Optimization problem in matlab

Nonconvex
1

objective function: min_{X} F(X)

where, A&C&Dis a matrix of N\times N and X is matrix of N\times N and F(X)=trace$(AX^HX)$

how this problem is solved using cvx?

consider, constraint as : V(X)=trace$(CXDX^H)/(trace(CXX^H)+\sigma) >=\alpha$
this is a non convex quadratic constraint.
So,it can be changed to convex quadratic as

Y>=\alpha*(trace(CXX^H)+\sigma) and Y=trace(CXDX^H)
now the constraint become convex quadratic in (X,Y)

sir, I studied in a paper that $$\eqalignno{ &\sigma_{s}^{2}{\bf Y}(i, i)\geq\cr&\alpha_{i}\Bigg(\sigma_{s}^{2}\sum_{j\neq i}\langle { l}{i}{ l}{i}^{H}{ Xh}{j}{ h}{j}^{H}{ X}^{H}\rangle+\sigma_{{\rm re}}^{2}\langle { l}{i}{ l}{i}^{H}{ XX}^{H}\rangle+\sigma_{{\rm de}}^{2}\Bigg),\cr&\quad i=1,2, \ldots, { M}. }$$ is a convex quadratic in (X,Y). where Y(i,i) plays the role of trace(l_il_i^HXh_ih_i^HX^H).

Is this a wrong statement? And if it is a right statement, thn how this constraint is written in cvx?
the constraint can be written as:

$$\eqalignno{ &\sigma_{s}^{2}{\bf Y}(i, i)-\cr&\alpha_{i}\Bigg(\sigma_{s}^{2}\sum_{j\neq i}\langle { l}{i}{ l}{i}^{H}{ Xh}{j}{ h}{j}^{H}{ X}^{H}\rangle+\sigma_{{\rm re}}^{2}\langle { l}{i}{ l}{i}^{H}{ XX}^{H}\rangle \Bigg)\geq \alpha_{i}*\sigma_{{\rm de}}^{2}\cr\quad i=1,2, \ldots, { M}. }$$

2

This problem is very likely non-convex, so CVX cannot solve it.

3

sir how its a non convex. here F(X) is a convex quadratic function.

4

Yes, if A is PSD. But your inequality is nonconvex.

5

if considering inequality as a convex by assuming anything, then how we can solve it. mainly I want to know to write objective function of such type (i.e X^HX)in cvx.

6

You are going about this entirely backwards. If you cannot prove this is convex, then you shouldn’t even be considering CVX. Prove to me that this model is convex first; then we will discuss how to express it in CVX. But as written, I am 99% certain it is not.

7

(Edit your question to provide the additional information needed to establish that this model is convex.)

8

Sir ,I have edited the question.

9

Y=trace(CXDX^H) is not convex in (X,Y). Example: Imagine N=1, C=D=1. Then y=x^2 is not a convex set.

10

sir , N>1 and C and D is a N$\times$N matrix C=b*b^H (b is a N$\times$1 matrix), similarly as D also.

11

In that case, trace(CXDX^H) is indeed convex, but the constraint is not, because it is on the wrong side of the inequality. This remains a non-convex problem.

12

I see you’ve re-written your constraint; but I remain unconvinced that it is convex. How do you know it is convex? It is never a good sign that there are quadratic terms on the left-hand side of \geq.

13

yes sir, I also don’t know but, it is given in that the constraint is convex…

14

Sir, I want to know why it is not a convex function.?

15

The question isn’t whether it’s a convex function but a convex constraint. You have a convex quadratic on the left-hand side of \geq. That usually results in a non-convex constraints; and it always violates CVX’s rules.

16

You also have a quadratic term on the right-hand side of the \geq that does not appear to be convex.