Convergence of the BFGS-SQP Method for Degenerate Problems

In this paper, we propose a BFGS (Broyden–Fletcher–Goldfarb–Shanno)-SQP (sequential quadratic programming) method for nonlinear inequality constrained optimization. At each step, the method generates a direction by solving a quadratic programming subproblem. A good feature of this subproblem is that it is always consistent. Moreover, we propose a practical update formula for the quasi-Newton matrix. Under mild conditions, we prove the global and superlinear convergence of the method. We also present some numerical results.

[1]  Klaus Schittkowski,et al.  Test examples for nonlinear programming codes , 1980 .

[2]  Richard H. Byrd,et al.  Practical Update Criteria for Reduced Hessian SQP: Global Analysis , 1999, SIAM J. Optim..

[3]  Jacques Gauvin,et al.  A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming , 1977, Math. Program..

[4]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[6]  D. Mayne,et al.  Exact penalty function algorithm with simple updating of the penalty parameter , 1991 .

[7]  Ya-Xiang Yuan,et al.  A recursive quadratic programming algorithm that uses differentiable exact penalty functions , 1986, Math. Program..

[8]  J. J. Moré,et al.  A Characterization of Superlinear Convergence and its Application to Quasi-Newton Methods , 1973 .

[9]  Masao Fukushima,et al.  A successive quadratic programming algorithm with global and superlinear convergence properties , 1986, Math. Program..

[10]  P. Boggs,et al.  On the Local Convergence of Quasi-Newton Methods for Constrained Optimization , 1982 .

[11]  Stephen J. Wright.  Modifying SQP for Degenerate Problems , 2002, SIAM J. Optim..

[12]  J. Nocedal,et al.  A tool for the analysis of Quasi-Newton methods with application to unconstrained minimization , 1989 .

[13]  John E. Dennis,et al.  On the Local and Superlinear Convergence of Quasi-Newton Methods , 1973 .

[14]  Stephen J. Wright.  Superlinear Convergence of a Stabilized SQP Method to a Degenerate Solution , 1998, Comput. Optim. Appl..

[15]  Xiaojun Chen,et al.  Convergence of the BFGS Method for LC 1 Convex Constrained Optimization , 1996 .

[16]  Masao Fukushima,et al.  On the Global Convergence of the BFGS Method for Nonconvex Unconstrained Optimization Problems , 2000, SIAM J. Optim..

[17]  Jorge Nocedal,et al.  An analysis of reduced Hessian methods for constrained optimization , 1991, Math. Program..