# Global Optimization of Polynomials over Real Algebraic Sets

Article

First Online:

## Abstract

Let *f*, *g*_{1},..., *g*_{s} be polynomials in R[*X*_{1},..., *X*_{n}]. Based on topological properties of generalized critical values, the authors propose a method to compute the global infimum *f** of *f* over an arbitrary given real algebraic set *V* = {*x* ∈ R^{n} | *g*_{1}(*x*) = 0,..., *g*_{s}(*x*) = 0}, where *V* is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing *f* over a basic closed semi-algebraic set *S* = {*x* ∈ R^{n} | *g*_{1}(*x*) ≥ 0,..., *g*_{s}(*x*) ≥ 0}.

## Keywords

Polynomial optimization real algebraic set generalized critical value## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgement

We would like to thank Mohab Safey El Din for valuable comments and suggestions on the topic of the paper.

## References

- [1]Kamyar R and Peet M, Polynomial optimization with applications to stability analysis and controlalternatives to sum of squares,
*Discrete and Continuous Dynamical Systems-Series B*, 2015, 20(8): 2383–2417.MathSciNetzbMATHGoogle Scholar - [2]Ahmadi A A and Majumdar A, Some applications of polynomial optimization in operations research and real-time decision making,
*Optimization Letters*, 2016, 10(4): 709–729.MathSciNetzbMATHGoogle Scholar - [3]Qi L and Teo K L, Multivariate polynomial minimization and its application in signal processing,
*Journal of Global Optimization*, 2003, 26(4): 419–433.MathSciNetzbMATHGoogle Scholar - [4]Kahl F and Henrion D, Globally optimal estimates for geometric reconstruction problems,
*International Journal of Computer Vision*, 2007, 74(1): 3–15.Google Scholar - [5]Nesterov Y,
*Squared functional systems and optimization problems*, High Performance Optimization, Springer, 2000, 405–440.zbMATHGoogle Scholar - [6]Wu W T, On problems involving inequalities,
*Mathematics-Mechanization Research Preprints*, 1992, 7: 103–138.Google Scholar - [7]Wu W T, On zeros of algebraic equations — An application of Ritt principle,
*Chinese Science Bulletin*, 1986, 31(1): 1–5.MathSciNetzbMATHGoogle Scholar - [8]Wu W T, A zero structure theorem for polynomial-equations-solving and its applications,
*Mathematics-Mechanization Research Preprints*, 1987, 1: 2–12.Google Scholar - [9]Wu W T, A zero structure theorem for polynomial-equations-solving and its applications,
*European Conference on Computer Algebra*, 1987.Google Scholar - [10]Wu W T, On a finiteness theorem about optimization problems,
*Mathematics-Mechanization Research Preprints*, 1992, 81–18.Google Scholar - [11]Wu W T, On a finite kernel theorem for polynomial-type optimization problems and some of its applications,
*Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation, ISSAC*’05, (New York, USA), ACM, 2005, 4–4.Google Scholar - [12]Morse A P, The behavior of a function on its critical set,
*Annals of Mathematics*, 1939, 40(1): 62–70.MathSciNetzbMATHGoogle Scholar - [13]Sard A, The measure of the critical values of differentiable maps,
*Bulletin of the American Mathematical Society*, 1942, 48(1942): 883–890.MathSciNetzbMATHGoogle Scholar - [14]Sard A, Hausdorff measure of critical images on banach manifolds,
*American Journal of Mathematics*, 1965, 87(1): 158–174.MathSciNetzbMATHGoogle Scholar - [15]Wu T, Some test problems on applications of Wu’s method in nonlinear programming problems,
*Chinese Quarterly Journal of Mathematics*, 1994, 9(2): 8–17.zbMATHGoogle Scholar - [16]Wu T, On a collision problem,
*Acta Mathematica Scientia*, 1995,**15**(Supp.): 32–38.MathSciNetGoogle Scholar - [17]Yang L, Recent advances in automated theorem proving on inequalities,
*Journal of Computer Science & Technology*, 1999, 14(5): 434–446.MathSciNetzbMATHGoogle Scholar - [18]Fang W, Wu T, and Chen J, An algorithm of global optimization for rational functions with rational constraints,
*Journal of Global Optimization*, 2000, 18(3): 211–218.MathSciNetzbMATHGoogle Scholar - [19]Yang L and Zhang J,
*A Practical Program of Automated Proving for a Class of Geometric Inequalities*, Springer Berlin Heidelberg, 2001, 41–57.zbMATHGoogle Scholar - [20]Xiao S J and Zeng G X, Algorithms for computing the global infimum and minimum of a polynomial function,
*Science China Mathematics*, 2012, 55(4): 881–891.MathSciNetzbMATHGoogle Scholar - [21]Zeng G and Xiao S, Global minimization of multivariate polynomials using nonstandard methods,
