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On Types of Isolated KKT Points in Polynomial Optimization

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Abstract

Let f be a real polynomial function with n variables and S be a basic closed semialgebraic set in ℝn. In this paper, the authors are interested in the problem of identifying the type (local minimizer, maximizer or not extremum point) of a given isolated KKT point x* of f over S. To this end, the authors investigate some properties of the tangency variety of f on S at x*, by which the authors introduce the definition of faithful radius of f over S at x*. Then, the authors show that the type of x* can be determined by the global extrema of f over the intersection of S and the Euclidean ball centered at x* with a faithful radius. Finally, the authors propose an algorithm involving algebraic computations to compute a faithful radius of x* and determine its type.

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References

  1. Murty K G and Kabadi S N, Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming, 1987, 39(2): 117–129.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bolis T S, Degenerate critical points, Mathematics Magazine, 1980, 53(5): 294–299.

    Article  MathSciNet  MATH  Google Scholar 

  3. Cushing J M, Extremal tests for scalar functions of several real variables at degenerate critical points, Aequationes Mathematicae, 1975, 13(1–2): 89–96.

    Article  MathSciNet  MATH  Google Scholar 

  4. Qi L, Extrema of a real polynomial, Journal of Global Optimization, 2004, 30(4): 405–433.

    Article  MathSciNet  MATH  Google Scholar 

  5. Nie J, The hierarchy of local minimums in polynomial optimization, Mathematical Programming, 2015, 151(2): 555–583.

    Article  MathSciNet  MATH  Google Scholar 

  6. Guo F and Pham T S, On types of degenerate critical points of real polynomial functions, Journal of Symbolic Computation, 2020, 99: 108–126.

    Article  MathSciNet  MATH  Google Scholar 

  7. Alonso M E, Becker E, Roy M F, et al., Zeros, multiplicities, and idempotents for zero-dimensional systems, Algorithms in Algebraic Geometry and Applications, Birkhauser Basel, Basel, 1996.

    Google Scholar 

  8. Faugere J, Gianni P, Lazard D, et al., Efficient computation of zero-dimensional Gröbner bases by change of ordering, Journal of Symbolic Computation, 1993, 16(4): 329–344.

    Article  MathSciNet  MATH  Google Scholar 

  9. Giusti M, Heintz J, Morais J E, et al., When polynomial equation systems can be “solved” fast? Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Eds. by Cohen G, Giusti M, and Mora T, Springer, Berlin, 1995.

    Google Scholar 

  10. Giusti M, Lecerf G, and Salvy B, A Gröbner free alternative for polynomial system solving, Journal of Complexity, 2001, 17(1): 154–211.

    Article  MathSciNet  MATH  Google Scholar 

  11. Rouillier F, Solving zero-dimensional systems through the rational univariate representation, Applicable Algebra in Engineering, Communication and Computing, 1999, 9(5): 433–461.

    Article  MathSciNet  MATH  Google Scholar 

  12. Benedetti R and Risler J J, Real Algebraic and semi-Algebraic Sets, Hermann, Paris, 1990.

    MATH  Google Scholar 

  13. Bochnak J, Coste M, and Roy M F, Real Algebraic Geometry, Springer-Verlag, New York, 1998.

    Book  MATH  Google Scholar 

  14. Van den Dries L and Miller C, Geometric categories and o-minimal structures, Duke Mathematical Journal, 1996, 84: 497–540.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hà H V and Pham T S, Genericity in Polynomial Optimization, World Scientific Publishing, London, 2017.

    Book  MATH  Google Scholar 

  16. Milnor J, Singular Points of Complex Hypersurfaces, volume 61 of Annals of Mathematics Studies, Princeton University Press, Princeton, 1968.

    Google Scholar 

  17. Guillemin V and Pollack A, Differential Topology, Prentiee-Hall, New Jersey, 1974.

    MATH  Google Scholar 

  18. Cox D A, Little J, and O’Shea D, Ideals, Varieties, and Algorithms — An Introduction to Computational Algebraic Geometry and Commutative Algebra, Graduate Texts in Mathematics, Springer-Verlag, New York, 2007.

