Abstract
We consider the semi-infinite system of polynomial inequalities of the form
where p(x, y) is a real polynomial in the variables x and the parameters y, the index set S is a basic semialgebraic set in \({{\mathbb {R}}}^n\), \(-p(x,y)\) is convex in x for every \(y\in S\). We propose a procedure to construct approximate semidefinite representations of \({{\mathbf {K}}}\). There are two indices to index these approximate semidefinite representations. As two indices increase, these semidefinite representation sets expand and contract, respectively, and can approximate \({{\mathbf {K}}}\) as closely as possible under some assumptions. In some special cases, we can fix one of the two indices or both. Then, we consider the optimization problem of minimizing a convex polynomial over \({{\mathbf {K}}}\). We present an SDP relaxation method for this optimization problem by similar strategies used in constructing approximate semidefinite representations of \({{\mathbf {K}}}\). Under certain assumptions, some approximate minimizers of the optimization problem can also be obtained from the SDP relaxations. In some special cases, we show that the SDP relaxation for the optimization problem is exact and all minimizers can be extracted.
Similar content being viewed by others
References
Ahmadi, A., Parrilo, P.: A complete characterization of the gap between convexity and SOS-convexity. SIAM J. Optim. 23(2), 811–833 (2013)
Ahmadi, A.A., Olshevsky, A., Parrilo, P.A., Tsitsiklis, J.N.: NP-hardness of deciding convexity of quartic polynomials and related problems. Math. Program. 137(1), 453–476 (2013)
Ahmadi, A.A., Parrilo, P.A.: A convex polynomial that is not SOS-convex. Math. Program. 135(1), 275–292 (2012)
Belousov, E.: Introduction to Convex Analysis and Integer Programming. Moscow University Publ, Moscow (1977)
Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2001)
Berg, C., Maserick, P.H.: Exponentially bounded positive definite functions. Ill. J. Math. 28, 162–179 (1984)
Bertsekas, D.P.: Convex Optimization Theory. Athena Scientific, Belmont (2009)
Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)
Borwein, J.M.: Direct theorems in semi-infinite convex programming. Math. Program. 21(1), 301–318 (1981)
Curto, R.E., Fialkow, L.A.: Truncated \(K\)-moment problems in several variables. J. Oper. Theory 54(1), 189–226 (2005)
Glashoff, K., Gustafson, S.A.: Linear Optimization and Approximation. Springer, Berlin (1983)
Goberna, M.A., López, M.A.: Recent contributions to linear semi-infinite optimization. 4OR 15(3), 221–264 (2017)
Gouveia, J., Parrilo, P., Thomas, R.: Theta bodies for polynomial ideals. SIAM J. Optim. 20(4), 2097–2118 (2010)
Guo, F., Wang, C., Zhi, L.: Semidefinite representations of noncompact convex sets. SIAM J. Optim. 25(1), 377–395 (2015)
Haviland, E.K.: On the momentum problem for distribution functions in more than one dimension. Am. J. Math. 57(3), 562–568 (1935)
Helton, J., Nie, J.: Sufficient and necessary conditions for semidefinite representability of convex hulls and sets. SIAM J. Optim. 20(2), 759–791 (2009)
Helton, J., Nie, J.: Semidefinite representation of convex sets. Math. Program. 122(1), 21–64 (2010)
Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35(3), 380–429 (1993)
Kriel, T., Schweighofer, M.: On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets. SIAM J. Optim. 28(2), 1796–1816 (2018)
Lasserre, J.: Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19(4), 1995–2014 (2009a)
Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)
Lasserre, J.B.: A semidefinite programming approach to the generalized problem of moments. Math. Program. 112(1), 65–92 (2008)
Lasserre, J.B.: Convex sets with semidefinite representation. Math. Program. Ser. A 120(2), 457–477 (2009b)
Lasserre, J.B.: An algorithm for semi-infinite polynomial optimization. TOP 20(1), 119–129 (2012)
Lasserre, J.B.: Tractable approximations of sets defined with quantifiers. Math. Program. 151(2), 507–527 (2015)
Lasserre, J.B., Netzer, T.: Sos approximations of nonnegative polynomials via simple high degree perturbations. Mathematische Zeitschrift 256(1), 99–112 (2007)
Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, pp. 157–270. Springer, New York (2009)
Levin, V .L.: Application of E. Helly’s theorem to convex programming, problems of best approximation and related questions. Math. USSR-Sbornik 8(2), 235 (1969)
Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), pp. 284–289 (2004)
López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180(2), 491–518 (2007)
Magron, V., Henrion, D., Lasserre, J.: Semidefinite approximations of projections and polynomial images of semialgebraic sets. SIAM J. Optim. 25(4), 2143–2164 (2015)
Nie, J.: Discriminants and nonnegative polynomials. J. Symbol. Comput. 47(2), 167–191 (2012)
Nie, J.: An exact Jacobian SDP relaxation for polynomial optimization. Math. Program. Ser. A 137(1–2), 225–255 (2013)
Nie, J.: Optimality conditions and finite convergence of lasserre’s hierarchy. Math. Program. Ser. A 146(1–2), 97–121 (2014)
Nie, J., Ranestad, K.: Algebraic degree of polynomial optimization. SIAM J. Optim. 20(1), 485–502 (2009)
Nie, J., Schweighofer, M.: On the complexity of Putinar’s positivstellensatz. J. Complex. 23(1), 135–150 (2007)
Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1(1), 15–22 (1991)
Parpas, P., Rustem, B.: An algorithm for the global optimization of a class of continuous minimax problems. J. Optim. Theory Appl. 141(2), 461–473 (2009)
Parrilo, P.A., Sturmfels, B.: Minimizing polynomial functions. In: Algorithmic and quantitative real algebraic geometry. Vol. 60 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, pp. 83–99 (2003)
Powers, V., Reznick, B.: Polynomials that are positive on an interval. Trans Am. Math. Soc. 352(10), 4677–4692 (2000)
Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)
Reznick, B.: Some concrete aspects of Hilbert’s 17th problem. In: Contemporary Mathematics. Vol. 253. American Mathematical Society, pp. 251–272 (2000)
Rostalski, P.: Bermeja—software for convex algebraic geometry (2010). http://math.berkeley.edu/~philipp/cagwiki
Scheiderer, C.: Spectrahedral shadows. SIAM J. Appl. Algebra Geom. 2(1), 26–44 (2018)
Schmüdgen, K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289(1), 203–206 (1991)
Wang, L., Guo, F.: Semidefinite relaxations for semi-infinite polynomial programming. Comput. Optim. Appl. 58(1), 133–159 (2013)
Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming—Theory, Algorithms, and Applications. Kluwer Academic Publisher, Dordrecht (2000)
Xu, Y., Sun, W., Qi, L.: On solving a class of linear semi-infinite programming by SDP method. Optimization 64(3), 603–616 (2015)
Acknowledgements
The authors are very grateful for the comments of two anonymous referees which helped to improve the presentation. The first author was supported by the Chinese National Natural Science Foundation under Grants 11401074 and 11571350. The second author was supported by the Chinese National Natural Science Foundation under Grant 11801064, the Foundation of Liaoning Education Committee under Grant LN2017QN043.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Guo, F., Sun, X. On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems. Comput Optim Appl 75, 669–699 (2020). https://doi.org/10.1007/s10589-020-00168-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-020-00168-0
Keywords
- Semi-infinite systems
- Convex polynomials
- Semidefinite representations
- Semidefinite programming relaxations
- Sum of squares
- Polynomial optimization