Abstract
Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality constraints are relaxed, a series of linear relaxation subproblems with auxiliary variables and bound constraints are generated, which can determine the effective lower bound of the global optimal value of QCQP. To enhance the compactness of sub-rectangles and improve the ability to remove sub-rectangles, two rectangle-reduction strategies are employed. Besides, two ϵ-subproblem deletion rules are introduced to improve the convergence speed of the algorithm. Therefore, a relaxation and bound algorithm based on auxiliary variables are proposed to solve QCQP. Numerical experiments show that this algorithm is effective and feasible.
Highlights
If all the matrices Qs, s = 0, 1, 2, · · ·, N are semi-positive definite, Quadratically constrained quadratic programs (QCQP) is a convex quadratic program with convex quadratic constraints (CQPCQC), which can be reconstructed into a second-order cone programming problem (SOCP) that can be solvable in polynomial time [14,15]
After the equality constraint is relaxed, a linear relaxation subproblem with auxiliary variables and bounded constraints is generated for QCQP, and the lower bound of the global optimal value of QCQP is determined
By embedding the two rectangular reduction algorithms given in the previous section into the branch-and-bound scheme, we develop a new global optimization algorithm for solving QCQP
Summary
QCQP is NP-hard, and its complexity is mainly reflected in the non-convexity of quadratic objective function and the feasible region with non-connectivity Most of these problems cannot be solved in polynomial time, and it is difficult to search for its global optimal solution. In [20,32], two branch-and-bound algorithms with linear relaxation, which are based on the method of dividing the feasible region, can solve QCQP globally. By adopting the quadratic proxy function, literature [33] convexifies all quadratic inequality constraint functions, and presents a new algorithm based on the quadratic convex reconstruction method All these methods mentioned above can solve QCQP and its variants well.
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