SUPERLINEARLY CONVERGENT EXACT PENALTY PROJECTED STRUCTURED HESSIAN UPDATING SCHEMES FOR CONSTRAINED NONLINEAR LEAST SQUARES: ASYMPTOTIC ANALYSIS

We present a structured algorithm for solving constrained non- linear least squares problems, and establish its local two-step Q-superlinear convergence. The approach is based on an adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method of Coleman and Conn for nonlinearly constrained optimization problems. The structured adaptation also makes use of the ideas of Nocedal and Overton for handling quasi-Newton updates of projected Hessians. We discuss the comparative results of the test- ing of our programs and three nonlinear programming codes from KNITRO on some randomly generated test problems due to Bartels and Mahdavi-Amiri. The results indeed conrm the practical signicance of our special considera- tions for the inherent structure of the least squares.

[1]  T. Pietrzykowski.  An Exact Potential Method for Constrained Maxima , 1969 .

[2]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

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

[4]  W. Murray,et al.  Projected Lagrangian Methods Based on the Trajectories of Penalty and Barrier Functions. , 1978 .

[5]  M. J. D. Powell,et al.  THE CONVERGENCE OF VARIABLE METRIC METHODS FOR NONLINEARLY CONSTRAINED OPTIMIZATION CALCULATIONS , 1978 .

[6]  John E. Dennis,et al.  An Adaptive Nonlinear Least-Squares Algorithm , 1977, TOMS.

[7]  Andrew R. Conn,et al.  Nonlinear programming via an exact penalty function: Global analysis , 1982, Math. Program..

[8]  Andrew R. Conn,et al.  Nonlinear programming via an exact penalty function: Asymptotic analysis , 1982, Math. Program..

[9]  T. Coleman,et al.  On the Local Convergence of a Quasi-Newton Method for the Nonlinear Programming Problem , 1984 .

[10]  R. Bartels,et al.  On generating test problems for nonlinear programming algorithms , 1986 .

[11]  Richard H. Bartels,et al.  Constrained nonlinear least squares: an exact penalty approach with projected structured quasi-Newton updates , 1989, TOMS.

[12]  J. Dennis,et al.  Convergence theory for the structured BFGS secant method with an application to nonlinear least squares , 1989 .

[13]  Héctor J. Martínez,et al.  Local and Superlinear Convergence for Partially Known Quasi-Newton Methods , 1991, SIAM J. Optim..

[14]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

[15]  Jorge Nocedal,et al.  A Reduced Hessian Method for Large-Scale Constrained Optimization , 1995, SIAM J. Optim..

[16]  Nicholas I. M. Gould,et al.  On the Convergence of Successive Linear-Quadratic Programming Algorithms , 2005, SIAM J. Optim..

[17]  Jorge Nocedal,et al.  Knitro: An Integrated Package for Nonlinear Optimization , 2006 .

[18]  Dianne P. O'Leary,et al.  Modified Cholesky algorithms: a catalog with new approaches , 2008, Math. Program..