Abstract
Proximal bundle methods for minimizing a convex functionf generate a sequence {x k} by takingx k+1 to be the minimizer of\(\hat f^k (x) + u^k |x - x^k |^2 /2\), where\(\hat f^k \) is a sufficiently accurate polyhedral approximation tof andu k > 0. The usual choice ofu k = 1 may yield very slow convergence. A technique is given for choosing {u k} adaptively that eliminates sensitivity to objective scaling. Some encouraging numerical experience is reported.
Similar content being viewed by others
References
A. Auslender, “Numerical methods for nondifferentiable convex optimizations,”Mathematical Programming Study 30 (1986) 102–126.
J. Chatelon, D. Hearn and T.J. Lowe, “A subgradient algorithm for certain minimax and minisum problems,”SIAM Journal on Control and Optimization 20 (1982) 455–469.
F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
M. Fukushima, “A descent algorithm for nonsmooth convex programming,”Mathematical Programing 30 (1984) 163–175.
W. Hock and K. Schittkowski,Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems 187 (Springer, Berlin, 1981).
K.C. Kiwiel, “An aggregate subgradient method for nonsmooth convex minimization,”Mathematical Programming 27 (1983) 320–341.
K.C. Kiwiel, “An algorithm for linearly constrained convex nondifferentiable minimization problems,”Journal of Mathematical Analysis and Applications 105 (1985) 452–465.
K.C. Kiwiel, “An exact penalty function method for nonsmooth constrained convex minimization problems,”IMA Journal of Numerical Analysis 5 (1985) 111–119.
K.C. Kiwiel,Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics 1133 (Springer, Berlin, 1985).
K.C. Kiwiel, “A method of linearizations for linearly constrained nonconvex nonsmooth optimization,”Mathematical Programming 34 (1986) 175–187.
K.C. Kiwiel, “A constraint linearization method for nondifferentiable convex minimization,”Numerische Mathematik 51 (1987) 395–414.
K.C. Kiwiel, “A dual method for solving certain positive semi definite quadratic programming problems,”SIAM Journal on Scientific and Statistical Computing (to appear).
C. Lemarechal, “Nonsmooth optimization and descent methods,” Research Report RR-78-4, International Institute of Applied Systems Analysis (Laxenburg, Austria, 1977).
C. Lemarechal, “Nonlinear programming and nonsmooth optimization-a unification,” Rapport de Recherche No. 332, Institut de Recherche d'Informatique et d'Automatique (Rocquencourt, Le Chesnay, 1978).
C. Lemarechal, “Numerical experiments in nonsmooth optimization,” in: E.A. Nurminski, ed.,Progress in Nondifferentiable Optimization (CP-82-S8, International Institute for Applied Systems Analysis (Laxenburg, Austria, 1982) pp. 61–84.
C. Lemarechal, “Constructing bundle methods for convex optimization,” in: J.B. Hiriart-Urruty, ed.,Fermat Days 85:Mathematics for Optimization (North-Holland, Amsterdam, 1986) pp. 201–240.
C. Lemarechal and R. Mifflin, eds.,Nonsmooth Optimization (Pergamon Press, Oxford, 1978).
R. Mifflin, “A modification and an extension of Lemarechal's algorithm for nonsmooth minimization,”Mathematical Programming Study 17 (1982) 77–90.
R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization 14 (1976) 877–898.
A. Ruszczyński, “A regularized decomposition method for minimizing a sum of polyhedral functions,”Mathematical Programming 35 (1986) 309–333.
N.Z. Shor,Minimization Methods for Nondifferentiable Functions (Springer, Berlin, 1985).
R.L. Streit, “Solution of systems of complex linear equations in thel ∞ norm with constraints on the unknowns,”SIAM Journal on Scientific and Statistical Computing 7 (1986) 132–149.
J. Zowe, “Nondifferentiable optimization,” in: K. Schittkowski, ed.,Computational Mathematical Programming (Springer, Berlin, 1985) pp. 323–356.
Author information
Authors and Affiliations
Additional information
This research was supported by Project CPBP.02.15.
Rights and permissions
About this article
Cite this article
Kiwiel, K.C. Proximity control in bundle methods for convex nondifferentiable minimization. Mathematical Programming 46, 105–122 (1990). https://doi.org/10.1007/BF01585731
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585731