I would like to solve a minimization problem of convex conjugates defined by the well-known Legendre-Fenchel transformation.
For example, a simple problem can be given as:
Minimize w.r.t. lambda Maximize w.r.t. x : lambda’x - 1/2norm(x)^2
subject to some interval constraint of lambda : a <= lambda <= b
This is actually a convex program w.r.t. lambda because the function of lambda:
"Maximize w.r.t. x : lambda’x - 1/2norm(x)^2 "
is the conjugate of quadratic functions, whose closed form is exactly given as 1/2*norm(lambda)^2.
However, CVX returns the error:
The second argument must be positive or negative semidefinite.
Is there any possible way to avoid this error?