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 - 1/2*norm(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 - 1/2*norm(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:

cvx/quad_form (line_230)

The second argument must be positive or negative semidefinite.

Is there any possible way to avoid this error?