Explicit Solutions for a General Class of Optimal Allocation Problems

We revisit the problem of minimizing a separable convex function with a linear constraint and box constraints. This optimization problem arises naturally in many applications in economics, insurance, and finance. Existing literature exclusively tackles this problem by using the traditional Kuhn-Tucker theory, which leads to either iterative schemes or yields explicit solutions only under some special classes of convex functions. Instead, we present a new approach of solving this constrained minimization problem explicitly by using the theory of comonotonicity. The key step is to apply an integral representation result to express each convex function as the expected stop-loss of some suitable random variable