Optimization problems and second order approximated optimization problems

Authors

  • Eugenia Duca Technical University Cluj-Napoca
  • Dorel I. Duca Babeş-Bolyai University, Cluj-Napoca

Keywords:

saddle points, invex functions, pseudoinvex functions, \(\eta\)-approximation

Abstract

In this paper, a so-called second order approximated optimization problem associated to an optimization problem is considered. The equivalence between the saddle points of the lagrangian of the second order approximated optimization problem and optimal solutions of the original optimization problem is established.

Downloads

Download data is not yet available.

References

T. Antczak, A modified objective function method in mathematical programming with second order invexity, Numerical Functional Analysis and Optimization, nos. 1-2, pp. 1-12, 2007, https://doi.org/10.1080/01630560701190265.

T. Antczak, Saddle-Point Criteria in an η-Approximation Method for Nonlinear Mathematical Programming Problems Involving Invex Functions, Journal of Optimization Theory and Applications, 132, pp. 71-87, 2007, https://doi.org/10.1007/s10957-006-9069-9

T. Antczak, Saddle Point Criteria and Duality in Multiobjective Programming via an η-Approximation Method, ANZIAM J., 47, pp. 155-172, 2005. https://doi.org/10.1017/s1446181100009962

T. Antczak, An η-Approximation Approach to Nonlinear Mathematical Programming Involving Invex Functions, Numerical Functional Analysis and Optimization, 25, pp. 423-438, 2004, https://doi.org/10.1081/nfa-200042183

A. Ben-Israel and B. Mond, What is Invexity ?, Journal of the Australian Mathematical Society, 28B, pp. 1-9, 1986, https://doi.org/10.1017/s0334270000005142

B.D. Craven, Invex Functions and Constrained local Minima, Bulletin of the Australian Mathematical Society, 24, pp. 357-366, 1981, https://doi.org/10.1017/s0004972700004895

D.I. Duca, On the Higher-Order in Nonlinear Programming in Complex Space, Seminar on Optimization Theory Cluj-Napoca, pp. 39-50, 1985, Preprint 85-5, Univ. Babeş-Bolyai, Cluj-Napoca, 1985.

D.I. Duca, Multicriteria Optimization in Complex Space, House of the Book of Science, Cluj-Napoca, 2006

D.I. Duca, Optimization Problems and η-Approximated Optimization Problems, Studia Univ. "Babeş-Bolyai", Mathematica, 54, no. 4, pp. 49-62, 2009.

M.A. Hanson, On Sufficiency of Kuhn-Tucker Conditions, Journal of Mathematical Analysis and Applications, 30, pp. 545-550, 1981, https://doi.org/10.1016/0022-247x(81)90123-2

O.L. Mangasarian, Nonlinear Programming, McGraw-Hill Book Company, New York, NY, 1969.

O.L. Mangasarian, Second-and Higher-Order Duality in Nonlinear Programming, Journal of Mathematical Analysis and Applications, 51, pp. 607-620, 1975, https://doi.org/10.1016/0022-247x(75)90111-0

D.H. Martin, The Essence of Invexity, Journal of Optimization Theory and Applications, 47, pp. 65-76, 1985, https://doi.org/10.1007/bf00941316

S.K. Mishra and K.K. Lai: Second Order Symmetric Duality in Multiobjective Programming Involving Generalized Cone-Invex Functions, European Journal of Operational Research, 178, no. 1, pp. 20-26, 2007.

S.K. Mishra, Second Order Generalized Invexity and Duality in Mathematical Programming, Optimization, 42, no. 1, pp. 51-69, 1997, https://doi.org/10.1016/j.ejor.2005.11.024

P. Wolfe, A Duality Theorem for Nonlinear Programming, Quart. Appl. Math., 19, pp. 239-244, 1961, https://doi.org/10.1090/qam/135625

J. Zhang and B. Mond, Second Order B-Invexity and Duality in Mathematical Programming, Utilitas Mathematica, 50, pp. 19-31, 1996.

Downloads

Published

2010-08-01

How to Cite

Duca, E., & Duca, D. I. (2010). Optimization problems and second order approximated optimization problems. Rev. Anal. Numér. Théor. Approx., 39(2), 107–121. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2010-vol39-no2-art2

Issue

Section

Articles