Nonconvex quadratic inventory control problems by conic optimization

Inventory control is to find a optimal strategy to minimum the cost or maximum the profit. There have been numerous research on inventory control with the linear or other convex objective functions. However, the nonconvex cases are the more frequently occurring in practice, which are more difficult to handle than the convex one. In this paper, a class of inventory control with nonconvex quadratic objective is considered. The Lagrange decomposition method is used for generating convex relaxations for the nonconvex quadratic inventory control problem. We show that the best decomposition can be identified by solving an semidefinite problem(SDP) through a network flow problem on a directed acyclic graph and the totally unimodular(TUM) structure. For practical large-scale inventory control, a second order cone problem(SOCP) is constructed to overcome the computational difficulty of SDP, which sacrifices a little tightness. Besides, the nonconvex linear-quadratic(LQ)control is discussed as an extension. The optimal solution of the nonconvex LQ can be computed by a conic relaxation problem.