Projected Newton methods for optimization problems with simple constraints

We consider the problem min {f(x)|x ? 0} and algorithms of the form xk+1 = [xk - ?k Dk?f(xk)]+ where [?]+ denotes projection on the positive orthant, ?k is a stepsize chosen by an Armijolike rule, and Dk is a positive definite symmetric matrix which is partly diagonal. We show that Dk can be calculated simply on the basis of second derivatives of f so that the resulting Newton-like algorithm has a typically superlinear rate of convergence. With other choices of Dk convergence at a typically linear rate is obtained. The algorithms are almost as simple as their unconstrained counterparts. They are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low-dimensional problems.

[1]  J. B. Rosen.  The Gradient Projection Method for Nonlinear Programming. Part I. Linear Constraints , 1960 .

[2]  A. Goldstein.  Convex programming in Hilbert space , 1964 .

[3]  Sanjoy K. Mitter,et al.  Successive approximation methods for the solution of optimal control problems , 1966, Autom..

[4]  D. Goldfarb.  Extension of Davidon’s Variable Metric Method to Maximization Under Linear Inequality and Equality Constraints , 1969 .

[5]  G. McCormick.  The Variable Reduction Method for Nonlinear Programming , 1970 .

[6]  Pravin Varaiya,et al.  Notes on Optimization , 1972 .

[7]  E. Polak,et al.  Computational methods in optimization : a unified approach , 1972 .

[8]  E. Polak.  Introduction to linear and nonlinear programming , 1973 .

[9]  Philip E. Gill,et al.  Numerical methods for constrained optimization , 1974 .

[10]  D. Bertsekas.  On the Goldstein-Levitin-Polyak gradient projection method , 1974, CDC 1974.

[11]  W. Murray.  Numerical Methods for Unconstrained Optimization , 1975 .

[12]  Olvi L. Mangasarian,et al.  Superlinearly convergent quasi-newton algorithms for nonlinearly constrained optimization problems , 1976, Math. Program..

[13]  R. Fletcher,et al.  A modified Newton method for minimization , 1977 .

[14]  J. Cullum,et al.  An algorithm for minimizing a differentiable function subject to box constraints and errors , 1979 .

[15]  Danny C. Sorensen,et al.  On the use of directions of negative curvature in a modified newton method , 1979, Math. Program..

[16]  Melanie L. Lenard,et al.  A computational study of active set strategies in nonlinear programming with linear constraints , 1979, Math. Program..

[17]  J. Dunn.  Newton’s Method and the Goldstein Step-Length Rule for Constrained Minimization Problems , 1980 .

[18]  J. Dunn.  Global and Asymptotic Convergence Rate Estimates for a Class of Projected Gradient Processes , 1981 .

[19]  D. Bertsekas.  Projected Newton methods for optimization problems with simple constraints , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.