A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs

T. D. Chuong (), V. Jeyakumar () and G. Li ()
Additional contact information
T. D. Chuong: University of New South Wales
V. Jeyakumar: University of New South Wales
G. Li: University of New South Wales

Journal of Global Optimization, 2019, vol. 75, issue 4, No 1, 885-919

Abstract: Abstract In this paper, we propose a bounded degree hierarchy of both primal and dual conic programming relaxations involving both semi-definite and second-order cone constraints for solving a nonconvex polynomial optimization problem with a bounded feasible set. This hierarchy makes use of some key aspects of the convergent linear programming relaxations of polynomial optimization problems (Lasserre in Moments, positive polynomials and their applications, World Scientific, Singapore, 2010) associated with Krivine–Stengle’s certificate of positivity in real algebraic geometry and some advantages of the scaled diagonally dominant sum of squares (SDSOS) polynomials (Ahmadi and Hall in Math Oper Res, 2019. https://doi.org/10.1287/moor.2018.0962; Ahmadi and Majumdar in SIAM J Appl Algebra Geom 3:193–230, 2019). We show that the values of both primal and dual relaxations converge to the global optimal value of the original polynomial optimization problem under some technical assumptions. Our hierarchy, which extends the so-called bounded degree Lasserre hierarchy (Lasserre et al. in Eur J Comput Optim 5:87–117, 2017), has a useful feature that the size and the number of the semi-definite and second-order cone constraints of the relaxations are fixed and independent of the step or level of the approximation in the hierarchy. As a special case, we provide a convergent bounded degree second-order cone programming (SOCP) hierarchy for solving polynomial optimization problems. We then present finite convergence at step one of the SOCP hierarchy for classes of polynomial optimization problems. This includes one-step convergence for a new class of first-order SDSOS-convex polynomial programs. In this case, we also show how a global solution is recovered from the level one SOCP relaxation. We finally derive a corresponding convergent conic linear programming hierarchy for conic-convex semi-algebraic programs. Whenever the semi-algebraic set of the conic-convex program is described by concave polynomial inequalities, we show further that the values of the relaxation problems converge to the common value of the convex program and its Lagrangian dual under a constraint qualification.

Keywords: Nonconvex polynomial optimization; Conic programming relaxations; Global optimization; Cone-convex polynomial programs; Convex semi-algebraic programs (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (8)

Downloads: (external link)
http://link.springer.com/10.1007/s10898-019-00831-9 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jglopt:v:75:y:2019:i:4:d:10.1007_s10898-019-00831-9

Ordering information: This journal article can be ordered from
http://www.springer. ... search/journal/10898

DOI: 10.1007/s10898-019-00831-9

Access Statistics for this article

Journal of Global Optimization is currently edited by Sergiy Butenko

More articles in Journal of Global Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().