Nonconvex sensitivity-based generalized Benders decomposition

This paper considers general separable pseudoconvex optimization problems with continuous complicating variables in which primal and projected problems are both pseudoconvex problems. A novel decomposition method based on generalized Benders decomposition, named nonconvex sensitivity-based generalized Benders decomposition, is developed and proved strictly to obtain optimal solutions of general separable pseudoconvex optimization problems of interest without constructing surrogate models. By the use of a reformulation strategy (introducing an extra equality constraint and constructing several subproblems), the algorithm handles the nonconvexity by direct manipulations of consistent linear Benders cuts and the check of optimality conditions and approximating the feasible region of complicating variables by supporting hyperplanes. The master problems of the new algorithm are always linear programming problems and the solution of the algorithm contains sensitivity information about complicating variables. Moreover, the new algorithm could also be used as a tool to check the nonconvexity of an optimization problem. Two cases are given to confirm the validity and applicability of the proposed algorithm.