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Tangencies and polynomial optimization

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Abstract

Given a polynomial function \(f :{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) and an unbounded closed semi-algebraic set \(S \subset {\mathbb {R}}^n,\) we show that the conditions listed below are characterized exactly in terms of the so-called tangency variety of the restriction of f on S:

  • f is bounded from below on S; 

  • f attains its infimum on S; 

  • The sublevel sets \(\{x \in S \ | \ f(x) \le \lambda \}\) for \(\lambda \in {\mathbb {R}}\) are compact;

  • f is coercive on S.

Besides, we also provide some stability criteria for boundedness and coercivity of f on S.

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Notes

  1. The computations are performed with the software Maple, using the command “puiseux” of the package “algcurves” for the rational Puiseux expansions.

References

  1. Ahmadi, A.A., Zhang, J.: On the complexity of testing attainment of the optimal value in nonlinear optimization. Math. Program. Ser. A 184, 221–241 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bajbar, T., Behrends, S.: How fast do coercive polynomials grow? Instituts für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Tech. rep. (2017)

    Google Scholar 

  3. Bajbar, T., Stein, O.: Coercive polynomials and their Newton polytopes. SIAM J. Optim. 25(3), 1542–1570 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bajbar, T., Stein, O.: Coercive polynomials: stability, order of growth, and Newton polytopes. Optimization 68(1), 99–124 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry, vol. 36. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  6. Bolte, J., Hochart, A., Pauwels, E.: Qualification conditions in semi-algebraic programming. SIAM J. Optim. 28(3), 2131–2151 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Demmel, J., Nie, J., Powers, V.: Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals. J. Pure Appl. Algebra 209(1), 189–200 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Grandjean, V.: On the limit set at infinity of a gradient trajectory of a semialgebraic function. J. Differ. Equ. 233(1), 22–41 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hà, H.V., Phạm, T.S.: Global optimization of polynomials using the truncated tangency variety and sums of squares. SIAM J. Optim. 19(2), 941–951 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hà, H.V., Phạm, T.S.: On the Łojasiewicz exponent at infinity of real polynomials. Ann. Polon. Math. 94(3), 197–208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hà, H.V., Phạm, T.S.: Solving polynomial optimization problems via the truncated tangency variety and sums of squares. J. Pure Appl. Algebra 213, 2167–2176 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hà, H.V., Phạm, T.S.: Representation of positive polynomials and optimization on noncompact semialgebraic sets. SIAM J. Optim. 20, 3082–3103 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hà, H.V., Phạm, T.S.: Genericity in Polynomial Optimization, vol. 3. Ser. Optim. Appl. World Scientific, Singapore (2017)

    MATH  Google Scholar 

  14. Hardt, R.M.: Semi-algebraic local-triviality in semi-algebraic mappings. Amer. J. Math. 102(2), 291–302 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jeyakumar, V., Lasserre, J.B., Li, G.: On polynomial optimization over non-compact semi-algebraic sets. J. Optim. Theory Appl. 163, 707–718 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. John, F.: Extremum problems with inequalities as side constraints. In: Studies and Essays, Courant Anniversary Volume (New York, 1948), Friedrichs, K. O., Neugebauer, O. E., Stoker, J. J. (eds.) Wiley (interscience), pp. 187–204

  17. Karush, W.: Minima of functions of several variables with inequalities as side constraints. Master’s thesis, Dept. of Mathematics, Univ. of Chicago, Chicago, Illinois (1939)

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Kuhn, H. W., Tucker, A. W.: Nonlinear programming. In: Proceedings of 2nd Berkeley Symposium (Berkeley, 1951), Neyman, J. (ed.) University of California Press, pp. 481–492

  20. Kurdyka, K., Mostowski, T., Parusiński, A.: Proof of the gradient conjecture of R. Thom. Ann. of Math. (2) 152(3), 763–792 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. Lasserre, J.B.: An Introduction to Polynomial and Semi-Algebraic Optimization. Cambridge University Press, Cambridge, UK (2015)

    Book  MATH  Google Scholar 

  23. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging Applications of Algebraic Geometry, Putinar, M., Sullivant, S. (eds.) vol. IMA Math. Appl. 149. Springer, New York, pp. 157–270 (2009)

  24. Loi, T.L., Zaharia, A.: Bifurcation sets of functions definable in o-minimal structures. Illinois J. Math. 43(3), 449–457 (1998)

    MathSciNet  MATH  Google Scholar 

  25. Mangasarian, O.L., Fromovitz, S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  26. Marshall, M.: Positive Polynomials and Sums of Squares, vol. 146 of Math. Surveys and Monographs. American Mathematical Society, Providence, RI (2008)

  27. Motzkin, T.: The arithmetic-geometric inequalities. In Inequalities (Academic Press, 1967), Shisha, O. Ed. pp. 205–224

  28. Nie, J., Demmel, J., Sturmfels, B.: Minimizing polynomials via sum of squares over the gradient ideal. Math. Program. Ser. A 106(3), 587–606 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Phạm, T.S.: Local minimizers of semi-algebraic functions from the viewpoint of tangencies. SIAM J. Optim. 30(3), 1777–1794 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shor, N.Z.: Class of global minimum bounds of polynomial functions. Cybernetics 23(6), 731–734 (1987)

    Article  MATH  Google Scholar 

  31. Spingarn, J.E., Rockafellar, R.T.: The generic nature of optimality conditions in nonlinear programming. Math. Oper. Res. 4, 425–430 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84, 497–540 (1996)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Jérôme Bolte for the useful discussions. He is also grateful to the anonymous referees for their careful reading with constructive comments and suggestions on the paper. The paper was partially written while the author had been visiting at the Vietnam Institute for Advanced Study in Mathematics (VIASM) from January 1 to 31 March, 2019. He would like to thank the Institute for hospitality and support.

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  1. Department of Mathematics, Dalat University, 1 Phu Dong Thien Vuong, Dalat, Vietnam

    Tiến-Sơn Phạm

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Correspondence to Tiến-Sơn Phạm.

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Dedicated to Professor Gue Myung Lee on the occasion of his 70th birthday.

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The author is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grand number 101.04-2019.302 and by International Centre for Research and Postgraduate Training in Mathematics (ICRTM) under grant number ICRTM01_2022.01.

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Phạm, TS. Tangencies and polynomial optimization. Math. Program. 199, 1239–1272 (2023). https://doi.org/10.1007/s10107-022-01869-6

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