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

Abstract

top
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.

How to cite

top

Agnieszka 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
  1. [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. 
  2. [1] B.P. Demidovitch and I.A. Maron, Basis of Computational Mathematics, Nauka, Moscow 1973. (in Russian) 
  3. [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. [4] A.F. Izmailov and A.A. Tret`yakov, Factor-Analysis of Non-Linear Mapping, Nauka, Moscow, Fizmatlit Publishing Company, 1994. 
  5. [5] L.V. Kantorovitch and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford 1982. 
  6. [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. 
  7. [M] K. Maurin, Analysis, Part I, Elements, PWN, Warsow 1971. (in Polish) 
  8. [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. [9] A.A. Tret'yakov, Necessary Conditions for Optimality of p-th Order, Control and Optimization, Moscow MSU (1983), 28-35 (in Russian). 
  10. [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. 
  11. [8] A.A. Tret'yakov, The implicit function theorem in degenerate problems, Russ. Math. Surv. 42 (1987), 179-180. Zbl0683.58008

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.