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Characterizing Existence of Minimizers and Optimality to Nonconvex Quadratic Integrals

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Abstract

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete description about the existence or nonexistence of solution in terms of the (strict) copositivity of the matrix involved in the integrand. In addition, we characterize optimality via the Hamiltonian function.

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References

  1. Tonelli, L.: Fondamenti di Calcolo delle Variazioni. Zanichelli, Bologna (1921)

    MATH  Google Scholar 

  2. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics. SIAM, New Delhi (1999)

    Book  Google Scholar 

  3. Bomze, I.M.: On standard quadratic optimization problems. J. Glob. Optim. 13, 369–387 (1998)

    Article  MathSciNet  Google Scholar 

  4. Bomze, I.M., de Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Glob. Optim. 24, 163–185 (2002)

    Article  MathSciNet  Google Scholar 

  5. Bomze, I.M., Dür, M., Roos, C., de Klerk, E., Quist, A.J., Terlaky, T.: On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18, 301–320 (2000)

    Article  MathSciNet  Google Scholar 

  6. Flores-Bazán, F., Cárcamo, G., Caro, S.: Extensions of the standard quadratic optimization problem: strong duality, optimality, hidden convexity and S-lemma. Appl. Math. Optim. 81, 383–408 (2020)

    Article  MathSciNet  Google Scholar 

  7. Flores-Bazán, F., Jourani, A., Mastroeni, A.: On the convexity of the value function for a class of nonconvex variational problems: existence and optimaly conditions. SIAM J. Control Optim. 52, 3673–3693 (2014)

    Article  MathSciNet  Google Scholar 

  8. Mordukhovich, B.S., Sagara, N.: Subdifferentials of nonconvex integral functionals in Banach spaces with applications to stochastic dynamic programming. J. Convex Anal. 25, 643–673 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Mordukhovich, B.S., Sagara, N.: Subdifferential of value functions in nonconvex dynamic programming for nonstationary stochastic processes. Commun. Stoch. Anal. 13, 3 (2019). Article 5

    MathSciNet  Google Scholar 

  10. Crasta, G.: Existence of minimizers for nonconvex variational problems with slow growth. J. Optim. Theory Appl. 99, 381–401 (1998)

    Article  MathSciNet  Google Scholar 

  11. Artstein, Z.: A variational problem determined by probability measures. Optimization 68(1), 81–98 (2019)

    Article  MathSciNet  Google Scholar 

  12. Artstein, Z.: Convexity and closure in optimal allocations determined by decomposable measures. Vietnam J. Math. 47, 563–577 (2019)

    Article  MathSciNet  Google Scholar 

  13. Bouras, A., Giner, E.: Kuhn–Tucker conditions and integral functionals. J. Convex Anal. 8, 533–553 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Ioffe, A.D., Tikhomirov, V.M.: On minimization of integral functionals. Funct. Anal. Appl. 3, 218–227 (1969)

    Article  MathSciNet  Google Scholar 

  15. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, Part I: quasi relative interiors and duality theory. Math. Program. 57, 15–48 (1992)

    Article  Google Scholar 

  16. Olech, O.: The Lyapunov theorem: its extensions and applications. In: Cellina, A. (ed.) Methods of Nonconvex Analysis. Springer, New York (1990)

    Google Scholar 

  17. Sagara, N.: Decomposability, convexity and continuous linear operators in \(L^1(\mu, E)\): the case for saturated measure spaces. Linear Nonlinear Anal. 5, 113–19 (2019)

    MathSciNet  Google Scholar 

  18. Sagara, N.: Optimality conditions for nonconvex variational problems with integral constraints in Banach spaces. J. Convex Anal. 27, 567–583 (2020)

    MathSciNet  MATH  Google Scholar 

  19. Wheeden, R.L., Zygmund, A.: Measure and Integral: An Introduction to Real Analysis. Chapman and Hall/CRC Pure and Applied Mathematics, 2nd edn. Chapman and Hall/CRC, Boca Raton (2015)

    Book  Google Scholar 

  20. Aubin, J., Frankowka, H.: Set-Valued Analysis. Birkhauser, Boston (1990)

    Google Scholar 

  21. Hiai, F., Umegaki, H.: Integrals, conditional expectations, and martingales of multivalued functions. J. Multivar. Anal. 7, 149–182 (1977)

    Article  MathSciNet  Google Scholar 

  22. Hadeler, K.P.: On copositive matrices. Linear Algorithm Appl. 49, 79–89 (1983)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are indebted to two anonymous referees and to Associate Editor for their helpful remarks and suggestions that allowed us to improve the presentation of the paper. The research, for the first author, was supported in part by ANID-Chile through FONDECYT 118-1316 and PIA/Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento AFB170001.

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Correspondence to Fabián Flores-Bazán.

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Communicated by Boris S. Mordukhovich.

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Flores-Bazán, F., González-Valencia, L. Characterizing Existence of Minimizers and Optimality to Nonconvex Quadratic Integrals. J Optim Theory Appl 188, 497–522 (2021). https://doi.org/10.1007/s10957-020-01794-8

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  • DOI: https://doi.org/10.1007/s10957-020-01794-8

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