    MATH  Google Scholar 

  19. Greuel G M and Pfister G, A Singular Introduction to Commutative Algebra, Springer-Verlag, Berlin, 2002.

    Book  MATH  Google Scholar 

  20. Basu S, Pollack R, and Roy M F, Algorithms in Real Algebraic Geometry, Springer-Verlag, Berlin, 2006.

    Book  MATH  Google Scholar 

  21. Bertsekas D P, Nonlinear Programming, 3rd Edition, Athena Scientific, Belmont, 2016.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Bank B, Giusti M, Heintz J, et al., Polar varieties and efficient real equation solving: The hypersurface case, Journal of Complexity, 1997, 13: 5–27.

    Article  MathSciNet  MATH  Google Scholar 

  24. Bank B, Giusti M, Heintz J, et al., Polar varieties and efficient real elimination, Mathematische Zeitschrift, 2001, 238(1): 115–144.

    Article  MathSciNet  MATH  Google Scholar 

  25. Safey El Din M and Schost E, Polar varieties and computation of one point in each connected component of a smooth real algebraic set, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation–ISSAC 2003, New York, 2003.

  26. Nesterov Y, Random walk in a simplex and quadratic optimization over convex polytopes, https://ideas.repec.org/p/cor/louvco/2003071.html, 2003.

  27. Demazure M, Bifurcations and Catastrophes, Geometry of Solutions to Nonlinear Problems, Uni-versitext, Springer, Berlin, 2000.

    MATH  Google Scholar 

  28. Bank B, Giusti M, Heintz J, et al., On the geometry of polar varieties, Applicable Algebra in Engineering, Communication and Computing, 2010, 21(1): 33–83.

    Article  MathSciNet  MATH  Google Scholar 

  29. Safey El Din M and Schost E, A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets, Journal of the ACM, 2017, 63(6): 1–48.

    Article  MathSciNet  MATH  Google Scholar 

  30. Aubry P, Rouillier F, and Safey El Din M, Real solving for positive dimensional systems, Journal of Symbolic Computation, 2002, 34(6): 543–560.

    Article  MathSciNet  MATH  Google Scholar 

  31. Chen C and Moreno Maza M, Cylindrical algebraic decomposition in the RegularChains library, Proceedings of the Mathematical Software–ICMS 2014, Eds. by Hong H and Yap C, Berlin, 2014.

  32. Chen C and Moreno Maza M, Quantifier elimination by cylindrical algebraic decomposition based on regular chains, Journal of Symbolic Computation, 2016, 75: 74–93.

    Article  MathSciNet  MATH  Google Scholar 

  33. Caviness B F and Johnson J R, Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1998.

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Feng Guo

  2. Academy for Advanced Interdisciplinary Studies, Northeast Normal University, Changchun, 130024, China

    Liguo Jiao

  3. Department of Applied Mathematics, Pukyong National University, Busan, 48513, South Korea

    Do Sang Kim

  4. Department of Mathematics, Dalat University, Dalat, 96300, Vietnam

    Tien-Son Pham

Authors
  1. Feng Guo
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  2. Liguo Jiao
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  3. Do Sang Kim
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  4. Tien-Son Pham
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Corresponding author

Correspondence to Liguo Jiao.

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The authors declare no conflict of interest.

Additional information

This research was supported by the Chinese National Natural Science Foundation under Grant No. 11571350, the Science and Technology Development Plan Project of Jilin Province, China under Grant No. YDZJ202201ZYTS302, the National Research Foundation of Korea (NRF) Grant Funded by the Korean Government under Grand No. NRF-2019R1A2C1008672, and the International Centre for Research and Postgraduate Training in Mathematics (ICRTM) under Grant No. ICRTM01.2022.01.

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Guo, F., Jiao, L., Kim, D.S. et al. On Types of Isolated KKT Points in Polynomial Optimization. J Syst Sci Complex 36, 2186–2213 (2023). https://doi.org/10.1007/s11424-023-2119-7

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  • DOI: https://doi.org/10.1007/s11424-023-2119-7

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