I recently encountered an optimization problem and looking for some technical paper for the same.
The problem is give as below (For $ f \left( \cdot \right) $ which is Convex and $ r \left( \cdot \right) $ which is Concave),
$\min f(x)+\lambda*r(x) $
$\ s.t \ x \geq 0, ||x||_1 = 1$.
where $x$ is a n-dimensional vector. Here $\ x \geq 0$ implies each component of $x$ is greater than 0. The role of regularization term is to push the entries to zero while incrementing the other entries. In the end, as the $\lambda$ increases, the $x$ will have all zeros and only one 1. This paper solved the above optimization problem without regularization term.
Could anyone point me to any algorithm for solving the above problem?