Abstract
A broad spectrum of complex problems of mathematical programming can be rather easily reduced to problems of minimization of nondifferentiable functions without constraints or with simple constraints. Thus, when decomposition schemes are used to solve structured optimization problems of large dimension or with a large number of constraints, the coordination problems with respect to linking variables (or dual estimates of linking constraints) as a rule prove to be problems of nondifferentiable optimization. The use of exact nonsmooth penalty functions in problems of nonlinear programming, maximum functions to estimate discrepancies in constraints, piecewise smooth approximation of technical-economic characteristics in practical problems of optimal planning and design, minimax compromise functions in problems of multi-criterion optimization, all of these generate problems of nondifferentiable optimization. Therefore, the efficiency of procedures for the solution of various complex problems of mathematical programming greatly depends on the efficiency of the algorithms employed to minimize nondifferentiable functions.
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References
A. A. Bakaev, V. S. Mikhalevich, S. V. Branovickaja, N. Z. Shor. Methodology and Attempted Solution of Large Network Transportation Problems by Computer. - In the Book: Mathematical Methods for Production Problems, M., 1963, pp. 247–257.
L. V. Beljaeva, N. T. Zhurbenko, N. Z. Shor. Concerning Methods of Solution of One Class of Dynamic Distribution Problems. - Economics and Mathematical Methods, 1978, Vol. 14, Issue 1, pp. 137–146.
V. A. Bulavskij. Iterative Method of Solution of General Problem of Linear Programming. - In Coll.: Numerical Methods of Optimal Planning, Issue 1, Econ.- Math. Ser., Siberian Branch of Ac. Sci. U.S.S.R., 1962, pp. 35–64.
V. S. Mikhalevich, N. Z. Shor, L. A. Galustova, et al. Computational Methods of Choice of Optimal Design Decisions. - Kiev, Naukova Dumka, 1977, 178 p.
V. I. Gershovich. About a Cutting Method with Linear Space Transformations. In the Book: Theory of Optimal Solutions, Kiev, Inst, of Cybernetics. Ac. Sci. Ukr.S.S.R., 1979, pp. 15–23.
V. I. Gershovich. One Way of Representing Space Transformation Operators in High-Speed Versions of Generalized Gradient Methods. In the Book: Numerical Methods of Nonlinear Programming. Summaries of Papers at the III. All-Union Seminar. - Kharkov, 1979, pp. 64–66.
V. I. Gershovich. Efficient Algorithm for Solving Problems of Nonsmooth Optimization that Combines Cutting Methods and Linear Transformations in Variable Space. - In the Book: Systems of Software Support for Solution of Optimal Planning Problems. Summaries of Papers at the IV. Ail-Union Symposium.-M., 1980, pp. 50–51.
V. I. Gershovich. One Optimization Method Using Linear Space Transformations. - In the Book: Theory of Optimal Solutions. Kiev, Inst, of Cybernetics, Ac. Sci. Ukr. S.S.R., 1980, pp. 38–45.
V. I. Gershovich. About an Ellipsoid Algorithm. In the Book: Some Algorithms of Nonsmooth Optimization and Discrete Programming. Preprint 81-6 Inst, of Cybernetics, Ac. Sci. Ukr.S.S.R., 1981, pp. 8–13.
V. I. Gershovich. Quadratic Smoothing in Iterative Decomposition Algorithms of Solution of Large-Scale Linear-Programming Problems. In the Book: II. Republican Symposium on Methods of Solution of Nonlinear Equations and Optimization Problems. Tallin, 1981, pp. 188–190.
A. M. Gupal. Stochastic Methods of Solution of Nonsmooth Extremum Problems. - Kiev, Naukova Dumka, 1979, 149 p.
V. V. Dem’janov, V. N. Malozemov. Introduction to Minimax - M., Nauka, 1972, 368 p.
V. F. Dem’janov, L. V. Vasil’ev. Non-Differentiable Optimization. - M., Nauka, 1981, 384 p.
I. I. Eremin. Iterative Method for Tchebycheff Approximations of Inconsistent Systems of Linear Inequalities. - Dokl. AN S.S.S.R., 1962, Vol. 143, No. 6, pp. 1254–1256.
I. I. Eremin. About “Penalty ” Method in Convex Programming. - Kybernetika, 1967, No. 4, pp. 63–67.
I. I. Eremin. Methods of Fejer Approximations in Convex Programming. Mathem. Notes, 1968, Vol. 3, No. 2, pp. 217–234.
Ju. M. Ermol’jev. Stochastic Programming Methods, M., Nauka, 1976, 240 p.
