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The conditions of minimum for a smooth function on the boundary of a quasidifferntiable set

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In this paper, we consider problems of mathematical programming with nonsmooth constraints of equality type given by quasidifferentiable functions. By using the technique of upper convex approximations, developed by B. N. Pshenichy, necessary conditions of extremum for such problems are established. The Lagrange multipliers signs are specified by virtue of the fact that one can construct whole familers of upper convex approximations for quasidifferentiable function and thus the minimum points in such extremal problems are characterized more precisely. Also the simplest problem of calculus of variations with free right hand side is considered, where the left end of the trajectory starts on the boundary of the convex set. The transversality condition at the left end of the trajectory is improved provided sertain sufficient conditons hold.

Keywords

upper convex approximation; quasidifferentiable function; subdifferential; tent

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DOI

10.20310/2686-9667-2020-25-130-165-182

UDC

519.6

Pages

165-182

References

[1] F.H. Clarke, “A new approah to Lagrange multipliers”, Mathematics of Operations Research, 1:2 (1976), 165–174. [2] R.A. Khachatryan, “On Necessary Optimality Conditions in Non-Smooth Problems with Constraints”, Vladikavkazian Mathematical Journal, 18:3 (2016), 72–83 (In Russian). [3] A.D. Ioffe, “Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional costraints”, Mathematical Programming, 58 (1993), 137–145. [4] E.S. Polovinkin, “Subdifferential for the difference of two convex functions”, J. Math. Sci., 218:5 (2016), 664–677. [5] V.F. Dem’yanov, L.V. Vasilev, Nedifferenciruemaja Optimizacia, Nauka Publ., Moscow, 1981 (In Russian). [6] V.F. Dem’yanov, A.M. Rubinov, Osnovi Negladkogo Analisa i Kvazidifferentialnogo Ischislenya, Nauka Publ., Moscow, 1990 (In Russian). [7] V.F. Dem’yanov, L.N. Polyakova, “Minimization of a quasi-differentiable function in a quasi-differntiable set”, Computational Mathematics and Mathematical Physics, 20:4 (1980), 34–43. [8] V.F. Dem’yanov, B.N. Malozemov, Vvedenie v Minimax, Nauka Publ., Moscow, 1972 (In Russian). [9] F. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. [10] B. N. Pshenycnii, Vypuklij Analiz i Extremalnye Zadachi, Nauka Publ., Moscow, 1980 (In Russian). [11] V.G. Boltyanskii, “The method of tents in the theory of extremal problems”, Russian Math. Surveys, 30:3 (1975), 1–54. [12] R. Ivanachi, “On the intersection of Continous local Tents”, Proc. Japan Acad., 69:A (1993), 308–311. [13] B.N. Pshenychnii, R. A. Khachatryan, “On the necessary extremum conditions for a nonsmooth functions”, Izwestya NAN Armenii, Mathematika, 18:4(1983), 318–325 (In Russian). [14] B.N. Pshenychnii, Neobkhodymie Uslovya Extremuma, Nauka Publ., Moscow, 1982 (In Russian). [15] A.G. Sukharev, A.V. Timokhov, V.V. Fedorov, Methodi Optimizacii, Nauka Publ., Moscow, 1986 (In Russian). [16] R.A. Khachatryan, “On regular tangent cones”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 52:2 (2017), 74–80. [17] V. M. Alekseev, E.M. Galeev, V. M. Tikhomirov, Sbornik Zadach po Optimizacii, Teoria i Primeri-zadachy, Nauka Publ., Moscow, 1984 (In Russian).

Received

2020-03-18

Section of issue

Scientific articles

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