IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v75y2020i3d10.1007_s10589-020-00168-0.html
   My bibliography  Save this article

On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems

Author

Listed:
  • Feng Guo

    (Dalian University of Technology)

  • Xiaoxia Sun

    (Dongbei University of Finance and Economics)

Abstract

We consider the semi-infinite system of polynomial inequalities of the form $$\begin{aligned} {{\mathbf {K}}}:=\{x\in {{\mathbb {R}}}^m\mid p(x,y)\ge 0,\quad \forall y\in S\subseteq {{\mathbb {R}}}^n\}, \end{aligned}$$K:={x∈Rm∣p(x,y)≥0,∀y∈S⊆Rn},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 $${{\mathbb {R}}}^n$$Rn, $$-p(x,y)$$-p(x,y) is convex in x for every $$y\in S$$y∈S. We propose a procedure to construct approximate semidefinite representations of $${{\mathbf {K}}}$$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 $${{\mathbf {K}}}$$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 $${{\mathbf {K}}}$$K. We present an SDP relaxation method for this optimization problem by similar strategies used in constructing approximate semidefinite representations of $${{\mathbf {K}}}$$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.

Suggested Citation

  • Feng Guo & Xiaoxia Sun, 2020. "On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems," Computational Optimization and Applications, Springer, vol. 75(3), pages 669-699, April.
  • Handle: RePEc:spr:coopap:v:75:y:2020:i:3:d:10.1007_s10589-020-00168-0
    DOI: 10.1007/s10589-020-00168-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-020-00168-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10589-020-00168-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    2. J. Lasserre, 2012. "An algorithm for semi-infinite polynomial optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 119-129, April.
    3. M. A. Goberna & M. A. López, 2017. "Recent contributions to linear semi-infinite optimization," 4OR, Springer, vol. 15(3), pages 221-264, September.
    4. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    5. P. Parpas & B. Rustem, 2009. "An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 461-473, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Feng Guo & Liguo Jiao, 2021. "On solving a class of fractional semi-infinite polynomial programming problems," Computational Optimization and Applications, Springer, vol. 80(2), pages 439-481, November.
    2. Goberna, M.A. & Jeyakumar, V. & Li, G. & Vicente-Pérez, J., 2022. "The radius of robust feasibility of uncertain mathematical programs: A Survey and recent developments," European Journal of Operational Research, Elsevier, vol. 296(3), pages 749-763.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Feng Guo & Liguo Jiao, 2021. "On solving a class of fractional semi-infinite polynomial programming problems," Computational Optimization and Applications, Springer, vol. 80(2), pages 439-481, November.
    2. Li Wang & Feng Guo, 2014. "Semidefinite relaxations for semi-infinite polynomial programming," Computational Optimization and Applications, Springer, vol. 58(1), pages 133-159, May.
    3. M. A. Goberna & M. A. López, 2018. "Recent contributions to linear semi-infinite optimization: an update," Annals of Operations Research, Springer, vol. 271(1), pages 237-278, December.
    4. Bo Wei & William B. Haskell & Sixiang Zhao, 2020. "An inexact primal-dual algorithm for semi-infinite programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(3), pages 501-544, June.
    5. Sönke Behrends & Anita Schöbel, 2020. "Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 911-935, September.
    6. Holger Berthold & Holger Heitsch & René Henrion & Jan Schwientek, 2022. "On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 1-37, August.
    7. Olga Kostyukova & Tatiana Tchemisova, 2017. "Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 76-103, October.
    8. M. A. Goberna & M. A. López, 2017. "Recent contributions to linear semi-infinite optimization," 4OR, Springer, vol. 15(3), pages 221-264, September.
    9. Bo Wei & William B. Haskell & Sixiang Zhao, 2020. "The CoMirror algorithm with random constraint sampling for convex semi-infinite programming," Annals of Operations Research, Springer, vol. 295(2), pages 809-841, December.
    10. Chuong, T.D. & Jeyakumar, V., 2017. "Convergent hierarchy of SDP relaxations for a class of semi-infinite convex polynomial programs and applications," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 381-399.
    11. Yves Crama & Michel Grabisch & Silvano Martello, 2022. "Preface," Annals of Operations Research, Springer, vol. 314(1), pages 1-3, July.
    12. Xiao Wang & Xinzhen Zhang & Guangming Zhou, 2020. "SDP relaxation algorithms for $$\mathbf {P}(\mathbf {P}_0)$$P(P0)-tensor detection," Computational Optimization and Applications, Springer, vol. 75(3), pages 739-752, April.
    13. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
    14. Laurent, Monique & Vargas, Luis Felipe, 2022. "Finite convergence of sum-of-squares hierarchies for the stability number of a graph," Other publications TiSEM 3998b864-7504-4cf4-bc1d-f, Tilburg University, School of Economics and Management.
    15. Polyxeni-Margarita Kleniati & Panos Parpas & Berç Rustem, 2010. "Partitioning procedure for polynomial optimization," Journal of Global Optimization, Springer, vol. 48(4), pages 549-567, December.
    16. Laurent, M. & Rostalski, P., 2012. "The approach of moments for polynomial equations," Other publications TiSEM f08f3cd2-b83e-4bf1-9322-a, Tilburg University, School of Economics and Management.
    17. Souvik Das & Ashwin Aravind & Ashish Cherukuri & Debasish Chatterjee, 2022. "Near-optimal solutions of convex semi-infinite programs via targeted sampling," Annals of Operations Research, Springer, vol. 318(1), pages 129-146, November.
    18. Jie Wang & Victor Magron, 2021. "Exploiting term sparsity in noncommutative polynomial optimization," Computational Optimization and Applications, Springer, vol. 80(2), pages 483-521, November.
    19. Aguiar, Victor H. & Kashaev, Nail & Allen, Roy, 2023. "Prices, profits, proxies, and production," Journal of Econometrics, Elsevier, vol. 235(2), pages 666-693.
    20. Tomohiko Mizutani & Makoto Yamashita, 2013. "Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables," Journal of Global Optimization, Springer, vol. 56(3), pages 1073-1100, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:75:y:2020:i:3:d:10.1007_s10589-020-00168-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.