A short elementary proof of the Lagrange multiplier theorem

Optimization Letters

Volumn 6, Issue 8, 2012, Pages 1597-1601

  Brezhneva, Olga ^a   Tret'yakov, Alexey A ^b,c,d   Wright, Stephen E ^a  

^a MIAMI UNIVERSITY   (United States)

^b DORODNICYN COMPUTING CENTER   (Russian Federation)

^c SYSTEMS RESEARCH INSTITUTE   (Poland)

^d SIEDLCE UNIVERSITY OF NATURAL SCIENCES AND HUMANITIES   (Poland)

Author keywords

First order necessary conditions; Lagrange multiplier theorem; Nonlinear programming

Indexed keywords

ADVANCED ANALYSIS; CONTINUOUS FUNCTIONS; CRITICAL POINTS; DIFFERENTIABILITY; ELEMENTARY PROOF; EQUALITY-CONSTRAINED OPTIMIZATION; FIRST-ORDER NECESSARY CONDITIONS; IMPLICIT FUNCTION THEOREM; LAGRANGE MULTIPLIER THEOREM;

LAGRANGE MULTIPLIERS; LINEAR ALGEBRA; NONLINEAR PROGRAMMING;

THEOREM PROVING;

EID: 84869505267     PISSN: 18624472     EISSN: 18624480     Source Type: Journal    
DOI: 10.1007/s11590-011-0349-4     Document Type: Article

References (11)

1. Beltrami E. J.: A constructive proof of the Kuhn-Tucker multiplier rule. J. Math. Anal. Appl. 26, 297-306 (1969).
2. Bertsekas D. P.: Nonlinear programming. Athena Scientific, Boston (1995).
3. Birbil S. I., Frenk J. B. G., Still G. J.: An elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions in nonlinear programming. Euro. J. Oper. Res. 180, 479-484 (2007).
4. Brezhneva O. A., Tret'yakov A. A., Wright S. E.: A simple and elementary proof of the Karush-Kuhn-Tucker theorem for inequality-constrained optimization. Optim. Lett. 3, 7-10 (2009).
5. Clarke F. H.: Optimization and nonsmooth analysis. Wiley, New York (1983).
6. Craven B. D.: Mathematical programming and control theory. Chapman and Hall, London (1978).
7. Halkin H.: Implicit functions and optimization problems without continuous differentiability of the data. SIAM J. Control 12, 229-236 (1974).
8. Luenberger D. G.: Introduction to linear and nonlinear programming. Addison-Wesley, Reading (1973).
9. McShane E. J.: The Lagrange multiplier rule. Amer. Math. Monthly 80, 922-925 (1973).
10. Pourciau B. H.: Modern multiplier rules. Amer. Math. Monthly 87, 433-452 (1980).
11. Rockafellar R. T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183-238 (1993).