Skip to main content
SpringerLink
Log in

Convex and convex-like optimization over a range inclusion problem and first applications

  • Published:
Decisions in Economics and Finance Aims and scope Submit manuscript

Abstract

The paper deals with the minimization of a function over the solution set of a range inclusion problem determined by a multifunction. A strong Lagrange duality is provided first in terms of a quasirelative interior condition and then under a so-called Assumption (S). When the function and the multifunction are convex, we improve this duality under a closed cone condition. The stability analysis is investigated. In addition, if the multifunction is a convex process, then the Fenchel dual is performed in terms of its conjugate. As a first application, we provide a unified approach to the optimization of general discrete inclusions systems; in particular, we improve several results on optimal control, strong Lagrange duality and Fenchel duality for some classes of convex controlled discrete processes.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley Interscience, New York (1984)

    Google Scholar 

  • Borwein, J.M.: Multivalued convexity and optimization : a unified approach to inequality and equality constraints. Math. Prog. 13, 183–189 (1977)

    Article  Google Scholar 

  • Borwein, J.M., Lewis, A.S.: Partially finite convex programming. Part I: Quasi relative interior and duality theory. Math. Prog. 57, 15–48 (1992)

    Article  Google Scholar 

  • Bot, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin (2010)

  • Bot, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61(1), 35–65 (2012)

    Article  Google Scholar 

  • Bot, R.I., Grad, S.M., Wanka, G.: On strong and total Lagrange duality for convex optimization problems. J. Math. Anal. Appl. 337, 1315–1325 (2008)

    Article  Google Scholar 

  • Bot, R.I., Csetnek, E.R.: Conjugate duality and the control of linear discrete systems. J. Optim. Theory Appl. 159(3), 576–589 (2013)

    Article  Google Scholar 

  • Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)

    Google Scholar 

  • Cruceanu, S.: Duality and optimality for convex extremal problems decribed by discrete inclusions. Math. Operationsforsch. Sratist. Ser. Optim. 11(1), 13–30 (1980)

    Article  Google Scholar 

  • Giannessi, F.: Constrained Optimization and Image Space Analysis, Separation of sets and Optimality Conditions, vol. I. Springer, Berlin (2005)

    Google Scholar 

  • Jeyakumar, V.: Constraint qualifications characterizing Lagrange duality in convex optimization. J. Optim. Theory Appl. 136, 31–41 (2008)

    Article  Google Scholar 

  • Kaczorek, T.: Singular two dimensional continuous-discrete linear systems. Dyn. Contin. Discret. Impuls. Syst. 2, 193–204 (1996)

    Google Scholar 

  • Lee, K.Y., Show, S., Barr, R.O.: On the control of discret-time distributed parameter systems. SIAM J. Control 10, 361–376 (1972)

    Article  Google Scholar 

  • Luenberger, D.G.: Dynamical systems in descriptor form. IEEE. Trans. Automat. Control AC 22, 312–321 (1977)

    Article  Google Scholar 

  • Luenberger, D.G.: Time invariant descriptor systems. Automatica 14, 473–480 (1978)

    Article  Google Scholar 

  • Mahmudov, E.N.: On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type. J. Math. Anal. Appl. 307, 628–640 (2005)

    Article  Google Scholar 

  • Mahmudov, E.N.: Necessary and sufficient conditions for discrete and differential inclusions of elliptic type. J. Math. Anal. Appl. 323, 768–789 (2006)

    Article  Google Scholar 

  • Mahmudov, E.N.: Duality in the problems of optimal control described by first-order partial differential inclusions. Optimization 59(4), 589–599 (2010)

    Article  Google Scholar 

  • Mahmudov, E.N.: Duality in problems described by convex multivalued mappings. Cybernetic 23(2), 238–244 (1987)

    Article  Google Scholar 

  • Mokhtar-Kharroubi, H.: Sur quelques fonctions marginales et leurs applications, Thèse Doctorat Es-sciences, Lille I (1987)

  • Mokhtar-Kharroubi, H.: Convex and Paraconvex multimaps. Vietnam J Math (Preprint)

  • Mordukhovich, B.S.: Vatriational Analysis and Generalized Differentiation II. Springer, Berlin (2006)

    Google Scholar 

  • Phat, V.N.: Constrained Control Problems of Discrete Processes (Series in Advances in Mathematics for Applied Sciences) vol. 42 (1996)

  • Pshenichnyi, B.M.: Convex multi-valued Mapping and their conjugates. In: Los, J., Los, M.W. (eds.) Mathematical Models in Economics, pp. 333–349. North Holland, Amsterdam (1974)

    Google Scholar 

  • Reich, S.: An existence theorem for a difference inclusion in general Banach Space. J. Math. Anal. Appl. 160, 406–412 (1991)

    Article  Google Scholar 

  • Rubinov, A.M., Makarov, V.L.: Mathematical Theory of Economic dynamics and Equilibria. Springer, Berlin (1977)

    Google Scholar 

  • Zabczyk, J.: Remarks on the control of dicrete time distributed parameter systems. SIAM J. Control 12, 721–735 (1974)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université Oran I, BP l524, Elmn’aouer, 31000, Oran, Algeria

    Hocine Mokhtar-Kharroubi

Authors
  1. Hocine Mokhtar-Kharroubi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hocine Mokhtar-Kharroubi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mokhtar-Kharroubi, H. Convex and convex-like optimization over a range inclusion problem and first applications. Decisions Econ Finan 40, 277–299 (2017). https://doi.org/10.1007/s10203-017-0190-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10203-017-0190-z

Keywords

Mathematics Subject Classification

JEL Classification

Search

Navigation