Second-order optimality conditions in nonsmooth vector optimization

Priyanka Yadav

doi:10.2478/candc-2023-0031

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Second-order optimality conditions in nonsmooth vector optimization

Control and Cybernetics

Volume 52 (2023): Issue 1 (March 2023)

By: Priyanka Yadav

Open Access

|Feb 2024

Abstract

In this paper, we introduce new classes of nonsmooth second-order cone-convex functions and respective generalizations in terms of first and second-order directional derivative. These classes encapsulate several already existing classes of cone-convex functions and their weaker variants. Second-order KKT type sufficient optimality conditions and duality results for a nonsmooth vector optimization problem are proved using these functions. The results have been supported by examples.

References

Aggarwal, S. (1998) Optimality and duality in mathematical programming involving generalized convex functions. Ph.D. thesis, University of Delhi, Delhi.
Search in Google Scholar Back to article
Aghezzaf, B. (2003) Second order mixed type duality in multiobjective programming problem. Journal of Mathematical Analysis and Applications 285, 97–106.
Search in Google Scholar Back to article
Ahmad, I. and Husain, Z. (2006) Second order (F; α ρ d)-convexity and duality in multiobjective programming. Information Sciences 176, 3094–3103.
Search in Google Scholar Back to article
Auslender, A. (1979) Penalty methods for computing points that satisfy second order necessary conditions. Mathematical Programming 17, 229–238.
Search in Google Scholar Back to article
Ben-Tal, A. and Zowe, J. (1985) Directional derivatives in nonsmooth optimization. Journal of Optimization Theory and Applications 47, 483–490.
Search in Google Scholar Back to article
Coladas, L., Li, Z. and Wang, S. (1994) Optimality conditions for multiobjective and nonsmooth minimisation in abstract spaces. Bulletin of Australian Mathematical Society 50, 205–218.
Search in Google Scholar Back to article
Cominetti, R. and Correa, R. (1990) A generalized second-order derivative in nonsmooth optimization. SIAM Journal on Control and Optimization 28, 789-809.
Search in Google Scholar Back to article
Demyanov, W.F. and Pevnyi, A.B. (1974) Expansion with Respect to a Parameter of the Extremal Values of Game Problems. USSR Computational Mathematics and Mathematical Physics 14, 33-45.
Search in Google Scholar Back to article
Facchinei, F. and Lucidi, S. (1998) Convergence to Second Order Stationary Points in Inequality Constrained Optimization. Mathematics of Operations Research 23, 746—766.
Search in Google Scholar Back to article
Flores-Bazan, F., Hadjisavvas, N. and Vera, C. (2007) An Optimal Alternative Theorem and Applications to Mathematical Programming. Journal of Global Optimization 37, 229–243.
Search in Google Scholar Back to article
Giorgi, G. and Guerraggio, A. (1996) The notion of invexity in vector optimization: Smooth and nonsmooth case. In: J. P. Crouzeix, J. E. Martinez-Legaz and M. Volle (Eds), Generalized Convexity, Generalized Monotonicity: Recent Results. Nonconvex Optimization and Its Applications, 27, Kluwer Academic Publishers, Dordrecht, 389–401.
Search in Google Scholar Back to article
Hanson, M.A. (1993) Second order invexity and duality in mathematical programming. Opsearch 30, 313–320.
Search in Google Scholar Back to article
Kumar, P. and Sharma, B. (2017) Higher order e ciency and duality for multiobjective variational problem. Control and Cybernetics, 46, 137-145.
Search in Google Scholar Back to article
Luenberger, D. G. and Ye, Y. (2008) Linear and Nonlinear Programmming. Springer, New York.
Search in Google Scholar Back to article
Mangasarian, O.L. (1975) Second and higher-order duality in nonlinear programming. Journal of Mathematical Analysis and Applications 51, 607–620.
Search in Google Scholar Back to article
Mishra, S.K. (1997) Second order generalized invexity and duality in mathematical programming. Optimization 42, 51–69.
Search in Google Scholar Back to article
Mond, B. (1974) Second-order duality for nonlinear programs. Opsearch 11, 90–99.
Search in Google Scholar Back to article
Mond, B. and Weir, T. (1981-1983) Generalized convexity and higher order duality. Journal of Mathematical Sciences, 16–18, 74–94.
Search in Google Scholar Back to article
Nocedal, J. and Wright, S. J. (2006) Numerical Optimization. Springer, New York.
Search in Google Scholar Back to article
Suneja, S. K., Sharma, S. and Vani (2008) Second-order duality in vector optimization over cones. Journal of Applied Mathematics and Informatics 26, 251–261.
Search in Google Scholar Back to article
Yuan, G. X., Chang, K. W., Hsieh, C. J. and Lin, C. J. (2010) A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research 11, 3183–3234.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/candc-2023-0031 | Journal eISSN: 2720-4278 | Journal ISSN: 0324-8569

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Language: English

Page range: 35 - 51

Submitted on: Aug 1, 2021

Accepted on: Sep 1, 2023

Published on: Feb 24, 2024

Published by: Systems Research Institute Polish Academy of Sciences

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

vector optimization,

cones,

nonsmooth second-order cone-convexity,

second-order optimality,

duality

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Electrical engineering,

Fundamentals of electrical engineering,

Mechanical engineering,

Fundamentals of mechanical engineering,

Mathematics,

General mathematics

© 2024 Priyanka Yadav, published by Systems Research Institute Polish Academy of Sciences
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 52 (2023): Issue 1 (March 2023)