DC (Difference of Convex) programming and DCA (CD Algorithms) for globally solving the value-at-risk

The value-at-risk is an important risk measure that has been used extensively in recent years in portfolio selection and in risk analysis. This problem, with its known bilevel linear program, is reformulated as a polyhedral Difference of Concave (DC) program with the help of exact penalty techniques in DC programming and solved by a Difference of Convex Algorithm (DCA). To check globality of computed solutions, a global method combining the local algorithm DCA with a well adapted branch-and-bound algorithm is investigated. An illustrative example and numerical simulations are reported, which show the robustness, the globality and the efficiency of DCA.