Abstract
This paper presents new versions of proximal bundle methods for solving convex constrained nondifferentiable minimization problems. The methods employϑ 1 orϑ ∞ exact penalty functions with new penalty updates that limit unnecessary penalty growth. In contrast to other methods, some of them are insensitive to problem function scaling. Global convergence of the methods is established, as well as finite termination for polyhedral problems. Some encouraging numerical experience is reported. The ideas presented may also be used in variable metric methods for smooth nonlinear programming.
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References
A. Auslender, J.T. Crouzeix and P. Fedit, “Penalty-proximal methods in convex programming,”Journal of Optimization Theory and Applications 55 (1987) 1–21.
D. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982).
J. Chatelon, D. Hearn and T.J. Lowe, “A subgradient algorithm for certain minimax and minisum problems,”SIAM Journal of Control and Optimization 20 (1982) 455–469.
F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
W. Hock and K. Schittkowski,Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems No. 187 (Springer, Berlin, 1981).
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 No. 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, “An exact penalty function method for nondifferentiable constrained minimization,” Prace IBS PAN 155 (Warszawa, 1987).
K.C. Kiwiel, “A dual method for solving certain positive semidefinite quadratic programming problems,”SIAM Journal on Scientific and Statistical Computing 10 (1989) 175–186.
K.C. Kiwiel, “Proximity control in bundle methods for convex nondifferentiable minimization,”Mathematical Programming 46 (1990) 105–122.
K.C. Kiwiel, “A survey of bundle methods for nondifferentiable optimization,” in: M. Iri and K. Tanabe, eds.,Mathematical Programming: Recent Developments and Applications (KTT/Kluwer, Tokyo, 1989) pp. 263–282.
C. Lemaréchal, “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. Lemaréchal, “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. Lemaréchal, “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. Lemaréchal, “Nondifferentiable optimization,” in: G.L. Nemhauser, A.H.G. Rinooy Kan and M.J. Todd, eds.,Handbooks in Operations Research and Management Science, Vol. 1, Optimization (North-Holland, Amsterdam, 1989) pp. 529–572.
C. Lemaréchal and R. Mifflin, eds.,Nonsmooth Optimization (Pergamon Press, Oxford, 1978).
O.L. Mangasarian, “Sufficiency of exact penalty minimization,”SIAM Journal on Control and Optimization 23 (1985) 30–37.
R. Mifflin, “A modification and an extension of Lemarechal's algorithm for nonsmooth minimization,”Mathematical Programming Study 17 (1982) 77–90.
E. Polak, D.Q. Mayne and Y. Wardi, “On the extension of constrained optimization methods from differentiable to non-differentiable problems,”SIAM Journal on Control and Optimization 25 (1983) 179–203.
B.N. Pshenichnyi and Yu.M. Danilin,Numerical Methods for Extremal Problems (Mir, Moscow, 1978).
R.L. Streit, “Solution of systems of complex linear equations in theϑ ∞ norm with constraints on the unknowns,”SIAM Journal of 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.
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This research was supported by the Polish Academy of Sciences.
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Kiwiel, K.C. Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization. Mathematical Programming 52, 285–302 (1991). https://doi.org/10.1007/BF01582892
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DOI: https://doi.org/10.1007/BF01582892
Key words
- Nondifferentiable minimization
- convex programming
- exact penalty functions
- numerical methods
- descent methods
- proximal bundle methods