Study On Global Optimization Algorithm For Optimal Deleveraging Problem

After the 2008 financial crisis,people have paid more attention to the harmful effects of excessive leverage for the health and development of financial institutions.Companies and investors in a wide range of industries have been beginning the process of deleveraging.In this paper,we consider the optimal deleveraging model that investors maximize the net assets of the portfolio under the provisions of debt/equity limit.We also consider investors not allowing short-selling and buying a new asset,and the asset price subjects to the influence of price shocks such as permanent and temporary price impact,the optimal deleveraging problem can be reduced to a nonconvex quadratic optimization problem with a quadratic constraint and box constraint.Generally,the Lagrange method is not always able to obtain the global optimal solution of the optimal deleveraging problem.In this paper,we first propose a successive convex optimization algorithm for the optimal deleveraging problem and show that the proposed algorithm converges to a KKT point of the original problem.By combining the successive convexoptimization technique and the Lagrange breakpoint algorithm,we present an improved Lagrange algorithm for the deleveraging problem.Computational results show that the improved Lagrange algorithm will obtain a better solution when the Lagrange algorithm finds a sub-optimal solution.Second,by combining the successive convex optimization algorithm and quadratic convex relaxation technique,we present a new branch-and-bound algorithm to find the global optimal solution of the optimal deleveraging problem,where the upper bound is obtained by the successive convex optimization algorithm and the lower bound is computed by the quadratic convex relaxation.Then we prove the global convergence of the new global algorithm.Computational results indicate that the new global algorithm can effectively find the global optimal solution of the optimal deleveraging problem.