Quadratic Convergence of a Primal-Dual Interior Point Method for Degenerate Nonlinear Optimization Problems
- 108 Downloads
Recently studies of numerical methods for degenerate nonlinear optimization problems have been attracted much attention. Several authors have discussed convergence properties without the linear independence constraint qualification and/or the strict complementarity condition. In this paper, we are concerned with quadratic convergence property of a primal-dual interior point method, in which Newton’s method is applied to the barrier KKT conditions. We assume that the second order sufficient condition and the linear independence of gradients of equality constraints hold at the solution, and that there exists a solution that satisfies the strict complementarity condition, and that multiplier iterates generated by our method for inequality constraints are uniformly bounded, which relaxes the linear independence constraint qualification. Uniform boundedness of multiplier iterates is satisfied if the Mangasarian-Fromovitz constraint qualification is assumed, for example. By using the stability theorem by Hager and Gowda (1999), and Wright (2001), the distance from the current point to the solution set is related to the residual of the KKT conditions.
By controlling a barrier parameter and adopting a suitable line search procedure, we prove the quadratic convergence of the proposed algorithm.
Unable to display preview. Download preview PDF.
- 1.R.H. Byrd, G. Liu, and J. Nocedal, “On the local behaviour of an interior point method for nonlinear programming,” in Numerical Analysis 1997, D.F. Griffiths, D.J. Higham, and G.A. Watson (Eds.), Longman, 1998, pp. 37–56.Google Scholar
- 2.G.Di Pillo, S. Lucidi, and L. Palagi, “A superlinearly convergent primal-dual algorithm for constrained optimization problems with bounded variables,” Technical Report 02-99, Dipartimento di Informatica e Sistemistica, Universita di Roma “La Sapienza,” Roma, Italy, 1999.Google Scholar
- 3.A.S. El-Bakry, R.A. Tapia, T. Tsuchiya, and Y. Zhang, “On the formulation and theory of the Newton interior-point method for nonlinear programming,” Journal of Optimization Theory and Applications, vol. 89, pp. 507–541, 1996.Google Scholar
- 4.A.S. El-Bakry, R.A. Tapia, and Y. Zhang, “On the convergence rate of Newton interior-point methods in the absence of strict complementarity,” Computational Optimization and Applications, vol. 6, pp. 157–167, 1996.Google Scholar
- 10.H.J. Martínez, Z. Parada, and R.A. Tapia, “On the characterization of Q-superlinear convergence of quasi-Newton interior-point methods for nonlinear programming,” Boletín de la Sociedad Matemática Mexicana, vol. 1, pp. 137–148, 1995.Google Scholar
- 15.S.J. Wright, “Modifying SQP for degenerate problems,” Technical Report ANL/MCS-P699-1097, Mathematics and Computer Science Division, Argonne National Laboratory, Illinois, USA, Oct. 1997.Google Scholar
- 18.S.J. Wright and D. Orban, “Properties of the log-barrier function on degenerate nonlinear problems,” Technical Report ANL/MCS-P772-0799, Mathematics and Computer Science Division, Argonne National Laboratory, Illinois, USA, Sept. 1999.Google Scholar
- 19.H. Yabe and H. Yamashita, “Q-superlinear convergence of primal-dual interior point quasi-Newton methods for constrained optimization,” Journal of the Operations Research Society of Japan, vol. 40, pp. 415–436, 1997.Google Scholar
- 20.H. Yamashita, “A globally convergent primal-dual interior point method for constrained optimization,” Optimization Methods and Software, vol. 10, pp. 443–469, 1998.Google Scholar