Ju. M. Ermol’jev. Methods of Solution of Nonlinear Extremum Problems. - Kiber- netika, 1966, No. 4, pp. 1–17.
A. Ju. Levin. On an Algorithm for Minimization of Convex Functions. - Dokl. AN S.S.S.R., 1965, Vol. 160, No. 6, pp. 1244–1247.
A. S. Nemirovskij, D. B. Judin. Optimization Methods Adaptive to “Considerable ” Dimension of a Problem. - Avtomatika i Telemekhanika, 1977, No. 4, pp. 75–87.
N. S. Nemirovskij, D. B. Judin. Complexity of Problems and Efficiency of Optimization Methods. - M., Nauka, 1979, 383 p.
E. A. Nurminskij. Numerical Methods of Solution of Deterministic and Stochastic Minimax Problems. - Kiev. Naukova Dumka, 1979, 159 p.
E. A. Nurminskij. About Continuity of e-Subgradient Mappings. - Kibernetika, 1977, No. 5, pp. 148–149.
B. T. Poljak. A General Method of Solution of Extremum Problems. - Dokl. AN S.S.S.R., 1967, Vol. 174, No. 1, pp. 33–36.
B. T. Poljak. Minimization of Unsmooth Functionals. - J. Vychisl. Matem. i Mat. Fiziki, 1969, Vol. 9, No. 3, pp. 509–521.
B. N. Pshenichnyj, Ju. M. Danilin. Numerical Methods for Extremum Problems.–M., Nauka, 1975, 319 p.
S. V. Rzhevskij. About a Unified Approach to Solution of Unconditional Minimization Problems. - J. Vychisl. Matem. i Matem. Fiziki, 1969, Vol. 20, No. 4, pp. 857–863.
S. V. Rzhevskij. e-Subgradient Method of Solution of Convex Programming Problem. - J. Vychisl. Matem. i Matem. Fiziki, Vol. 21, No. 5, 1981, pp. 1126–1132.
S. V. Rzhevskij. Monotone Algorithm of Search for Saddle Point of Nonsmooth Function. - Kibernetika, 1982, No. 1, pp. 95–98.
V. A. Skokov. Note on Minimization Methods Using Operation of Space Dilatation. - Kibernetika, 1974, No. 4, pp. 115–117.
L. G. Khachijan. Polynomial Algorithm in Linear Programming. - Dokl. AN S.S.S.R., 1979, Vol. 244, No. 5, pp. 1093–1096.
L. G. Khachijan. Polynomial Algorithms in Linear Programming. - J. Vychisl. Matem. i Matem. Fiziki, 1980, Vol. 20, No. 1, pp. 51–68.
M. K. Kozlov, S. P. Tarasov, L. G. Khachijan. Polynomial Solvability of Convex Quadratic Programming. - Dokl. AN S.S.S.R., 1979, Vol. 248, No. 5.
N. Z. Shor. Application of Gradient Descent Method for Solution of Network Transportation Problems. In the Book: Notes, Scientific Seminar on Theory and Application of Cybernetics and Operations Research. Nauchn. Sovet po Kibernetika AN U.S.S.R, Kiev, 1962, Issue 1, pp. 9–17.
N. Z. Shor. Rate of Convergence of Generalized Gradient Descent Method. Kibernetika, 1968, No. 3, pp. 98–99.
N. Z. Shor. Utilization of Space Dilatation Operation in Minimization of Convex Functions. - Kibernetika, 1970, No. 1, pp. 6–12.
N. Z. Shor. About Rate of Convergence of Method of Generalized Gradient Descent with Space Dilatation. - Kibernetika, 1970, No. 2, pp. 80–85.
N. Z. Shor, N. G. Zhurbenko. Minimization Method Using Operation of Space Dilatation in the Direction of Difference of Two Sequential Gradients. - Kibernetika, 1971, No. 3, pp. 51–59.
N. Z. Shor, L. P. Shabashova. Solution of Minimax Problems by Generalized Gradient Descent Method with Space Dilatation. - Kibernetika, 1972, No. 1, pp. 82– 88.
N. Z. Shor, V. I. Gershovich. Family of Algorithms for Solving Problems of Convex Programming. - Kibernetika, 1979, No. 4, pp. 62–67.
N. Z. Shor. Method of Minimization of Almost Differentiable Functions. - Kibernetika, 1972, No. 4, pp. 65–70.
N. Z. Shor, V. I. Gershovich. About One Modification of Gradient-Type Algorithms with Space Dilatation for Solution of Large-Scale Problems. - Kibernetika, 1981, No. 5, pp. 67–70.
