Abstract
A readily implementable algorithm is given for minimizing a (possibly nondifferentiable and nonconvex) locally Lipschitz continuous functionf subject to linear constraints. At each iteration a polyhedral approximation tof is constructed from a few previously computed subgradients and an aggregate subgradient, which accumulates the past subgradient information. This aproximation and the linear constraints generate constraints in the search direction finding subproblem that is a quadratic programming problem. Then a stepsize is found by an approximate line search. All the algorithm's accumulation points are stationary. Moreover, the algorithm converges whenf happens to be convex.
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References
A. Bihain, “Optimization of upper-semidifferentiable functions“,Journal of Optimization Theory and Applications 44 (1984) 545–568.
F.H. Clarke,Optimization and nonsmooth analysis (Wiley-Interscience, New York, 1983).
G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, New Jersey, 1963).
K.C. Kiwiel, “An aggregate subgradient method for nonsmooth convex minimization“,Mathematical Programming 27 (1983) 320–341.
K.C. Kiwiel, “A linearization algorithm for constrained nonsmooth minimization“, in: P. Thoft-Christensen, ed.,System modelling and optimization, Lecture Notes in Control and Information Sciences 59 (Springer-Verlag, Berlin, 1984) pp. 311–320.
K.C. Kiwiel, “An algorithm for linearly constrained convex nondifferentiable minimization problems“,Journal of Mathematical Analysis and Applications 105 (1985) 111–119.
K.C. Kiwiel,Methods of descent for nondifferentiable optimization, Lecture Notes in Mathematics 1133 (Springer-Verlag, Berlin, 1985).
K.C. Kiwiel, “A linearization algorithm for nonsmooth minimization“,Mathematics of Operations Research 10 (1985) 185–194.
K.C. Kiwiel, “A method for minimizing the sum of a convex function and a continuously differentiable function”,Journal of Optimization Theory and Applications, to appear.
K.C. Kiwiel, “A method for solving certain quadratic programming problems arising in nonsmooth optimization”,IMA Journal of Numerical Analysis, to appear.
C. Lemarechal, “Numerical experiments in nonsmooth optimization“, in: E.A. Nurminski, ed.,Progress in nonsmooth optimization, CP-82-S8 (International Institute for Applied Systems Analysis, Laxenburg, Austria, 1982) pp. 61–84.
C. Lemaréchal and R. Mifflin, eds.,Nonsmooth optimization (Pergamon Press, Oxford, 1978).
C. Lemaréchal, J.-J. Strodiot and A. Bihain, “On a bundle algorithm for nonsmooth minimization“, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear programming 4 (Academic Press, New York, 1981) pp. 245–281.
K. Madsen and J. Schjaer-Jacobsen, “Linearly constrained minimax optimization“,Mathematical Programming 14 (1978) 208–223.
R. Mifflin, “An algorithm for constrained optimization with semismooth functions“,Mathematics of Operations Research 2 (1977) 191–207.
R. Mifflin, “A modification and an extension of Lemaréchal'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 algorithms from differentiable to nondifferentiable problems“,SIAM Journal on Control and Optimization 21 (1983) 179–203.
R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, New Jersey,. 1970).
J.-J. Strodiot, V. Hien Nguyen and N. Heukems, “ε-Optimal solutions in nondifferentiable convex programming and related questions“,Mathematical Programming 25 (1982) 307–328.
J.-J. Strodiot and V. Hien Nguyen, “An algorithm for minimizing nondifferentiable convex functions under linear constraints“, in: P. Thoft-Christensen, ed.,System modelling and optimization, Lecture Notes in Control and Information Sciences 59 (Springer-Verlag, Berlin, 1984) pp. 338–344.
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Kiwiel, K.C. A method of linearizations for linearly constrained nonconvex nonsmooth minimization. Mathematical Programming 34, 175–187 (1986). https://doi.org/10.1007/BF01580582
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DOI: https://doi.org/10.1007/BF01580582