Cascade optimization including bilinear part inside

Dear all

Thank you very much for your time.


I formulate the problem as:

\max_{x\in\mathcal{X}} ~\{\lambda^{T}x ~-~\max_{y\in\mathcal{Y}} x^{T}y \},

where the inside part \mathcal{F}(x):=\max_{y\in\mathcal{Y}} x^{T}y is proved to be convex in x for each y\in\mathcal{Y}. (refers to Boyd’s book, example 3.7)

Then, the total objective function is one concave function in x.

However, I failed to apply cvx to solve the cascade optimization problem, because of the inner bilinear structure, I guess.

Could anybody show me some hint to solve this problem?