*Journal of Global Optimization*, 2012, 53(3): 391–415.MathSciNetzbMATHGoogle Scholar - [22]Collins G E, Quantifier elimination for real closed fields by cylindrical algebraic decomposition,
*Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern*, May 20–23, 1975, Springer, 1975, 134–183.Google Scholar - [23]Hong H, An improvement of the projection operator in cylindrical algebraic decomposition,
*Proceedings of the International Symposium on Symbolic and Algebraic Computation*, ISSAC’90, (New York, USA), ACM, 1990, 261–264.Google Scholar - [24]Collins G E and Hong H, Partial cylindrical algebraic decomposition for quantifier elimination,
*Journal of Symbolic Computation*, 1991, 12(3): 299–328.MathSciNetzbMATHGoogle Scholar - [25]Hong H, Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination,
*Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC*’92, (New York, USA), ACM, 1992, 177–188.Google Scholar - [26]McCallum S, An improved projection operation for cylindrical algebraic decomposition,
*Quantifier Elimination and Cylindrical Algebraic Decomposition*, Eds. by Caviness B F and Johnson J R, Springer Vienna, 1998, 242–268.Google Scholar - [27]McCallum S, On projection in CAD-based quantifier elimination with equational constraint,
*Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, ISSAC*’99, (New York, USA), ACM, 1999, 145–149.Google Scholar - [28]Brown C W, Improved projection for cylindrical algebraic decomposition,
*Journal of Symbolic Computation*, 2001, 32(5): 447–465.MathSciNetzbMATHGoogle Scholar - [29]Brown C W, QEPCAD B: A program for computing with semi-algebraic sets using CADs,
*SIGSAM Bull.*, 2003, 37(4): 97–108.zbMATHGoogle Scholar - [30]Han J, Dai L, and Xia B, Constructing fewer open cells by GCD computation in CAD projection,
*Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ISSAC*’14, (New York, USA), ACM, 2014, 240–247.Google Scholar - [31]Han J, Jin Z, and Xia B, Proving inequalities and solving global optimization problems via simplified CAD projection,
*Journal of Symbolic Computation*, 2016, 72: 206–230.MathSciNetzbMATHGoogle Scholar - [32]Han J, Dai L, Hong H, et al., Open weak CAD and its applications,
*Journal of Symbolic Computation*, 2017, 80: 785–816.MathSciNetGoogle Scholar - [33]Basu S, Pollack R, and Roy M F,
*Algorithms in Real Algebraic Geometry*, Springer Berlin Heidelberg, Berlin, 2006.zbMATHGoogle Scholar - [34]Hong H and Safey El Din M, Variant real quantifier elimination: Algorithm and application,
*Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ISSAC*’09, (New York, USA), ACM, 2009, 183–190.Google Scholar - [35]Hong H and Safey El Din M, Variant quantifier elimination,
*Journal of Symbolic Computation*, 2012, 47(7): 883–901, International Symposium on Symbolic and Algebraic Computation (ISSAC 2009).MathSciNetzbMATHGoogle Scholar - [36]Safey El Din M, Computing the global optimum of a multivariate polynomial over the reals,
*Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation, ISSAC*’08, (New York, NY, USA), ACM, 2008, 71–78.Google Scholar - [37]Greuet A and Safey El Din M, Probabilistic algorithm for polynomial optimization over a real algebraic set,
*SIAM Journal on Optimization*, 2014, 24(3): 1313–1343.MathSciNetzbMATHGoogle Scholar - [38]Lasserre J B, Global optimization with polynomials and the problem of moments,
*SIAM Journal on Optimization*, 2001, 11(3): 796–817.MathSciNetzbMATHGoogle Scholar - [39]Nie J, Demmel J, and Sturmfels B, Minimizing polynomials via sum of squares over the gradient ideal,
*Mathematical Programming*, 2006, 106(3): 587–606.MathSciNetzbMATHGoogle Scholar - [40]Demmel J, Nie J, and Powers V, Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals,
*Journal of Pure and Applied Algebra*, 2007, 209(1): 189–200.MathSciNetzbMATHGoogle Scholar - [41]Nie J, An exact Jacobian SDP relaxation for polynomial optimization,
*Mathematical Programming*, 2013, 137(1–2): 225–255.MathSciNetzbMATHGoogle Scholar - [42]Schweighofer M, Global optimization of polynomials using gradient tentacles and sums of squares,
*SIAM Journal on Optimization*, 2006, 17(3): 920–942.MathSciNetzbMATHGoogle Scholar - [43]Hà H V and Phạm T S, Solving polynomial optimization problems via the truncated tangency variety and sums of squares,
*J. Pure Appl. Algebra*, 2009, 213(11): 2167–2176.MathSciNetzbMATHGoogle Scholar - [44]Hà H V and Phạm T S, Representations of positive polynomials and optimization on noncompact semialgebraic sets,
