Cone product reformulation for global optimization

In this paper, we study nonconvex optimization problems involving sum of linear times convex (SLC) functions as well as conic constraints belonging to one of the five basic cones, that is, linear cone, second order cone, power cone, exponential cone, and semidefinite cone. By using the Reformulation Perspectification Technique, we can obtain a convex relaxation by forming the perspective of each convex function and linearizing all product terms with newly introduced variables. To further tighten the approximation, we can pairwise multiply the conic constraints. In this paper, we analyze all possibilities of multiplying conic constraints. Especially the results for the cases in which a power cone or an exponential cone is involved are new. Moreover, in case of an exponential cone we generate valid inequalities that can be used to further strengthen the approximation and in case of a power cone we generate additional valid inequalities. Numerical experiments on a quadratic optimization problem over exponential cone constraints and on a robust palatable diet problem over power cone constraints, demonstrate that including additional inequalities generated from the proposed pairwise multiplications improve the approximation. Moreover, when incorporated in branch and bound the global optimal solution of the original nonconvex optimization problem can often be obtained faster than BARON.



View Cone product reformulation for global optimization