How CVX can be applied in Game theory?How to write two optimizations as a part of a single CVX?

1

Suppose the optimization variable is x, and we need to find \max f(x) and \min g(x). Note that f and g are the linear function of x. Any suggestion how to proceed?

2

Please state your complete problem mathematically.

3

Let 0<x<10 be the optimization variable of size 1\!\times\!N.

f(x) = \text{average} \big(\sum_{k=1}^{K} Ax - Bx\big) and g(x) = \text{sum} (Cx), where K, A, B, \text{and }C are scalar constant.

Now, my optimization problem is to find \max f(x) and \min g(x) at the same time.

4

I have no idea what your problem is. I have some familiarity with mini-max problems in game theory

In your case, i don’t know what you mean. g(x) is not even a scalar. And I don’t think f(x) is either, although it is too vaguely written for me to be sure.

When you have a well-formulated CONVEX optimization problem, and you need help entering it into CVX, readers on this forum might be able to help you. But if you need help formulating a well-defined convex problem, this forum is not the right place.

5

Edited the above. Kindly have a look.

 cvx_begin
 variable x(1,N)
 maximize(mean(sum(A*x)-B*x))
 minimize(sum(C*x))
 cvx_end
6

That is not a convex optimization problem. You have two separate optimization problems. And unless all the coefficients are zero, they will both be unbounded.