*SIAM Journal on Optimization*, 2010, 20(6): 3082–3103.MathSciNetzbMATHGoogle Scholar - [45]Guo F, Safey El Din M, and Zhi L, Global optimization of polynomials using generalized critical values and sums of squares,
*Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation*, ACM, 2010, 107–114.Google Scholar - [46]Greuet A, Guo F, Safey El Din M, et al., Global optimization of polynomials restricted to a smooth variety using sums of squares,
*Journal of Symbolic Computation*, 2012, 47(5): 503–518.MathSciNetzbMATHGoogle Scholar - [47]Rabier P J, Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds,
*Annals of Mathematics*, 1997, 647–691.Google Scholar - [48]Jelonek Z and Kurdyka K, Quantitative generalized Bertini-Sard theorem for smooth affine varieties,
*Discrete and Computational Geometry*, 2005, 34(4): 659–678.MathSciNetzbMATHGoogle Scholar - [49]Bochnak J, Coste M, and Roy M F,
*Real Algebraic Geometry*, 36, Springer Science and Business Media, 1998.zbMATHGoogle Scholar - [50]Greuel G M and Pfister G,
*A Singular Introduction to Commutative Algebra*, Springer Science and Business Media, 2012.zbMATHGoogle Scholar - [51]Eisenbud D,
*Commutative Algebra: With a View Toward Algebraic Geometry*, 150, Springer New York, 1995.zbMATHGoogle Scholar - [52]Yang Z H,
*Computation of real radicals and global optimization of polynomials*, PhD thesis, University of Chinese Academy of Sciences, 2018.Google Scholar - [53]Valette A and Valette G, A generalized Sard theorem on real closed fields,
*Mathematische Nachrichten*, 2016, 289(5–6): 748–755.MathSciNetzbMATHGoogle Scholar - [54]Kurdyka K, Orro P, and Simon S, Semialgebraic Sard theorem for generalized critical values,
*Journal of Differential Geometry*, 2000, 56(1): 67–92.MathSciNetzbMATHGoogle Scholar - [55]Jelonek Z, On the generalized critical values of a polynomial mapping,
*Manuscripta Mathematica*, 2003, 110(2): 145–157.MathSciNetzbMATHGoogle Scholar - [56]Jelonek Z, On asymptotic critical values and the Rabier theorem,
*Banach Center Publications*, 2004, 1(65): 125–133.MathSciNetzbMATHGoogle Scholar - [57]Cohen R L,
*The Topology of Fiber Bundles Lecture Notes*, Standford University, San Francisco, 1998.Google Scholar - [58]Cox D, Little J, and O’Shea D,
*Ideals, Varieties, and Algorithms*, Springer, New York, 2007.zbMATHGoogle Scholar - [59]Neuhaus R, Computation of real radicals of polynomial ideals II,
*Journal of Pure and Applied Algebra*, 1998, 124(1): 261–280.MathSciNetzbMATHGoogle Scholar - [60]Becker E and Neuhaus R,
*Computation of real radicals of polynomial ideals*, Computational Algebraic Geometry, Springer, 1993, 1–20.zbMATHGoogle Scholar - [61]Spang S J,
*On the Computation of the Real Radical*, PhD thesis, Thesis, Technische Universität Kaiserslautern, 2007.Google Scholar - [62]Spang S J, A zero-dimensional approach to compute real radicals,
*The Computer Science Journal of Moldova*, 2008, 16(1): 64–92.MathSciNetzbMATHGoogle Scholar - [63]Safey El Din M, Yang Z H, and Zhi L, On the complexity of computing real radicals of polynomial systems,
*Proceedings of the 2018 International Symposium on Symbolic and Algebraic Computation, ISSAC*’18, (New York, USA), ACM, 2018, 351–358.Google Scholar - [64]Aubry P, Rouillier F, and Safey El Din M, Real solving for positive dimensional systems,
*Journal of Symbolic Computation*, 2002, 34(6): 543–560.MathSciNetzbMATHGoogle Scholar - [65]Safey El Din M and Schost É, Properness defects of projections and computation of at least one point in each connected component of a real algebraic set,
*Discrete & Computational Geometry*, 2004, 32(3): 417–430.MathSciNetzbMATHGoogle Scholar - [66]Safey El Din M, Finding sampling points on real hypersurfaces is easier in singular situations,
*MEGA (Effective Methods in Algebraic Geometry) Electronic Proceedings*, 2005.Google Scholar - [67]Safey El Din M, Testing sign conditions on a multivariate polynomial and applications,
*Mathematics in Computer Science*, 2007, 1(1): 177–207.MathSciNetzbMATHGoogle Scholar - [68]Krick T and Logar A, An algorithm for the computation of the radical of an ideal in the ring of polynomials,
*International Symposium on Applied Algebra*, Algebraic Algorithms, and Error-Correcting Codes, Springer, 1991, 195–205.Google Scholar

## Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019