Abstract
A method of truncated codifferential descent for minimizing continuously codifferentiable functions is suggested. The convergence of the method is studied. Results of numerical experiments are presented. Application of the suggested method for the solution of some problems of cluster analysis are discussed. In numerical experiments Wisconsin Diagnostic Breast Cancer database was used.
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References
Andramonov, M. Yu., Rubinov, A.M. and Glover, B. M. (1999), Cutting angle methods in global optimization, Applied Mathematics Letters 12: 95–100.
Armijo, L. (1966), Minimization of functions having continuous partial derivatives, Pacific J. Math. 16: 1–13.
Bagirov, A.M. (2000), Numerical methods for minimizing quasidifferentiable functions: a survey and comparison, In: Quasidifferentiability and Related Topics, Demyanov, V.F. and Rubinov, A.M. (eds.), Ser.: Nonconvex Optimization and Its Applications, Vol. 43, 33–71, Kluwer Academic Publishers, Dordrecht.
Bagirov, A.M. and Rubinov, A.M. (2000), Global minimization of increasing positively homogeneous functions over the unit simplex, Annals of Operations Research 98: 171–187.
Bagirov, A.M., Rubinov, A. M., Stranieri, A. and Yearwood, J. (1999), The global optimization approach to the clustering analysis, Research Report 45/99, University of Ballarat, Australia.
Demyanov, V.F., Gamidov, S. and Sivelina, T.I. (1986), An algorithm for minimizing a certain class of quasidifferentiable functions, Math. Program. Study 29: 74–85.
Demyanov, V.F. and Rubinov, A.M. (1990), Foundations of Nonsmooth Analysis and Quasidifferential Calculus, Nauka, Moscow (in Russian)
Demyanov, V.F. and Rubinov, A.M. (1995), Constructive Nonsmooth Analysis, Springer, Frankfurt am Main.
Demyanov, V.F., Stavroulakis, G.E., Polyakova, L.N. and Panagiotopoulos, P.D. (1996), Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics, Kluwer Academic Publishers, Dordrecht.
Hiriart-Urruty, J.-B. (1989), From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality, In: Clarke F.H., Demyanov V.F. and Gianessi F. (eds.), Nonsmooth Optimization and Related Topics,Plenum Publ., New York, 219-239.
Hiriart-Urruty, J.B. and Lemarechal, C. (1993a), Convex Analysis and Minimization Algorithms, Springer, Heidelberg, Vol. 1.
Hiriart-Urruty, J.B. and Lemarechal, C. (1993b), Convex Analysis and Minimization Algorithms, Springer, New York, Vol. 2.
Kiwiel, K.C. (1986), A linearization method for minimizing certain quasidifferentiable functions, Math. Program. Study29: 86–94.
Mangasarian, O.L. (1997), Mathematical programming in data mining, Data Mining and Knowledge Discovery 1: 183–201.
Mangasarian, O.L., Street, W.N. and Wolberg, W.H. (1995), Breast cancer diagnosis and prognosis via linear programming, Operations Research 43(4): 570–577.
Polak, E., Mayne, D.Q. and Wardi, Y. (1983), On the extension of constrained optimization algorithms from differentiable to nondifferentiable problems, SIAM J. Control and Optimization 21(2): 179–203.
Polak, E. and Mayne, D.Q. (1985), Algorithm models for nondifferentiable optimization, SIAM J. Control and Optimization 23(3): 477–491.
Tuy, H., (1998), Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht.
Wolberg, W.H., Street, W.N. and Mangasarian, O.L. (1994), Machine learning techniques to diagnose breast cancer from fine-needle aspirates, Cancer Letters 77: 163–171.
Wolberg, W.H., Street, W.N. and Mangasarian, O.L. (1995a), Image analysis and machine learning applied to breast cancer diagnosis and prognosis, Analytical and Quantitative Cytology and Histology 17(2): 77–87.
Wolberg, W.H., Street, W.N., Heisey, D.M. and Mangasarian, O.L. (1995b), Computerized breast cancer diagnosis and prognosis from fine needle aspirates, Archives of Surgery 130: 511–516.
Wolberg, W.H., Street, W.N., Heisey, D.M. and Mangasarian, O.L. (1995c), Computer-derived nuclear features distinguish malignant from benign breast cytology, Human Pathology 26: 792–796.
Wolfe, P.H. (1976), Finding the nearest point in a polytope. Math. Programm. 11(2): 128–149.
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Demyanov, V., Bagirov, A. & Rubinov, A. A method of truncated codifferential with application to some problems of cluster analysis. Journal of Global Optimization 23, 63–80 (2002). https://doi.org/10.1023/A:1014075113874
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DOI: https://doi.org/10.1023/A:1014075113874