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Conditions for Global Optimality 2

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Abstract

In this paper bearing the same title as our earlier survey-paper [11] we pursue the goal of characterizing the global solutions of an optimization problem, i.e. getting at necessary and sufficient conditions for a feasible point to be a global minimizer (or maximizer) of the objective function. We emphasize nonconvex optimization problems presenting some specific structures like ‘convex-anticonvex’ ones or quadratic ones.

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References

  1. Danninger, G. and Bomze, I. (1993), Using copositivity for global optimality criteria in concave quadratic programming problems, Mathematical Programming 62: 575–580.

    Google Scholar 

  2. Dedieu, J.-P. (1995), Third-and fourth-order optimality conditions in optimization, Optimization 33: 97–104.

    Google Scholar 

  3. Dedieu, J.-P. and Janin, R. (1996), On third-and fourth-order optimality conditions in unconstrained programming, Technical report.

  4. Dür, M., Horst, R. and Locatelli, M. (1998), Necessary and sufficient global optimality conditions for convex maximization revisited, Journal of Mathematical Analysis and Applications 217: 637–649.

    Google Scholar 

  5. Ferrier, C. (1997), Bornes duales de problèmes d'optimisation polynomiaux, Ph. D. Thesis, Laboratoire Approximation et Optimisation, Université Paul Sabatier, Toulouse.

  6. Flippo, O. E. and Jansen, B. (1996), Duality and sensitivity in nonconvex quadratic optimization over an ellipsoid, European Journal of Operational Research 94: 167–178.

    Google Scholar 

  7. Floudas, C. A. and Visweswaran, V. (1995), Quadratic optimization, in: Handbook for Global Optimization, pp. 217–269. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  8. Giner, E. (1995), Local minimizers of integral functionals are global minimizers, Proceedings of the American Mathematical Society 123(3): 755–757.

    Google Scholar 

  9. Glover, B. M., Ishizuka, Y., Jeyakumar, V. and Tuan, H. D. (1996), Complete characterization of global optimality for problems involving the pointwise minimum of sublinear functions, SIAM J. Optimization 6(2): 362–372.

    Google Scholar 

  10. Hiriart-Urruty, J.-B. (1989), From convex optimization to nonconvex optimization. Part 1: Necessary and sufficient conditions for global optimality, in: Nonsmooth Optimization and Related Topics, pp. 219–239, Plenum Press, New York.

    Google Scholar 

  11. Hiriart-Urruty, J.-B. (1995), Conditions for global optimality, in: Handbook for Global Optimization, pp. 1–26. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  12. Hiriart-Urruty, J.-B. (1996), L'optimisation, in Que sais-je?, Presses Universitaires de France.

  13. Hiriart-Urruty, J.-B. and Ledyaev, J. S. (1996), A note on the characterization of the global maxima of a (tangentially) convex function over a convex set, J. of Convex Analysis 3(1): 55–61.

    Google Scholar 

  14. Hiriart-Urruty, J.-B. and Lemaréchal, C. (1993), Convex Analysis and Minimization Algorithms, Vols. 305 and 306 of Grundlehren der mathematischen Wissenschaften. Springer Verlag.

  15. Jiang, Y., Smith, W. R. and Chapman, G. R. (1995), Global optimality conditions and their geometric interpretation for the chemical and phase equilibrium problem, SIAM J. Optimization 5(4): 813–834.

    Google Scholar 

  16. Levy, A. V. and Montalvo, A. (1985), The tunneling algorithm for the global minimization of functions, SIAM J. Sci. Stat. Comp. 6(1): 15–29.

    Google Scholar 

  17. Martinez, J. M. (1994), Local minimizers of quadratic functions on Euclidean balls and spheres, SIAM J. Optimization 4(1): 159–176.

    Google Scholar 

  18. McKinnon, K. and Mongeau, M. (1998), A generic global optimization algorithm for the chemical and phase equilibrium problem [Technical Report 94–05, Laboratoire Approximation et Optimization], J. of Global Optimization 12(4): 325–351.

    Google Scholar 

  19. Moré, J. J. (1993), Generalizations of the trust region problem, Optimization Methods and Software 2: 189–209.

    Google Scholar 

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

    Google Scholar 

  21. Pham Dinh Tao and Le Thi Hoai An (1996), Difference of convex functions optimization algorithms (dca) for globally minimizing nonconvex quadratic forms on euclidean balls and spheres, Operations Research Letters 19: 207–216.

    Google Scholar 

  22. Rendl, F. and Wolkowicz, H. (1997), A semidefinite framework for trust region subproblems with applications to large scale minimization, Mathematical Programming 77: 273–299.

    Google Scholar 

  23. Stern, R. J. and Wolkowicz, H. (1995), Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations, SIAM J. Optimization 5(2): 286–313.

    Google Scholar 

  24. Strekalovsky, A. (1993), Extremal problems on complements of convex sets, Translated from Kibernetika i Sistemnyi Analiz 1: 113–126.

    Google Scholar 

  25. Strekalovsky, A. (1996), On d. c.-programming theory, Technical report, Irkutsk State University.

  26. Yuan, Y. (1990), On a subproblem of trust region algorithms for constrained optimization, Mathematical Programming 47: 53–63.

    Google Scholar 

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Hiriart-Urruty, JB. Conditions for Global Optimality 2. Journal of Global Optimization 13, 349–367 (1998). https://doi.org/10.1023/A:1008365206132

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  • DOI: https://doi.org/10.1023/A:1008365206132

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