The conditions of minimum for a smooth function on the boundary of a quasidifferntiable set. (Russian. English summary) Zbl 1480.90229

Summary: 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.


90C30 Nonlinear programming
Full Text: DOI MNR


[1] F. H. Clarke, “A new approah to Lagrange multipliers”, Mathematics of Operations Research, 1:2 (1976), 165-174 · Zbl 0404.90100 · doi:10.1287/moor.1.2.165
[2] R. A. Khachatryan, “On Necessary Optimality Conditions in Non-Smooth Problems with Constraints”, Vladikavkazian Mathematical Journal, 18:3 (2016), 72-83 (In Russian) · Zbl 1488.49036
[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 · Zbl 0782.90095 · doi:10.1007/BF01581262
[4] E. S. Polovinkin, “Subdifferential for the difference of two convex functions”, J. Math. Sci., 218:5 (2016), 664-677 · Zbl 1353.49024 · doi:10.1007/s10958-016-3049-x
[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 · Zbl 0466.90068 · doi:10.1016/0041-5553(80)90269-4
[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 · Zbl 0582.49001
[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 · Zbl 0334.49014 · doi:10.1070/RM1975v030n03ABEH001411
[12] R. Ivanachi, “On the intersection of Continous local Tents”, Proc. Japan Acad., 69:A (1993), 308-311 · Zbl 0798.49025 · doi:10.3792/pjaa.69.308
[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) · Zbl 0527.90087
[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) · Zbl 0602.90091
[16] R. A. Khachatryan, “On regular tangent cones”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 52:2 (2017), 74-80 · Zbl 1367.26054
[17] V. M. Alekseev, E. M. Galeev, V. M. Tikhomirov, Sbornik Zadach po Optimizacii, Teoria i Primeri-zadachy, Nauka Publ., Moscow, 1984 (In Russian)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.