N. Z. Shor. Convergence of Gradient-Type Method with Space Dilatation in the Direction of Difference of Two Sequential Gradients. - Kibernetika, 1975, No. 4, pp. 48–53.
N. Z. Shor. Cutting Method with Space Dilatation for Solving Problems of Convex Programming. - Kibernetika, 1977, No. 1, pp. 94–95.
N. Z. Shor. New Directions in Development of Nonsmooth Optimization Methods. - Kibernetika, 1977, No. 6, pp. 87–91.
N. Z. Shor. Methods of Optimization of Non-Differentiable Functions and Their Applications. - Kiev, Naukova Dumka, 1979, 200 p.
D. B. Judin, A. S. Nemirovskij. Evaluation of Information Complexity and Effective Methods for Solving of Complex Extremum Problems. - Ekonomika i Matema- ticheskie Metody, 1976, Vol. 12, Issue 2, pp. 357–369.
D. Agmon. The Relaxation Method for Linear Inequalities. - Can. J. Math. 1954, Vol. 6, No. 2, pp. 381–392.
B. Aspvall, G. Shiloach. A Polynomial Time Algorithm for Solving Systems of Linear Inequalities with Two Variables per Inequality. SIAM J. Comput., Vol. 9, No. 4, Nov. 1980.
R. Bland, D. Goldfarb, M. Todd. The Ellipsoid Method: A Survey. Cornell University, Ithaca, New York, Technical Report No. 476, 79 p.
F. Clarke. Generalized Gradients and Applications. - Trans. Amer. Math. Soc., 1975, Vol. 205, pp. 247–262.
M. Grötschel, L. Lovâsz, A. Schrijver. The Ellipsoid Method and its Consequences in Combinatorial optimization. - Combinatorica, 1981, Vol. I, No. 2, pp. 169–197.
M. Grötschel, L. Lovâsz, A. Schrijver. Polynomial Algorithms for Perfect Graphs. Report No. 81176-OR, Bonn University, Febr. 1981.
J. Kelley. The “Cutting Plane” Method for Solving Convex Programs. - SIAM Journal, 1960, Vol. 8, No. 4, pp. 703–712.
B. Körte, R. Schräder. A Note on Convergence Proofs for Shor-Khachijan Methods. - In “Optimization and Optimal Control”. Lecture Notes in Control and Information Sciences 30, Springer (1980), pp. 51–57.
C. Lemarechal. An Extension of Davidon Methods to Nondifferentiable Problems. - Math. Program. Study 3, (1975), pp. 95–100.
L. Lovâsz. On the Shannon Capacity of a Graph. IEEE Trans. Inform. Theory 11-25 (1979), pp. 1–7.
J. F. Maurras, K. Truemper, M. Akgül. Polynomial Algorithms for a Class of Linear Programs. Math. Progr., 21 (1981), pp. 121–136.
R. Mifflin. Semismooth and Semiconvex Functions in Constrained Optimization. RR-76-21. HAS A. - Laxenburg, Austria, 1976, 23 p.
R. Mifflin. An Algorithm for Constrained Optimization with Semismooth Functions. RR-73-3. IIASA. - Laxenburg, Austria, 1977, 32 p.
T. Motzkin, I. Schoenberg. The Relaxation Method for Linear Inequalities. - Can. J. Math., 1954, Vol. 6, No. 2, pp. 393–404.
R. Schräder. Ellipsoid Methods. In the Book: Modern Applied Mathematics: Optimization and Operations Research, B. Körte (ed.), North-Holland, Amsterdam, 1982, pp. 265–311.
B. Schwartz. Zur Minimierung konvexer sub differenzierbarer Funktionen über dem Rn. Math. Operationsforschung und Statistik, Series Optimization, 1978, 9, No. 4, pp. 545–557.
P. Wolfe. A Method of Conjugate Subgradients for Minimizing Nondifferentiable Functions. - Math. Program. Study 3, (1975), pp. 145–173.
P. Wolfe. A Bibliography for the Ellipsoid Algorithm. IBM Research Center, York-town Heights. New York, 1980, 6 p.
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Shor, N.Z. (1983). Generalized Gradient Methods of Nondifferentiable Optimization Employing Space Dilatation Operations. In: Bachem, A., Korte, B., Grötschel, M. (eds) Mathematical Programming The State of the Art. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68874-4_19
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