Optimizing over coherent risk measures and non-convexities: a robust mixed integer optimization approach

Recently, coherent risk measure minimization was formulated as robust optimization and the correspondence between coherent risk measures and uncertainty sets of robust optimization was investigated. We study minimizing coherent risk measures under a norm equality constraint with the use of robust optimization formulation. Not only existing coherent risk measures but also a new coherent risk measure is investigated by setting a new uncertainty set. The norm equality constraint itself has a practical meaning or plays a role to prevent a meaningless solution, the zero vector, in the context of portfolio optimization or binary classification in machine learning, respectively. For such advantages, the convexity is sacrificed in the formulation. However, we show a condition for an input of our problem which guarantees that the nonconvex constraint is convexified without changing the optimality of the problem. If the input does not satisfy the condition, we propose to solve a mixed integer optimization problem by using the 1 or ∞-norm. The numerical experiments show that our approach has good performance for portfolio optimization and binary classification and also imply its flexibility of modelling that makes it possible to deal with various coherent risk measures.