Abstract
The sensitivity function induced by a convex programming problem is examined. Its monotonicity, subdifferentiability, and closure properties are analyzed. A relation to the Pareto optimal solution set of the multicriteria convex optimization problem is established. The role of the sensitivity function in systems describing optimization problems is clarified. It is shown that the solution of these systems can often be reduced to the minimization of the sensitivity function on a convex set. Numerical methods for solving such problems are proposed, and their convergence is proved.
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References
D. Gale, The Theory of Linear Economic Models (McGraw-Hill, New York, 1960).
A. C. Williams, “Marginal Values in Linear Programming,” J. Soc. Ind. Appl. Math. 11, 82–94 (1963).
S. Zlobec, Stable Parametric Programming (Kluwer Academic, Dordrecht, 2001).
I. I. Eremin and N. N. Astaf’ev, Introduction to the Theory of Linear Convex Programming (Nauka, Moscow, 1976) [in Russian].
K.-H. Elster, R. Reinhardt, M. Schäuble, and G. Donath, Einführung in die nichtlineare Optimierung (Teubner, Leipzig, 1977; Nauka, Moscow, 1985).
S. V. Rzhevskii, Monotone Methods of Convex Programming (Naukova Dumka, Kiev, 1993) [in Russian].
A. I. Golikov, “Characterization of Optimal Estimates Set for Multicriterial Optimization Problems,” USSR Comput. Math. Math. Phys. 28, 1285–1296 (1988).
V. G. Zhadan, “An Augmented Lagrange Function Method for Multicriterial Optimization Problems,” Zh. USSR Comput. Math. Math. Phys. 28(6), 1–11 (1988).
Yu. G. Evtushenko, Solution Methods for Optimization Problems and Applications to Optimization Systems (Nauka, Moscow, 1982) [in Russian].
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis (Springer-Verlag, Berlin, 1998).
F. P. Vasil’ev, Optimization Methods (Faktorial, Moscow, 2002) [in Russian].
A. S. Antipin, “Saddle Point Problem and Optimization Problem as a Unified System,” Tr. Inst. Mat. Mekh. 14(2), 5–15 (2008).
A. S. Antipin, “Methods of Solving Systems of Convex Programming Problems,” USSR Comput. Math. Math. Phys. 27(2), 30–35 (1987).
A. S. Antipin, “Models of Interaction between Manufactures, Consumers, and the Transportation System,” Autom. Remote Control, No. 10, 1391–1398 (1990).
S. Karlin, Mathematical Methods and Theory in Games, Programming, and Economics (Pergamon, London, 1959; Mir, Moscow, 1964).
A. S. Antipin, “Inverse Optimization Problems,” in Large Russian Encyclopedia (INFRA-M, Moscow, 2003), pp. 346–347 [in Russian].
A. S. Antipin, “An Extraproximal Method for Solving Equilibrium Programming Problems and Games with Coupled Constraints,” Comput. Math. Math. Phys. 45, 2020–2028 (2005).
A. S. Antipin, L. A. Artem’eva, and F. P. Vasil’ev, “Multicriteria Equilibrium Programming: Extragradient Method,” Comput. Math. Math. Phys. 50, 224–230 (2010).
A. S. Antipin, L. A. Artem’eva, and F. P. Vasil’ev, “Regularized Extragradient Method for Solving Parametric Multicriteria Equilibrium Programming Problem,” Comput. Math. Math. Phys. 50, 1975–1989 (2010).
A. S. Antipin, “The Convergence of Proximal Methods to Fixed Points of Extremal Mappings and Estimates of Their Rate of Convergence,” Comput. Math. Math. Phys. 35, 539–551 (1995).
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Original Russian Text © A.S. Antipin, A.I. Golikov, E.V. Khoroshilova, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 12, pp. 2126–2142.
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Antipin, A.S., Golikov, A.I. & Khoroshilova, E.V. Sensitivity function: Properties and applications. Comput. Math. and Math. Phys. 51, 2000–2016 (2011). https://doi.org/10.1134/S0965542511120049
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DOI: https://doi.org/10.1134/S0965542511120049