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Multi-objective convex polynomial optimization and semidefinite programming relaxations

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Abstract

This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem \(({\mathrm{P}}_{z})\) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem \((P_{z})\), we establish two kinds of representations of non-negativity of convex polynomials over convex semi-algebraic sets, and propose two kinds of finite convergence results of the Lasserre-type hierarchy of semidefinite programming relaxations for problem \(({\mathrm{P}}_{z})\) under suitable assumptions. Finally, we show that finding efficient solutions to problem (MP) can be achieved successfully by solving hierarchies of semidefinite programming relaxations and checking a flat truncation condition.

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Notes

  1. A polynomial f is called SOS-convex if there exists a matrix polynomial F(x) such that the Hessian \(\nabla ^2f(x)\) factors as \(F(x)F(x)^T;\) see [1, 2, 12].

References

  1. Ahmadi, A.A., Parrilo, P.A.: A convex polynomial that is not SOS-convex. Math. Program. 135(1–2), 380–429 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Ahmadi, A.A., Parrilo, P.A.: A complete characterization of the gap between convexity and SOS-convexity. SIAM J. Optim. 23(2), 811–833 (2013)

    Article  MathSciNet  Google Scholar 

  3. Belousov, E.G., Klatte, D.: A Frank–Wolfe type theorem for convex polynomial programs. Comput. Optim. Appl. 22(1), 37–48 (2002)

    Article  MathSciNet  Google Scholar 

  4. Blanco, V., Puerto, J., Ali, S.E.H.B.: A semidefinite programming approach for solving multiobjective linear programming. J. Glob. Optim. 58(3), 465–480 (2014)

    Article  MathSciNet  Google Scholar 

  5. Chankong, V., Haimes, Y.Y.: Multiobjective decision making: theory and methodology. North-Holland, Amsterdam (1983)

    MATH  Google Scholar 

  6. Ehrgott, M.: Multicriteria optimization, 2nd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  7. Fernando, J.F., Gamboa, J.M.: Polynomial and regular images of \(\mathbb{R}^n\). Israel J. Math. 153, 61–92 (2006)

    Article  MathSciNet  Google Scholar 

  8. Fernando, J.F., Ueno, C.: On complements of convex polyhedra as polynomial and regular images of \(\mathbb{R}^n\). Int. Math. Res. Not. 2014(18), 5084–5123 (2014)

    Article  Google Scholar 

  9. Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities. Image space analysis and separation. In: F. Giannessi (ed.) Vector variational inequalities and vector equilibria: mathematical theories. Nonconvex Optimization and Its Applications, vol. 38, pp. 141–215. Springer, Boston, MA (2000)

  10. Gorissen, B.L., Hertog, D.D.: Approximating the Pareto set of multiobjective linear programs via robust optimization. Oper. Res. Lett. 40(5), 319–324 (2012)

    Article  MathSciNet  Google Scholar 

  11. Hà, H.V., Phạm, T.S.: Genericity in polynomial optimization. World Scientific Publishing, Singapore (2017)

    Book  Google Scholar 

  12. Helton, J.W., Nie, J.W.: Semidefinite representation of convex sets. Math. Program. 122(1), 21–64 (2010)

    Article  MathSciNet  Google Scholar 

  13. Henrion, D., Lasserre, J.B.: Detecting global optimality and extracting solutions in gloptipoly. In: Henrion, D., Garulli, A. (eds.) Positive polynomials in control. Lecture notes in control and information science, vol. 312, pp. 293–310. Springer, Berlin (2005)

    Chapter  Google Scholar 

  14. Henrion, D., Lasserre, J.B., Loefberg, J.: Gloptipoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)

    Article  MathSciNet  Google Scholar 

  15. Jeyakumar, V., Phạm, T.S., Li, G.: Convergence of the Lasserre hierarchy of SDP relaxations for convex polynomial programs without compactness. Oper. Res. Lett. 42(1), 34–40 (2014)

    Article  MathSciNet  Google Scholar 

  16. Jiao, L.G., Lee, J.H.: Finding efficient solutions in robust multiple objective optimization with SOS-convex polynomial data. Ann. Oper. Res. (2019). https://doi.org/10.1007/s10479-019-03216-z

    Article  MATH  Google Scholar 

  17. Jiao, L.G., Lee, J.H., Ogata, Y., Tanaka, T.: Multi-objective optimization problems with SOS-convex polynomials over an LMI constraint. Taiwan. J. Math. 24(4), 1021–1043 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Jiao, L.G., Lee, J.H., Zhou, Y.Y.: A hybrid approach for finding efficient solutions in vector optimization with SOS-convex polynomials. Oper. Res. Lett. 48(2), 188–194 (2020)

