# On existence of solutions to degenerate nonlinear optimization problems

Agnieszka Prusińska; Alexey Tret'yakov

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2007)

- Volume: 27, Issue: 1, page 151-164
- ISSN: 1509-9407

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topAgnieszka Prusińska, and Alexey Tret'yakov. "On existence of solutions to degenerate nonlinear optimization problems." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 27.1 (2007): 151-164. <http://eudml.org/doc/271183>.

@article{AgnieszkaPrusińska2007,

abstract = {
We investigate the existence of the solution to the following problem
min φ(x) subject to G(x)=0,
where φ: X → ℝ, G: X → Y and X,Y are Banach spaces. The question of existence is considered in a neighborhood of such point x₀ that the Hessian of the Lagrange function is degenerate. There was obtained an approximation for the distance of solution x* to the initial point x₀.
},

author = {Agnieszka Prusińska, Alexey Tret'yakov},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {Lagrange function; necessary condition of optimality; p-regularity; contracting mapping; p-factor operator; -regularity; -factor operator},

language = {eng},

number = {1},

pages = {151-164},

title = {On existence of solutions to degenerate nonlinear optimization problems},

url = {http://eudml.org/doc/271183},

volume = {27},

year = {2007},

}

TY - JOUR

AU - Agnieszka Prusińska

AU - Alexey Tret'yakov

TI - On existence of solutions to degenerate nonlinear optimization problems

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2007

VL - 27

IS - 1

SP - 151

EP - 164

AB -
We investigate the existence of the solution to the following problem
min φ(x) subject to G(x)=0,
where φ: X → ℝ, G: X → Y and X,Y are Banach spaces. The question of existence is considered in a neighborhood of such point x₀ that the Hessian of the Lagrange function is degenerate. There was obtained an approximation for the distance of solution x* to the initial point x₀.

LA - eng

KW - Lagrange function; necessary condition of optimality; p-regularity; contracting mapping; p-factor operator; -regularity; -factor operator

UR - http://eudml.org/doc/271183

ER -

## References

top- [0] V.M. Alexeev, V.M. Tihomirov and S.V. Fomin, Optimal Control, Consultants Bureau, New York, 1987. Translated from Russian by V.M. Volosov.
- [1] B.P. Demidovitch and I.A. Maron, Basis of Computational Mathematics, Nauka, Moscow 1973. (in Russian)
- [I-T] A.D. Ioffe and V.M. Tihomirov, Theory of extremal problems, North-Holland, Studies in Mathematics and its Applications, Amsterdam 1979. Zbl0407.90051
- [4] A.F. Izmailov and A.A. Tret`yakov, Factor-Analysis of Non-Linear Mapping, Nauka, Moscow, Fizmatlit Publishing Company, 1994.
- [5] L.V. Kantorovitch and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford 1982.
- [6] M.A. Krasnosel'skii, G.M. Wainikko, P.P. Zabreiko, Yu.B. Rutitskii and V.~Yu. Stetsenko, Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen (1972), 39.
- [M] K. Maurin, Analysis, Part I, Elements, PWN, Warsow 1971. (in Polish)
- [A-T] A. Prusińska and A.A. Tret'yakov, The theorem on existence of singular solutions to nonlinear equations, Trudy PGU, seria Mathematica, 12 (2005).
- [9] A.A. Tret'yakov, Necessary Conditions for Optimality of p-th Order, Control and Optimization, Moscow MSU (1983), 28-35 (in Russian).
- [11] A.A. Tret'yakov, Necessary and Sufficient Conditions for Optimality of p-th Order, USSR Comput. Math. and Math Phys. 24 (1984), 123-127.
- [8] A.A. Tret'yakov, The implicit function theorem in degenerate problems, Russ. Math. Surv. 42 (1987), 179-180. Zbl0683.58008

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