Equivalent sufficient conditions for global optimality of quadratically constrained quadratic programs

Sunyoung Kim () and Masakazu Kojima ()
Additional contact information
Sunyoung Kim: Ewha W. University
Masakazu Kojima: Chuo University

Mathematical Methods of Operations Research, 2025, vol. 101, issue 1, No 4, 73-94

Abstract: Abstract We study the equivalence of several well-known sufficient optimality conditions for a general quadratically constrained quadratic program (QCQP). The conditions are classified in two categories. The first is for determining an optimal solution and the second is for finding an optimal value. The first category includes the existence of a saddle point of the Lagrangian function and the existence of a rank-1 optimal solution of the primal SDP relaxation of QCQPs. The second category includes $$\eta _p = \zeta $$ η p = ζ , $$\eta _d = \zeta $$ η d = ζ , and $$\varphi = \zeta $$ φ = ζ , where $$\zeta $$ ζ , $$\eta _p$$ η p , $$\eta _d$$ η d , and $$\varphi $$ φ denote the optimal values of QCQPs, the primal SDP relaxation, the dual SDP relaxation and the Lagrangian dual, respectively. We show the equivalence of these conditions with or without assuming the existence of an optimal solution of QCQP and/or the Slater constraint qualification for the primal SDP relaxation. The results on the conditions are also extended to the doubly nonnegative relaxation of equality constrained QCQPs in nonnegative variables.

Keywords: Quadratically constrained quadratic program; Global optimality condition; Saddle point of Lagrangian function; Exact SDP relaxation; Rank-1 optimal solution of SDP relaxation; KKT condition; 90C20; 90C22; 90C25; 90C26 (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s00186-024-00885-w 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:mathme:v:101:y:2025:i:1:d:10.1007_s00186-024-00885-w

Ordering information: This journal article can be ordered from
http://www.springer.com/economics/journal/00186

DOI: 10.1007/s00186-024-00885-w

Access Statistics for this article

Mathematical Methods of Operations Research is currently edited by Oliver Stein

More articles in Mathematical Methods of Operations Research from Springer, Gesellschaft für Operations Research (GOR), Nederlands Genootschap voor Besliskunde (NGB)
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().