Local Optimization of Nonconvex Mixed-Integer Quadratically Constrained Quadratic Programming Problems

This paper develops a heuristic relaxation and iterative search combined algorithm to search for a local optimum of the nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) problem. Inspired by the branch and bound method, a heuristic relaxation approach is proposed to relax a MIQCQP problem as a mixed-binary QCQP problem that can be equivalently converted into a continuous/general QCQP. Next, to efficiently solve a general QCQP, an iterative optimization algorithm combined with an intersection cutting plane method is developed, where each iteration is formulated as a second-order cone programming problem. The proposed algorithm will guarantee global convergence to a local optimum of the original MIQCQP problem. Numerical examples are provided and compared to the state-of-the-art method to verify the effectiveness and efficiency of the proposed algorithm.