An Efficient Splitting Algorithm for Solving the CDT Subproblem

The CDT subproblem, which minimizes a quadratic function over the intersection of two ellipsoids, is a classical quadratic programming problem. In this paper, we study a method of solving CDT with a positive Lagrangian duality gap. An efficient splitting algorithm is proposed for finding the global optimal solutions. A cutting plane is firstly added to divide the feasible set of CDT into two subsets, and then two new quadratic programming problems with ellipsoidal and linear constraints are generated accordingly. Using the newly developed technique - second-order cone constraints to enhance the efficiencies of the SDP relaxation-based algorithms on the two subproblems, an optimal solution of CDT can be acquired by comparing the objective values of the two subproblems. Numerical experiments show that the new algorithm outperforms the two recent SDP relaxation-based algorithms, the two-parameter eigenvalue-based algorithm and the solver Gurobi 9.5 for certain types of CDT.