X)\leq\rho_n becomes norm(X*Q,'fro') \lt= sqrt(rho(n)). That doesn’t help your other problems though.