Abstract
Interest in the variable metric proximal point algorithm (VMPPA) is fueled by the desire to accelerate the local convergence of the proximal point algorithm without requiring the divergence of the proximation parameters. In this paper, the local convergence theory for matrix secant versions of the VMPPA is applied to a known globally convergent version of the algorithm. It is shown under appropriate hypotheses that the resulting algorithms are locally super-linearly convergent when executed with the BFGS and the Broyden matrix secant updates. This result unifies previous work on the global and local convergence theory for this class of algorithms. It is the first result applicable to general monotone operators showing that a globally convergent VMPPA with bounded proximation parameters can be accelerated using matrix secant techniques. This result clears the way for the direct application of these methods to constrained and non-finite-valued convex programming. Numerical experiments are included illustrating the potential gains of the method and issues for further study.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.F. Bonnans, J.C. Gilbert, C. Lemaréchal, and C. Sagastizâbal. A family of variable metric proximal point methods. Mathematical Programming, 68: 15–47, 1995.
M. A. Branch and A. Grace. Optimization Toolbox. The Math Works, Inc., Natick, MA (1996).
J.V. Burke and M. Qian. On the super-linear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating. Submitted to Mathematical Programming, August 1996.
J.V. Burke and M. Qian. A variable metric proximal point algorithm for monotone operators. To appear in SIAM J. Control and Optimization,1998.
P.H. Calamai, L.N. Vicente, and J.J. Judice. A new technique for generating quadratic programming test problems. Mathematical Programming, 61: 215–231, 1993.
X. Chen and M. Fukushima. Proximal quasi-Newton methods for nondifferentiable convex optimization. Technical Report AMR 95/32, Dept. of Applied Math., University of New South Wales, Sydney, Australia, 1995.
M. Fukushima and L. Qi. A globally and superlinearly convergent algorithm for nonsmooth convex minimization. SIAM J. Optim., 30: 1106 1120, 1996.
W. Hock. Test Examples for Nonlinear Programming Codes. Springer-Verlag, New York, 1981
C. Lemaréchal and C. Sagastizâbal. An approach to variable metric bundle methods. In J. Henry and J.P. Yuan, editors, IFIP Proceedings, Systems Modeling and Optimization, pages 144–162. Springer, Berlin, 1994.
C. Lemaréchal and C. Sagastizâbal. Variable metric bundle methods: from conceptual to implementable forms. Preprint, INRIA, BP 105, 78153 Le Chesnay, France, 1995.
R. Mifflin. A quasi-second-order proximal bundle algorithm. Mathematical Programming, 73: 51–72, 1996.
R. Mifflin, D. Sun, and L. Qi. Quasi-Newton bundle-type methods for nondifferentiable convex optimization. SIAM J. Optimization, 8: 563–603, 1998.
H. Mine, K. Ohno, and M. Fukushima. A conjugate interior penalty method for certain convex programs. SIAM J. Control and Optimization, 15: 747–755, 1977.
G.J. Minty. Monotone (nonlinear) operators in Hilbert space. Duke Math. J., 29: 341–346, 1962.
J.J. Moreau. Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. France, 93: 273–299, 1965.
L. Qi. Second-order analysis of the Moreau-Yosida regularization of a convex function. Technical Report AMR 94/20, Dept. of Applied Math., University of New South Wales, Sydney, Australia, 1994.
L. Qi and X. Chen. A preconditioning proximal Newton method for nondifferentiable convex optimization. Mathematical Programming, 76: 411–429, 1997.
M. Qian. The Variable Metric Proximal Point Algorithm: Theory and Application. Ph.D., University of Washington, Seattle, WA, 1992.
R.T. Rockafellar. Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. of Operations Research, 1: 97–116, 1976.
E. Zeidler. Nonlinear Functional Analysis and its Applications: II/A, Linear Monotone Operators. Springer-Verlag, New York, 1990.
E. Zeidler. Nonlinear Functional Analysis and its Applications: II/B, Nonlinear Monotone Operators. Springer—Verlag, New York, 1990.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Qian, M., Burke, J.V. (1998). On the Local Super-Linear Convergence of a Matrix Secant Implementation of the Variable Metric Proximal Point Algorithm for Monotone Operators. In: Fukushima, M., Qi, L. (eds) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods. Applied Optimization, vol 22. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6388-1_16
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6388-1_16
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4805-2
Online ISBN: 978-1-4757-6388-1
eBook Packages: Springer Book Archive