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Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions

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Abstract

In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems.

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References

  1. Beck A., Teboulle M. (2000). Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optim. 11:179–188

    Article  MATH  MathSciNet  Google Scholar 

  2. Ben-Tal A., Nemirovski A. (2000). Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications. SIAM-MPS, Philadelphia

    Google Scholar 

  3. Dahl G. (2000). A note on digonally dominat matrices. Linear Algebra Appl. 317:217–224

    Article  MATH  MathSciNet  Google Scholar 

  4. Derinkuyu, K., Pınar, M.C.: On the S-procedure and some variants, Math. Methods Oper. Res. (see http://www.ie.bilkent.edu.tr/~mustafap/pubs/)

  5. Dür M., Horst R., Locatelli M. (1998). Necessary and sufficient global optimality conditions for convex maximization revisited. J. Math. Anal. Appl. 217(2):637–649

    Article  MATH  MathSciNet  Google Scholar 

  6. Glover B.M., Jeyakumar V., Rubinov A.M. (1999). Dual conditions characterizing optimality for convex multi-objective programs. Math. Progr. 84:201–217

    MATH  MathSciNet  Google Scholar 

  7. Glover B.M., Ishizuka Y., Jeyakumar V., Tuan H.D. (1996). Complete characterizations of global optimality for problems involving the pointwise minimum of sublinear functions. SIAM J. Optim. 6:362–372

    Article  MATH  MathSciNet  Google Scholar 

  8. Hiriart-Urruty J.B. (2001). Global optimality conditions in maximizing a convex quadratic function under convex quadratic constraints. J. Global Optim. 21:445–455

    Article  MathSciNet  Google Scholar 

  9. Hiriart-Urruty J.B. (1998). Conditions for global optimality 2. J. Global Optim. 13:349–367

    Article  MATH  MathSciNet  Google Scholar 

  10. Hiriart-Urruty J.B., Lemarechal C. (1993). Convex Analysis and Minimization Algorithms. Springer, Berlin Heidelberg New York

    Google Scholar 

  11. Horst R., Pardalos P. (1995). Handbook of Global Optimization, Nonconvex Optimization and its Applications. Kluwer, Dordrecht

    Google Scholar 

  12. Jeyakumar V. (2000). Farkas Lemma: Generalizations, Encyclopedia of Optimization, Vol. 2. pp. 87–91, Kluwer Boston, USA

  13. Jeyakumar V. (2006). The strong conical hull intersection property for convex programming. Math. Program. Ser A, 106:81–92

    Article  MATH  MathSciNet  Google Scholar 

  14. Jeyakumar, V., Rubinov, A.M., Wu, Z.Y.: Sufficient global optimality conditions for non-convex quadratic minimization problems with box constraints. J. Global Optim. (in press)

  15. Moré J. (1993). Generalizations of the trust region problem. Optim. Meth. Soft. 2:189–209

    Google Scholar 

  16. Pallaschke, D., Rolewicz, S.: Foundations of Mathematical Optimization: Convex Analysis without Linearity. Dordrechet (1997)

  17. Pardalos P.M. (1991). Construction of test problems in quadratic bivalent programming. ACM Trans. Math. Software 17(1):74–87

    Article  MATH  MathSciNet  Google Scholar 

  18. Peng J.M., Yuan Y. (1997). Optimization conditions for the minimization of a quadratic with two quadratic constraints. SIAM J. Optim. 7(3):579–594

    Article  MATH  MathSciNet  Google Scholar 

  19. Pınar M.C. (2004). Sufficient global optimality conditions for bivalent quadratic optimization. J. Optim. Theor. Appl. 122(2):433–440

    Article  Google Scholar 

  20. Polyak B.T. (1998). Convexity of quadratic transformations and its use in control and optimization. J. Optim. Theor. Appl. 99:553–583

    Article  MATH  MathSciNet  Google Scholar 

  21. Rubinov A.M. (2000). Abstract Convexity and Global Optimization. Kluwer, Netherlands

    MATH  Google Scholar 

  22. Stern R., Wolkowicz H. (1995). Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim. 5:286–313

    Article  MATH  MathSciNet  Google Scholar 

  23. Strekalovsky A. (1998). Global optimality conditions for nonconvex optimization. J. Global Optim. 12(4):415–434

    Article  MATH  MathSciNet  Google Scholar 

  24. Yakubovich V.A. (1977). The S-procedure in nonlinear control theory. Vestnik Leningr. Univ. 4:73–93

    Google Scholar 

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Author notes

  1. Z. Y. Wu

    Present address: School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, 3353, VIC, Australia

Authors and Affiliations

  1. Department of Applied Mathematics, University of New South Wales, Sydney, 2052, Australia

    V. Jeyakumar

  2. School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, 3353, VIC, Australia

    A. M. Rubinov

  3. Department of Mathematics, Chongqing Normal University, Chongqing, 400047, People’s Republic of China

    Z. Y. Wu

Authors
  1. V. Jeyakumar
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  2. A. M. Rubinov
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  3. Z. Y. Wu
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Corresponding author

Correspondence to V. Jeyakumar.

Additional information

The work of Z. Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.

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Jeyakumar, V., Rubinov, A.M. & Wu, Z.Y. Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program. 110, 521–541 (2007). https://doi.org/10.1007/s10107-006-0012-5

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  • DOI: https://doi.org/10.1007/s10107-006-0012-5

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