Construction of convex continuations for functions defined on a hypersphere
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Construction of convex continuations for functions defined on the vertices of some combinatorial polyhedra, in particular the permutation polyhedron and the arrangement polyhedron, has been studied in [1, 2]. Subsequently this result has been generalized to functions defined at the extreme points of an arbitrary polyhedron . For purposes of combinatorial optimization [4-6] it is relevant to consider the existence and construction of convex continuations from continua, in particular, when the function is defined on a hypersphere in thek-dimensional space. Unfortunately, passage to the limit from discrete sets to continua does not produce positive results in this case. We are thus forced to develop special approaches to investigating the existence of convex continuations of functions defined on continua.
KeywordsPartial Derivative Quadratic Form Naukova Dumka Real Root Geometrical Design
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- 1.Yu. G. Stoyan and S. V. Yakovlev, “Construction of convex and concave functions on a permutation polyhedron,” Dokl. AN URSR, Ser. A, No. 5, 66–68 (1988).Google Scholar
- 2.S. V. Yakovlev and I. V. Grebennik, “On some classes of optimization problems on the set of arrangements and their applications,” Izv. Vyssh. Ucheb. Zaved., Ser. Mathem., No. 11, 74–86 (1991).Google Scholar
- 4.Yu. G. Stoyan and S. V. Yakovlev, Mathematical Models and Optimization Methods in Geometrical Design [in Russian], Naukova Dumka, Kiev (1986).Google Scholar
- 5.S. V. Yakovlev, N. I. Gil', V. M. Komyak, et al., Elements of Geometrical Design Theory [in Russian], Naukova Dumka, Kiev (1995).Google Scholar
- 6.Yu. G. Stoyan and O. A. Emets, Theory and Methods of Euclidean Combinatorial Optimization [in Russian], ISDO, Kiev (1993).Google Scholar
- 8.B. N. Pshenichnyi, Convex Analysis and Extremal Problems [in Russian], Nauka, Moscow (1980).Google Scholar
- 10.R. Bellman, Introduction to Matrix Analysis [Russian translation], Nauka, Moscow (1976).Google Scholar
- 11.K. Arrow, L. Hurwicz, and H. Uzawa, Studies in Linear and Nonlinear Programming [Russian translation], IL, Moscow (1962).Google Scholar
- 12.G. M. Fikhtengorts, A Course in Differential and Integral Calculus [in Russian], Vol. 1, Nauka, Moscow (1969).Google Scholar
- 13.N. Z. Shor and S. I. Stetsenko, Quadratic Extremal Problems and Nondifferentiable Optimization [in Russian], Naukova Dumka, Kiev (1989).Google Scholar