Abstract
We present a framework for the development of globally defined descent algorithms for the minimization of non-differentiable objective functionsF := h º f withh convex. Within our structure the global convergence properties of the Cauchy, Modified Newton, Gauss—Newton, and Variable-Metric methods are easily established along with that of several new approaches. Examples illustrating the calculational techniques are provided.
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References
D.H. Anderson and M.R. Osborne, “Discrete, nonlinear approximation problems in polyhedral norms”,Numerische Mathematik 28 (1977) 143–156.
H. Attouch and R.J.-B. Wets, “Approximation and convergence in nonlinear optimization”, in: O.L. Mangasarian, R.R. Meyer, and S.M. Robinson, eds.,Nonlinear programming 4 (Academic Press, New York, 1981) pp. 367–394.
D.P. Bertsekas and S. Mitter, “A descent numerical method for optimization problems with nondifferentiable cost functionals”,SIAM Journal on Control and Optimization 11 (1973) 637–652.
J.V. Burke and S.-P. Han, “A Gauss—Newton approach to solving generalized inequalities”, Preprint, Department of Mathematics, University of Kentucky (Lexington, Kentucky, 1984).
J.V. Burke,Methods for solving generalized inequalities with applications to nonlinear programming, Ph.D. Thesis, Department of Mathematics, University of Illinois at Urbana-Champaign (1983).
F.H. Clarke,Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley & Sons, NY, 1983).
L.C.W. Dixon, “Reflections on nondifferentiable optimization, Part I, Ball gradient”,Journal of Optimization Theory and Applications 32 (1980) 123–133.
L.C.W. Dixon and M. Gaviano, “Reflections on nondifferentiable optimization, Part 2, Convergences”,Journal of Optimization Theory and Applications 32 (1980) 259–275.
R. Fletcher, “A model algorithm for composite nondifferentiable optimization problems”,Mathematical Programming Study 17 (1982) 67–76.
U.M. Garcia-Palomares and A. Restuccia, “A global quadratic algorithm for solving a system of mixed equalities and inequalities”,Mathematical Programming 21 (1981) 290–300.
U.M. Garcia-Palomares and A. Restuccia, “Application of the Armijo stepsize rule to the solution of a nonlinear system of equalities and inequalities”,Journal of Optimization Theory and Applications 41 (1983) 405–415.
A.A. Goldstein, “Optimization of Lipschitz continuous functions”,Mathematical Programming 13 (1977) 14–22.
J.-B. Hiriart-Urruty, “ε-Subdifferential calculus”, Proceedings of the Colloquium “Convex Analysis and Optimization”, Imperial College (London, 1980).
C.L. Lawson and R. Hanson,Solving least squares problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).
D.G. Luenberger,Optimization by vector space methods (John Wiley & Sons, NY, 1969).
C. Lemarechal and R. Mifflin, “Global and superlinear convergence of an algorithm for onedimensional minimization of convex functions”,Mathematical Programming 24 (1982) 241–256.
R. Mifflin, “Semi-smooth and semi-convex functions in constrained optimization”,SIAM Journal on Control and Optimization 15 (1977) 959–972.
R. Mifflin, “An algorithm for constrained optimization with semi-smooth functions”,Mathematics of Operations Research 2 (1977) 191–207.
J.-J. Moreau, “Fonctionelles convex”, Lecture Notes, Séminaire “Equations aux derivées partielles”, Collège de France, 1966.
M.R. Osborne and G.A. Watson, “Nonlinear approximation problems in vector norms”, in: G.A. Watson, ed.,Numerical analysis, Dundee 1977, Lecture Notes in Mathematics 630 (Springer-Verlag, Berlin, 1978) pp. 115–132.
M.R. Osborne, “Algorithms for nonlinear approximation”, in: C.T.H. Baker and C. Phillips, eds.,The numerical solution of nonlinear problems (Clarendon Press, Oxford, 1981) pp. 270–286.
M.R. Osborne, “Strong uniqueness in nonlinear approximation”, in: C.T.H. Baker and C. Phillips, eds.,The Numerical Solution of Nonlinear Problems (Clarendon Press, Oxford, 1981) pp. 287–304.
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.
M.J.D. Powell, “General algorithms for discrete nonlinear approximation calculations”, in: C.K. Chui, L.L. Schumaker, and J.D. Ward, eds.,Approximation theory IV (Academic Press, NY, 1983) pp. 187–218.
M.J.D. Powell, “On the global convergence of trust-region algorithms for unconstrained minimization”,Mathematical Programming 29 (1984) 297–303.
M.J.D. Powell and Y. Yuan, “Conditions for superlinear convergence inl 1 andl ∞ solutions of overdetermined nonlinear equations”,IMA Journal of Numerical Analysis 4 (1984) 241–251.
R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, NJ, 1970).
K. Schittkowski, “Numerical solution of systems of nonlinear inequalities”, in:Optimization and operations research, Lecture Notes in Economics and Mathematical Systems 117 (Springer-Verlag, Berlin, 1975) pp. 259–272.
R.J.-B. Wets, “Convergence of convex functions, variational inequalities, and convex optimization problems”, in: R.W. Cottle, F. Giannessi, and J.-L. Lions, eds.,Variational inequalities and complementarity problems (John Wiley & Sons, NY, 1980) pp. 375–403.
P. Wolfe, “Convergence conditions for ascent methods”,SIAM Review 11 (1969) 226–235.
Y. Yuan, “Global convergence of trust region algorithms for nonsmooth optimization”, Report DAMTP 1983, Cambridge University (Cambridge, England, 1983).
Y. Yuan, “Some properties of trust region algorithms for non-smooth optimization”, Report DAMTP 1983, Cambridge University (Cambridge, England, 1983).
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Burke, J.V. Descent methods for composite nondifferentiable optimization problems. Mathematical Programming 33, 260–279 (1985). https://doi.org/10.1007/BF01584377
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DOI: https://doi.org/10.1007/BF01584377