Pow_pos( inv_pos(A) + inv_pos(B), 2 )

1

the constraint: A, B, and V are variables, C is a complex number.

V <= A^(-2) + B^(-2) + Re(C) * (AB)^(-1);

Relax the constraint as follows,
1.  A^(-2) >= a = 2A0 A - A0^(-2);
    B^(-2) >= b = 2B0 B - B0^(-2);

2.  Re(C) > 0, Flag = 1;
     (AB)^(-1) >= P1 =3(A0 B0)^(-1) - A0^(-2) B0^(-1) A - A0^(-1) B0^(-2) B;
    Re(C) < 0, Flag = 0;
     (AB)^(-1) <= N1 =0.5[ (A^(-1) + B^(-1))^2 + 2A0^(-3)A + 2B0^(-3)B - 3A0^(-2) - 3B0^(-2) ];

Then,

V <= a + b + Re(C) * ( Flag*P1 + (1-Flag)*N1 );

And I express N1 in CVX as follows, is it right ?

E = inv_pos(A) + inv_pos(B);
N1 = pow_pos( E, 2 );

2

1/A^2 is convex (for A>0), not concave, so you will not be able to model this inequality.

3

Yes, it is. But 1/A^2 is on the right of the unequality.

4

V\leq 1/A^2 + \ldots is not convex and not possible.

V\geq 1/A^2 + \ldots could be convex and could be possible.

5

Yes, so when V< 1/A^2, I reformulate 1/A^2 as:
1/A^2<a =-2A/A_0^3+3/A_0^2 ,A_0 is the feasible point of A.

6

For N_1, is it right in CVX?

E = inv_pos(A) + inv_pos(B);
2N1 = pow_pos(E,2) + 2A/(A0)^3 - 3/(A0)^2+ 2B/(B0)^3 - 3/(B0)^2;

A_0,B_0 are the feasible point of A,B,respectively.

Re(C)>0, Flag = 1,
1/(AB) > P_1=3/(A_0B_0)-A/(A_0^2B_0) - B/(A_0B_0^2)

Re(C)<0, Flag = 0,
1/(AB)<N_1 = 0.5[ (1/A+1/B)^2 + 2A/A_0^3-3/A_0^2 + 2B/B_0^3-3/B_0^2]

7

For N_1, is it right in CVX?

E = inv_pos(A) + inv_pos(B);
2N1 = pow_pos(E,2) + 2A/(A0)^3 - 3/(A0)^2+ 2B/(B0)^3 - 3/(B0)^2;

A_0,B_0 are the feasible point of A,B,respectively.

Re(C)>0, Flag = 1,
1/(AB) > P_1=3/(A_0B_0)-A/(A_0^2B_0) - B/(A_0B_0^2)

Re(C)<0, Flag = 0,
1/(AB)<N_1 = 0.5[ (1/A+1/B)^2 + 2A/A_0^3-3/A_0^2 + 2B/B_0^3-3/B_0^2]

8

It would be less confusing if you did not spread one question across multiple forum topics.

Does my answer to your other topic inv_pos(A)+inv_pos(B) (moderator note: the expression in the title is convex, but the problem in the question is not) - #2 by Mark_L_Stone r resolve this question?

9

Yes,Thank you!
I find my formulation wasn’t written well, so I create another topic .