Product of Optimization Variables

1

I have an optimization problem in CVX MATLAB:

min t
subject to
f(x) <= w^2zeta + wbeta - t*w
w >= 0
trace(x) <= P,

where t, w, and x are optimization parameters, and beta and zeta are positive constant scalars. x has to be a complex semidefinite matrix and f(x) is a convex function of x. The product t*w makes the problem non-convex. How could I convexify the problem?

2

If you have an upper bound on w, you could “grid” w For example w=0:.1:10; and make w input data to CVX rather than declaring as a variable. The pick the value of w which gives the smallest optimal objective value. You could grid in the vicinity of the nest w from the first gridding, and do a finer girdding.