Skip to main content
Log in

Solving mixed integer nonlinear programs by outer approximation

  • Published:
Mathematical Programming Submit manuscript


A wide range of optimization problems arising from engineering applications can be formulated as Mixed Integer NonLinear Programming problems (MINLPs). Duran and Grossmann (1986) suggest an outer approximation scheme for solving a class of MINLPs that are linear in the integer variables by a finite sequence of relaxed MILP master programs and NLP subproblems.

Their idea is generalized by treating nonlinearities in the integer variables directly, which allows a much wider class of problem to be tackled, including the case of pure INLPs. A new and more simple proof of finite termination is given and a rigorous treatment of infeasible NLP subproblems is presented which includes all the common methods for resolving infeasibility in Phase I.

The worst case performance of the outer approximation algorithm is investigated and an example is given for which it visits all integer assignments. This behaviour leads us to include curvature information into the relaxed MILP master problem, giving rise to a new quadratic outer approximation algorithm.

An alternative approach is considered to the difficulties caused by infeasibility in outer approximation, in which exact penalty functions are used to solve the NLP subproblems. It is possible to develop the theory in an elegant way for a large class of nonsmooth MINLPs based on the use of convex composite functions and subdifferentials, although an interpretation for thel 1 norm is also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. R. Breu and C.-A. Burdet, “Branch-and-bound experiments in zero—one programming,”Mathematical Programming Study 2 (1974) 1–50.

    Google Scholar 

  2. M. Duran and I.E. Grossmann, “An outer-approximation algorithm for a class of Mixed Integer Nonlinear Programs,”Mathematical Programming 36 (1986) 307–339.

    Google Scholar 

  3. R. Fletcher,Practical Methods of Optimization (John Wiley, Chichester, 1987).

    Google Scholar 

  4. O.E. Flippo et al., “Duality and decomposition in general mathematical programming,” Econometric Institute, Report 8747/B, University of Rotterdam (1987).

  5. A.M. Geoffrion, “Generalized Benders decomposition,”Journal of Optimization Theory and Applications 10 (1972) 237–262.

    Google Scholar 

  6. G.R. Kocis and I.E. Grossmann, “Global optimization of nonconvex MINLP in process synthesis,”Industrial and Engineering Chemistry Research 27 (1988) 1407–1421.

    Google Scholar 

  7. F. Körner, “A new branching rule for the branch-and-bound algorithm for solving nonlinear integer programming problems,”BIT 28 (1988) 701–708.

    Google Scholar 

  8. R. Lazimy, “Improved algorithm for mixed-integer quadratic programs and a computational study,”Mathematical Programming 32 (1985) 100–113.

    Google Scholar 

  9. J. Viswanathan and I.E. Grossmann, “A combined penalty function and outer-approximation method for MINLP optimization,”Computers and Chemical Engineering 14 (1990) 769–782.

    Google Scholar 

  10. X. Yuan, S. Zhang, L. Pibouleau and S. Domenech, “Une méthode d'optimization non linéaire en variables mixtes pour la conception de procédés,”Operations Research 22/4 (1988) 331–346.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

This work is supported by SERC grant no. SERC GR/F 07972.

Corresponding author.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fletcher, R., Leyffer, S. Solving mixed integer nonlinear programs by outer approximation. Mathematical Programming 66, 327–349 (1994).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: