Skip to main content
SpringerLink
Log in

New Generalized Invexity for Duality in Multiobjective Programming Problems Involving N-Set Functions

  • Conference paper
Generalized Convexity, Generalized Monotonicity and Applications

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 77))

Abstract

In this paper, we introduce four types of generalized convexity for an n -set function and discuss optimality and duality for a multiobjective programming problem involving n -set functions. Under some mild assumption on the new generalized convexity, we present a few optimality conditions for an efficient solution and a weakly efficient solution to the problem. Also we prove a weak duality theorem and a strong duality theorem for the problem and its Mond-Weir and general Mond-Weir dual problems respectively.

The research was supported by the University Grants Commission of India, the National Natural Science Foundation of China, Research Grants Council of Hong Kong and the Grantin-Aid (C-14550405) from the Ministry of Education, Science, Sports and Culture of Japan.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aghezzaf, B. and Hachimi, M. (2000), Generalized Invexity and Duality in Multiobjective Programming Problems, Journal of Global Optimization, vol. 18, pp. 91–101.

    Article  MathSciNet  MATH  Google Scholar 

  • Bector, C.R. and Singh, M. (1996), Duality for Multiobjective B-Vex Programming Involving n -set Functions, Journal of Mathematical Analysis and Applications, vol. 202, pp. 701–726.

    Article  MathSciNet  MATH  Google Scholar 

  • Chou, J.H., Hsia, W.S. and Lee, T.Y. (1985), On Multiple Objective Programming Problems with Set Functions, Journal of Mathematical Analysis and Applications, vol. 105, pp. 383–394.

    Article  MathSciNet  MATH  Google Scholar 

  • Chou, J.H., Hsia, W.S. and Lee, T.Y. (1986), Epigraphs of Convex Set Functions, Journal of Mathematical Analysis and Applications, vol. 118, pp. 247–254.

    Article  MathSciNet  MATH  Google Scholar 

  • Corley, H.W. (1987), Optimization Theory for n -set Functions, Journal of Mathematical Analysis and Applications, vol. 127, pp. 193–205.

    Article  MATH  MathSciNet  Google Scholar 

  • Hanson, M.A. and Mond, B. (1987), Convex Transformable Programming Problems and Invexity, Journal of Information and Optimization Sciences, vol. 8, pp. 201–207.

    MathSciNet  MATH  Google Scholar 

  • Hsia, W.S. and Lee, T.Y. (1987), Proper D Solution of Multiobjective Programming Problems with Set Functions, Journal of Optimization Theory and Applications, vol. 53, pp. 247–258.

    Article  MathSciNet  MATH  Google Scholar 

  • Jeyakumar, V. and Mond, B. (1992), On Generalized Convex Mathematical Programming, Journal of Australian Mathematical Society Ser. B, vol. 34, pp. 43–53.

    Article  MathSciNet  MATH  Google Scholar 

  • Kaul, R.N., Suneja, S.K. and Srivastava, M.K. (1994), Optimality Criteria and Duality in Multiple Objective Optimization Involving Generalized Invexity, Journal of Optimization Theory and Applications, vol. 80, pp. 465–482.

    Article  MathSciNet  MATH  Google Scholar 

  • Kim, D.S., Jo, C.L. and Lee, G.M. (1998), Optimality and Duality for Multiobjective Fractional Programming Involving n -set Functions, Journal of Mathematical Analysis and Applications, vol. 224, pp. 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  • Lin, L.J. (1990), Optimality of Differentiable Vector-Valued n -set Functions, Journal of Mathematical Analysis and Applications, vol. 149, pp. 255–270.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, L.J. (1991a), On the Optimality Conditions of Vector-Valued n -set Functions, Journal of Mathematical Analysis and Applications, vol. 161, pp. 367–387.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, L.J. (1991b), Duality Theorems of Vector Valued n -set Functions, Computers and Mathematics with Applications vol. 21, pp. 165–175.

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, L.J. (1992), On Optimality of Differentiable Nonconvex n -set Functions, Journal of Mathematical Analysis and Applications, vol. 168, pp. 351–366.

    Article  MATH  MathSciNet  Google Scholar 

  • Mangasarian, O.L. (1969), Nonlinear Programming, McGraw-Hill, New York.

    MATH  Google Scholar 

  • Mazzoleni, P. (1979), On Constrained Optimization for Convex Set Functions, in Survey of Mathematical Programming, Edited by A. Prekop, North-Holland, Amsterdam, vol. 1, pp. 273–290.

    Google Scholar 

  • Mishra, S.K. (1998), On Multiple-Objective Optimization with Generalized Univexity, Journal of Mathematical Analysis and Applications, vol. 224, pp. 131–148.

    Article  MATH  MathSciNet  Google Scholar 

  • Mond, B. and Weir, T. (1981), Generalized Concavity and Duality, in Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, pp. 263–280.

    Google Scholar 

  • Morris, R.J.T. (1979), Optimal Constrained Selection of a Measurable Subset, Journal of Mathematical Analysis and Applications, vol. 70, pp. 546–562.

    Article  MATH  MathSciNet  Google Scholar 

  • Mukherjee, R.N. (1991), Genaralized Convex Duality for Multiobjective Fractional Programs, Journal of Mathematical Analysis and Applications, vol. 162, pp. 309–316

    Article  MATH  MathSciNet  Google Scholar 

  • Preda, V. (1991), On Minimax Programming Problems Containing n -set Functions, Optimization, vol. 22, pp. 527–537.

    MATH  MathSciNet  Google Scholar 

  • Preda, V. (1995), On Duality of Multiobjective Fractional Measurable Subset Selection Problems, Journal of Mathematical Analysis and Applications, vol. 196, pp. 514–525.

    Article  MATH  MathSciNet  Google Scholar 

  • Preda, V. and Stancu-Minasian, I.M. (1997), Mond-Weir Duality for Multiobjective Mathematical Programming with n -set Functions, Analele Universitati Bucuresti, Matematica-Informatica, vol. 46, pp. 89–97.

    MathSciNet  Google Scholar 

  • Preda, V. and Stancu-Minasian, I.M. (1999), Mond-Weir Duality for Multiobjective Mathematical Programming with n -set Functions, Revue Roumaine de Mathématiques Pures et Appliquées, vol. 44, pp. 629–644.

    MathSciNet  MATH  Google Scholar 

  • Preda, V. and Stancu-Minasian, I.M. (2001), Optimality and Wolfe Duality for Multiobjective Programming Problems Involving n -set Functions, in Generalized Convexity and Generalized Monotonicity, Edited by Nicolas Hadjisavvas, J-E Martinez-Legaz and J-P Penot, Springer, Berlin, pp. 349–361.

    Google Scholar 

  • Rosenmuller, J. and Weidner, H.G. (1974), Extreme Convex Set Functions with Finite Carries: General Theory, Discrete Mathematics, vol. 10, pp. 343–382.

    Article  MathSciNet  Google Scholar 

  • Tanaka, K. and Maruyama, Y. (1984), The Multiobjective Optimization Problems of Set Functions, Journal of Information and Optimization Sciences, vol. 5, pp. 293–306.

    MathSciNet  MATH  Google Scholar 

  • Ye, Y.L. (1991), D-invexity and Optimality Conditions, Journal of Mathematical Analysis and Applications, vol. 162, pp. 242–249.

    Article  MATH  MathSciNet  Google Scholar 

  • Zalmai, G.J. (1989), Optimality Conditions and Duality for Constrained Measurable Subset Selection Problems with Minmax Objective Functions, Optimization, vol. 20, pp. 377–395.

    MATH  MathSciNet  Google Scholar 

  • Zalmai, G.J. (1990), Sufficiency Criteria and Duality for Nonlinear Programs Involving n -set Functions, Journal of Mathematical Analysis and Applications, vol. 149, pp. 322–338.

    Article  MATH  MathSciNet  Google Scholar 

  • Zalmai, G.J. (1991), Optimality Conditions and Duality for Multiobjective Measurable Subset Selection Problems, Optimization, vol. 22, pp. 221–238.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, G. B. Pant University of Agriculture and Technology, India

    S. K. Mishra

  2. Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, China

    S. Y. Wang

  3. Department of Management Sciences, City University of Hong Kong, Hong Kong

    K. K. Lai

  4. Department of Computer Science and Systems Engineering, Muroran Institute of Technology, Japan

    J. Shi

Authors
  1. S. K. Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Y. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. K. Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RMIT University, Australia

    Andrew Eberhard

  2. University of the Aegean, Greece

    Nicolas Hadjisavvas

  3. University of Avignon, France

    Dinh The Luc

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science + Business Media, Inc.

About this paper

Cite this paper

Mishra, S.K., Wang, S.Y., Lai, K.K., Shi, J. (2005). New Generalized Invexity for Duality in Multiobjective Programming Problems Involving N-Set Functions. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol 77. Springer, Boston, MA. https://doi.org/10.1007/0-387-23639-2_19

Download citation

Publish with us

Policies and ethics

Search

Navigation