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Local properties of inexact methods for minimizing nonsmooth composite functions.(English)Zbl 0622.90072

The author studies methods for the minimization of a class of composite nondifferentiable functions, based on the use of equality-constrained quadratic programming subproblems. The effect of inexactness in solving these subproblems on the local convergence is discussed in detail.

MSC:

90C30 Nonlinear programming
90C20 Quadratic programming
65K05 Numerical mathematical programming methods

Cite

References:

[1]D.H. Anderson and M.R. Osborne, ”Discrete, nonlinear approximation problems in polyhedral norms,”Numerische Mathematik 28 (1977) 143–156. ·Zbl 0342.65004 ·doi:10.1007/BF01394449
[2]D.H. Anderson and M.R. Osborne, ”Discrete, nonlinear approximation problems in polyhedral norms–A Levenberg-like algorithm,”Numerische Mathematik 28 (1977) 157–170. ·Zbl 0342.65005 ·doi:10.1007/BF01394450
[3]L. Cromme, ”Strong uniqueness: A far reaching criterion for the convergence analysis of iterative procedures,”Numerische Mathematik 29 (1978) 179–194. ·Zbl 0352.65012 ·doi:10.1007/BF01390337
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[6]R.S. Dembo and U. Tulowitzki, ”Sequential truncated quadratic programming methods,” in: P.T. Boggs, R.H. Byrd and R.B. Schabel, eds,Numerical Optimization 1984 (SIAM, Philadelphia, 1985) pp. 83–101. ·Zbl 0583.65042
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[8]R. Fletcher,Practical Methods of Optimization, Vol. 2: Constrained Optimization (John Wiley, Chichester and New York, 1981). ·Zbl 0474.65043
[9]R. Fletcher, ”Second-order corrections for nondifferentiable optimization,” in: G.A. Watson, ed.,Numerical Analysis, Proceedings, Dundee, 1981 (Springer-Verlag, 1982), pp. 85–114.
[10]R. Fletcher, ”Anl t penalty method for nonlinear constraints,” in: P.T. Boggs, R.H. Byrd and R.B. Schnabel, eds.,Numerical Optimization 1984 (SIAM, Philadelphia, 1985) pp. 102–118.
[11]R. Fontecilla, ”On inexact quasi-Newton methods for constrained optimization,” in: P.T. Boggs, R.B. Byrd and R.B. Schnabel, eds.,Numerical Optimization 1984 (SIAM, Philadelphia, 1985) pp. 102–118.
[12]J. Goodman, ”Newton’s method for constrained optimization,”Mathematical Programming 33 (1985) 162–171. ·Zbl 0589.90065 ·doi:10.1007/BF01582243
[13]K. Jittorntrum and M.R. Osborne, ”Strong uniqueness and second-order convergence in nonlinear discrete approximation,”Numerische Mathematik 34 (1980) 439–455. ·Zbl 0486.65008 ·doi:10.1007/BF01403680
[14]R.A. McLean and G.A. Watson, ”Numerical methods for nonlinear discretel t approximation problems,” in: L. Collatz, G. Meinardus, and H. Werner, eds.,Numerical Methods of Approximation Theory: Excerpts of the Conference at Oberwolfach, 1979 (Birkhauser-Verlag, Basel, 1980).
[15]W. Murray and M.L. Overton, ”A projected Lagrangian algorithm for nonlinear minimax optimization,”SIAM Journal of Scientific and Statistical Computing 1 (1980) 345–370. ·Zbl 0461.65052 ·doi:10.1137/0901025
[16]W. Murray and M.L. Overton, ”A projected Lagrangian algorithm for nonlinearl t optimization,”SIAM Journal on Scientific and Statistical Computing 2 (1981) 207–224. ·Zbl 0468.65036 ·doi:10.1137/0902018
[17]J. Nocedal and M.L. Overton, ”Projected Hessian updating algorithms for nonlinearly constrained optimization,”SIAM Journal on Numerical Analysis 22 (1985) 821–850. ·Zbl 0593.65043 ·doi:10.1137/0722050
[18]M.L. Overton, ”Algorithms for nonlinearl t andl fitting,” in: M.J.D. Powell, ed.,Nonlinear Optimization 1981, (Academic Press, 1982) pp. 91–102.
[19]R.A. Tapia, ”Diagonalized multiplier methods and quasi-Newton methods for constrained optimization,”Journal of Optimization Theory and its Applications 22 (1977) 135–194. ·Zbl 0336.65034 ·doi:10.1007/BF00933161
[20]G.A. Watson, ”The minimax solution of an overdetermined system of nonlinear equations,”Journal of the Institute of Mathematics and its Applications 23 (1979) 167–180. ·Zbl 0406.65025 ·doi:10.1093/imamat/23.2.167
[21]R.S. Womersley, ”Optimality conditions for piecewise smooth functions,”Mathematical Programming Study 17 (1982) 13–27. ·Zbl 0478.90059
[22]R.S. Womersley, ”Minimizing nonsmooth composite functions,” Centre for Mathematical Analysis Report CMA-R12-84, Australian National University (1984). ·Zbl 0571.90084
[23]R.S. Womersley, ”Local properties of algorithms for minimizing nonsmooth composite functions,”Mathematical Programming 32 (1985) 69–89. ·Zbl 0571.90084 ·doi:10.1007/BF01585659
[24]Y. Yuan, ”An example of only linear convergence of trust region algorithms for nonsmooth optimization,”IMA Journal of Numerical Analysis 4 (1984) 327–335. ·Zbl 0555.65037 ·doi:10.1093/imanum/4.3.327
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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