OSQP: an operator splitting solver for quadratic programs

Bartolomeo Stellato, Goran Banjac, Paul Goulart, Alberto Bemporad, Stephen Boyd

Research output: Contribution to journalArticlepeer-review

452 Scopus citations


We present a general-purpose solver for convex quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of a quasi-definite linear system with the same coefficient matrix at almost every iteration. Our algorithm is very robust, placing no requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. It can be configured to be division-free once an initial matrix factorization is carried out, making it suitable for real-time applications in embedded systems. In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates. The method also supports factorization caching and warm starting, making it particularly efficient when solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint, is library-free, and has been extensively tested on many problem instances from a wide variety of application areas. It is typically ten times faster than competing interior-point methods, and sometimes much more when factorization caching or warm start is used. OSQP has already shown a large impact with tens of thousands of users both in academia and in large corporations.

Original languageEnglish (US)
Pages (from-to)637-672
Number of pages36
JournalMathematical Programming Computation
Issue number4
StatePublished - Dec 1 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Software


  • First-order methods
  • Operator splitting
  • Optimization
  • Quadratic programming


Dive into the research topics of 'OSQP: an operator splitting solver for quadratic programs'. Together they form a unique fingerprint.

Cite this