Least-change quasi-Newton updates for equality-constrained optimization

Abstract.This paper investigates quasi-Newton updates for equality-constrained optimization. Using a least-change argument we derive a class of rank-3 updates to approximations of the one-sided projection of the Hessian of the Lagrangian which keeps the appropriate part symmetric (and possibly positive definite). By imposing the usual assumptions we are able to prove 1-step superlinear convergence for one of these updates. Encouraging numerical results and comparisons with other previously analyzed updates are presented.

[1]  Jean Charles Gilbert.  On the Realization of the Wolfe Conditions in Reduced Quasi-Newton Methods for Equality Constrained Optimization , 1997, SIAM J. Optim..

[2]  C. G. Broyden.  A Class of Methods for Solving Nonlinear Simultaneous Equations , 1965 .

[3]  J. Greenstadt.  Variations on Variable-Metric Methods , 1970 .

[4]  Thomas F. Coleman,et al.  Partitioned quasi-Newton methods for nonlinear equality constrained optimization , 1992, Math. Program..

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

[6]  J. J. Moré,et al.  Quasi-Newton Methods, Motivation and Theory , 1974 .

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

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

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

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

[11]  Roger Fletcher,et al.  A Rapidly Convergent Descent Method for Minimization , 1963, Comput. J..

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

[13]  H. Kunzi,et al.  Lectu re Notes in Economics and Mathematical Systems , 1975 .

[14]  Gene H. Golub,et al.  Matrix computations , 1983 .

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

[16]  William C. Davidon,et al.  Variable Metric Method for Minimization , 1959, SIAM J. Optim..

[17]  R. Fletcher,et al.  A New Approach to Variable Metric Algorithms , 1970, Comput. J..

[18]  Michael J. Todd.  Quasi-Newton Updates in Abstract Vector Spaces , 1984 .

[19]  D. Goldfarb.  A family of variable-metric methods derived by variational means , 1970 .

[20]  M. Powell.  A New Algorithm for Unconstrained Optimization , 1970 .

[21]  C. G. Broyden.  The Convergence of a Class of Double-rank Minimization Algorithms 2. The New Algorithm , 1970 .

[22]  Shih-Ping Han,et al.  Superlinearly convergent variable metric algorithms for general nonlinear programming problems , 1976, Math. Program..

[23]  Chaya Gurwitz.  Local Convergence of a Two-Piece Update of a Projected Hessian Matrix , 1994, SIAM J. Optim..

[24]  D. Shanno.  Conditioning of Quasi-Newton Methods for Function Minimization , 1970 .

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

[26]  Jonathan Goodman,et al.  Newton's method for constrained optimization , 1985, Math. Program..