    Article  MathSciNet  Google Scholar 

  19. Kim, D.S., Mordukhovich, B.S., Phạm, T.S., Tuyen, N.V.: Existence of efficient and properly efficient solutions to problems of constrained vector optimization. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01532-y

    Article  Google Scholar 

  20. Kim, D.S., Phạm, T.S., Tuyen, N.V.: On the existence of Pareto solutions for polynomial vector optimization problems. Math. Program. 177(1–2), 321–341 (2019)

    Article  MathSciNet  Google Scholar 

  21. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    Article  MathSciNet  Google Scholar 

  22. Lasserre, J.B.: Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19(4), 1995–2014 (2009)

    Article  MathSciNet  Google Scholar 

  23. Lasserre, J.B.: Moments. Positive Polynomials and their Applications. Imperial College Press, London (2010)

    MATH  Google Scholar 

  24. Lasserre, J.B.: An introduction to polynomial and semi-algebraic optimization. Cambridge University Press, Cambridge (2015)

    Book  Google Scholar 

  25. Lee, J.H., Jiao, L.G.: Solving fractional multicriteria optimization problems with sum of squares convex polynomial data. J. Optim. Theory Appl. 176(2), 428–455 (2018)

    Article  MathSciNet  Google Scholar 

  26. Lee, J.H., Jiao, L.G.: Finding efficient solutions for multicriteria optimization problems with SOS-convex polynomials. Taiwan. J. Math. 23(6), 1535–1550 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Luc, D.T.: Multiobjective linear programming: an introduction. Springer International Publishing, Switzerland (2016)

    Book  Google Scholar 

  28. Magron, V., Henrion, D., Lasserre, J.B.: Approximating Pareto curves using semidefinite relaxations. Oper. Res. Lett. 42(6–7), 432–437 (2014)

    Article  MathSciNet  Google Scholar 

  29. Nie, J.: Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program. 142(1–2), 485–510 (2013)

    Article  MathSciNet  Google Scholar 

  30. Nie, J., Ranestad, K.: Algebraic degree of polynomial optimization. SIAM J. Optim. 20(1), 485–502 (2009)

    Article  MathSciNet  Google Scholar 

  31. Phạm, T.S.: Optimality conditions for minimizers at infinity in polynomial programming. Math. Oper. Res. 44(4), 1381–1395 (2019)

    Article  MathSciNet  Google Scholar 

  32. Ploskas, N., Samaras, N.: Linear programming using MatLab. Springer, Switzerland (2017)

    Book  Google Scholar 

  33. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)

    Article  MathSciNet  Google Scholar 

  34. Putinar, M., Vasilescu, F.H.: Positive polynomials on semi-algebraic sets. C. R. Acad. Sci. I Math. 328(7), 585–589 (1999)

    MathSciNet  MATH  Google Scholar 

  35. Scheiderer, C.: Sums of squares on real algebraic curves. Math. Z. 245(4), 725–760 (2003)

    Article  MathSciNet  Google Scholar 

  36. Scheiderer, C.: Distinguished representations of non-negative polynomials. J. Algebra 289(2), 558–573 (2005)

    Article  MathSciNet  Google Scholar 

  37. Schmüdgen, K.: The \(K\)-moment problem for compact semi-algebraic sets. Math. Ann. 289(2), 203–206 (1991)

    Article  MathSciNet  Google Scholar 

  38. Schweighofer, M.: Optimization of polynomials on compact semialgebraic sets. SIAM J. Optim. 15(3), 805–825 (2005)

    Article  MathSciNet  Google Scholar 

  39. Ueno, C.: A note on boundaries of open polynomial images of \(\mathbb{R}^2\). Rev. Mat. Iberoam. 24(3), 981–988 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their sincere thanks to Prof. Tien-Son Pham of University of Dalat for his valuable suggestions and warm helps. They also would like to appreciate the anonymous referees for their very helpful and valuable suggestions and comments for the paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Pukyong National University, Busan, 48513, Republic of Korea

    Jae Hyoung Lee

  2. Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

    Nithirat Sisarat

  3. School of Mathematical Sciences, Soochow University, Suzhou, 215006, Jiangsu Province, China

    Liguo Jiao

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  1. Jae Hyoung Lee
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  2. Nithirat Sisarat
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  3. Liguo Jiao
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Correspondence to Liguo Jiao.

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Jae Hyoung Lee was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2018R1C1B6001842). Nithirat Sisarat was partially supported by a grant from the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program grant number PHD/0026/2555. Liguo Jiao was supported by Jiangsu Planned Projects for Postdoctoral Research Funds 2019 (no. 2019K151).

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Lee, J.H., Sisarat, N. & Jiao, L. Multi-objective convex polynomial optimization and semidefinite programming relaxations. J Glob Optim 80, 117–138 (2021). https://doi.org/10.1007/s10898-020-00969-x

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