| A parameterized eigenvalue strategy is applied to the lower-level algorithm of Quadratically Constrained Quadratic Programs (QCQP). This method recasts the trust-region subproblem in terms of a parameterized eigenvalue problem. In each step of the lower-level algorithm, only the smallest eigenvalue and the corresponding eigenvector of the bordered matrix B α need to be computed.; A constrained quasi-Newton-like procedure is employed to approximate the inverse Hessian of the dual objective function in the upper-level algorithm of QCQP, which can be used to update the Lagrange multipliers. Thus, we can reduce the computational burden.; The role of the spherical constraint is discussed and it is important in determining stepsizes is emphasized. From this discussion, we note the importance of generating solutions of the lower-level algorithm that are always on the boundary of the sphere. The modified QCQP algorithm employs changeable radius of the spherical trust-region constraint, which allow it to handle non-convex problems with duality gaps. The trust-region is expanded by increasing the radius of the spherical constraint when the QCQP model is tested to be a good approximation of the original problem. Conversely, if the approximations are bad, the radius is reduced. Some where in between, the same radius size is maintained. This helps reduce the number of iterations of the QCQP algorithm and save computation cost.; Put all the pieces together to form a successive QCQP algorithm that suitable for solving highly nonlinear optimization problems. The proposed successive QCQP finds a feasible solution of the highly nonlinear program using the lower-level algorithm of QCQP and improves the feasible solution toward optimum using the upper-level algorithm. The proposed successive QCQP is tested using several test problems. Results so far indicated that it handles highly nonlinear programs very well as intended. Also as expected, results show that it is not suitable for slightly-nonlinear or linear problems. |
