Abstract
This paper is concerned with the general polynomial programming problem with box constraints, including global optimality conditions and optimization methods. First, a necessary global optimality condition for a general polynomial programming problem with box constraints is given. Then we design a local optimization method by using the necessary global optimality condition to obtain some strongly or \(\varepsilon \)-strongly local minimizers which substantially improve some KKT points. Finally, a global optimization method, by combining the new local optimization method and an auxiliary function, is designed. Numerical examples show that our methods are efficient and stable.
Similar content being viewed by others
References
Sommese, A.J., Wampler, C.W.: Numerical algebraic geometry. In: Renegar, J., Shub, M., and Smale, S. (eds.) The Mathematics of Numerical Analysis, Proceedings of the AMS-SIAM Summer Seminar in Applied Mathematics, July 17-August 11, 1995, Park City, Utah. Lectures in Applied Mathematics, vol. 32, pp. 749–763 (1995)
Morgan, A.: Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, vol. 57. SIAM, Philadelphia (2009)
Roberts, A.P., Newmann, M.M.: Polynomial optimization of stochastic feedback control for stable plants. IMA J. Math. Control Inf. 5, 243–257 (1988)
Floudas, C.A., Pardalos, P.M.A.: Collection of test problems for constrained global optimization algorithms. In: Goos, G., Hartmanis, J. (eds.) Lecture Notes in Computer Science. Springer, New York (1990)
Sherali, H.D.: Global optimization of nonconvex polynomial programming problems having rational exponents. J. Glob. Optim. 12, 267–283 (1998)
Markowitz, H.M.: Portfolio selection. J. Finance 7, 79–91 (1952)
Qi, L., Teo, K.L.: Multivariate polynomial minimization and its application in signal processing. J. Glob. Optim. 26(4), 419–433 (2003)
Floudas, C.A., Pardalos, P.M., Adjiman, C.S., et al.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic, Dordrecht (1999)
Chang, Y.J., Wah, B.W.: Polynomial programming using Groebner bases. In: Computer Software and Applications Conference COMPSAC 94. Proceedings, Eighteenth Annual International, pp. 236–241 (1994)
Lasserre, J.B.: Moments and sums of squares for polynomial optimization and related problems. J. Glob. Optim. 45, 39–61 (2009)
Floudas, C.A., Visweswaran, V.: Quadratic optimization. Handb. Glob. Optim. 2, 217–269 (1995)
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V. (eds.): A-Discriminants. In: Discriminants, Resultants, and Multidimensional Determinants, pp. 271–296. Birkhäuser, Boston (1994)
Emiris, I.Z., Verschelde, J.: How to count efficiently all affine roots of a polynomial system. Discrete Appl. Math. 93, 21–32 (1999)
Li, T.Y.: Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numer. 6, 399–436 (1997)
Hanif, D.S., Cihan, H.T.: A global optimization algorithm for polynomial programming problems using a reformulation–linearization technique. J. Glob. Optim. 2, 101–112 (1992)
Sherali, H.D., Tuncbilek, C.H.: Comparison of two reformulation–linearization technique based linear programming relaxations for polynomial programming problems. J. Glob. Optim. 10, 381–390 (1997)
Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)
Nie, J., Demmel, J.W.: Minimum ellipsoid bounds for solutions of polynomial systems via sum of squares. J. Glob. Optim. 33(4), 511–525 (2005)
Nie, J., Demmel, J., Sturmfels, B.: Minimizing polynomials via sum of squares over the gradient. Math. Program. Ser. A 106, 587–-606 (2006)
Parrilo, P., Sturmfels, B.: Minimizing polynomial functions. In: Basu, S., Gonzalez-Vega, L. (eds.) Proceedings of the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, pp. 83–100 (March 2001). American Mathematical Society (2003)
Han, D.: Global optimization with polynomials. http://hdl.handle.net/1721.1/3883 (2004). Accessed 20 Dec 2013
Kojima, M., Kim, S., Waki, H.: A general framework for convex relaxation of polynomial optimization problems over cones. J. Oper. Res. Soc. Jpn. 46(2), 125–144 (2003)
Nie, J., Wang, L.: Regularization methods for SDP relaxations in large scale polynomial optimization. SIAM J. Optim. 22(2), 408–428 (2012)
Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17, 218–242 (2006)
Monteiro, R.D.: First-and second-order methods for semidefinite programming. Math. Program. 97(1–2), 209–244 (2003)
Jeyakumar, V., Li, G.: Necessary global optimality conditions for nonlinear programming problems with polynomial constraints. Math. Program. Ser. A 126(2), 393–399 (2011)
Jeyakumar, V., Li, G., Srisatkunarajah, S.: Global optimality principles for polynomial optimization over box or bivalent constraints by separable polynomial approximations. J. Glob. Optim. 58(1), 31–50 (2014)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2013)
Prasolov, V.V.: Polynomials. Springer, Berlin (2004)
Xie, L.J.: A note on Sturm theorem. Math. Pract. Theory 37(1), 121–125 (2007). (in Chinese)
Wu, Z.Y., Tian, J., Quan, J., Ugon, J.: Optimality conditions and optimization methods for quartic polynomial optimization. Appl. Math. Comput. 232, 968–982 (2014)
Wu, Z.Y., Quan, J., Li, G.Q., Tian, J.: Necessary optimality conditions and new optimization methods for cubic polynomial optimization problems with mixed variables. J. Optim. Theory Appl. 153(2), 408–435 (2012)
Li, G.Q., Wu, Z.Y., Quan, J.: A new local and global optimization method for mixed integer quadratic programming problems. Appl. Math. Comput. 217(6), 2501–2512 (2010)
Hansen, E.R.: Global optimization using interval analysis: the one-dimensional case. J. Optim. Theory Appl. 29(3), 331–344 (1979)
Klerk, E.D., Elabwabi, G., Hertog, D.D.: Optimization of univariate functions on bounded intervals by interpolation and semidefinite programming. CentER Discussion Paper Series No. 2006–26. SSRN: http://ssrn.com/abstract=900108 or doi:10.2139/ssrn.900108 (April 11, 2006). Accessed 11 Nov 2013
Floudas, C.A.: Deterministic Global Optimization: Theory, Methods and Applications. Kluwer Academic, Dordrecht (1999)
Visweswaran, V., Floudas, C.A.: Unconstrained and constrained global optimization of polynomial functions in one variable. J. Glob. Optim. 2(1), 73–99 (1992)
Fortune, S.: An iterated eigenvalue algorithm for approximating roots of univariate polynomials. J. Symb. Comput. 33, 627C646 (2002)
Bini, D.A., Gemignani, L., Pan, V.Y.: Inverse power and Durand–Kerner iterations for univaritate polynomial root-finding. Comput. Math. Appl. 47(2), 447–459 (2004)
Wu, Z.Y., Lee, H.W.J., Zhang, L.S., Yang, X.M.: A novel filled function method and quasi-filled function method for global optimization. Comput. Optim. Appl. 34, 249–272 (2005)
Laguna, M., Marti, R.: Experimental testing of advanced scatter search designs for global optimization of multimodal functions. J. Glob. Optim. 33(2), 235–255 (2005)
Henrion, D., Lasserre, J.B.: GloptiPoly: global optimization over polynomials with Matlab and SeDuMi. ACM Trans. Math. Softw. 29(2), 165–194 (2003)
Henrion, D., Lasserre, J.B., Lofberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is partially supported by cstc2013jjB00001, cstc2011jjA00010 and SRF for ROCS, SEM, NSFC 11471062.
Appendix
Appendix
1.1 Problem 1: Beale function
1.2 Problem 2: Booth function
1.3 Problem 3: Matyas function
1.4 Problem 4: Goldstein and Price function
1.5 Problem 5: Six-hump Camelback function
1.6 Problem 6: Perm(3, 0.5) function
1.7 Problem 7: Perm0(3, 10) function
1.8 Problem 8: Perm(4, 0.5) function
1.9 Problem 9: Perm0(4, 10) function
1.10 Problem 10: Colville function
1.11 Problem 11: Powersum function
1.12 Problem 12: Dixon and Price function
1.13 Problem 13: Dixon and Price function
1.14 Problem 14: Trid function
1.15 Problem 15: Rosenbrock function
1.16 Problem 16: Sum squares function
1.17 Problem 17: Zakharov function
1.18 Problem 18: Powell function
1.19 Problem 19: Sphere function
1.20 Problem 20: Example 4.1 in [23]
1.21 Problem 21: Example 5.2 in [23]
where \(n=30\) and the polynomials \(f_{i}\) are defined as follows:
and \(f_{30}=x_{1}\), \(f_{31}=x_{2}-x_{1}^2-1\).
Rights and permissions
About this article
Cite this article
Wu, Z.Y., Tian, J. & Ugon, J. Global optimality conditions and optimization methods for polynomial programming problems. J Glob Optim 62, 617–641 (2015). https://doi.org/10.1007/s10898-015-0292-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-015-0292-5