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Linearity Embedded in Nonconvex Programs

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Abstract

Nonconvex programs involving bilinear terms and linear equality constraints often appear more nonlinear than they really are. By using an automatic symbolic reformulation we can substitute some of the bilinear terms with linear constraints. This has a dramatically improving effect on the tightness of any convex relaxation of the problem, which makes deterministic global optimization algorithms like spatial Branch-and-Bound much more eff- cient when applied to the problem.

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References

  1. N. Adhya M. Tawarmalani N. Sahinidis (1999) ArticleTitleA Lagrangian approach to the pooling problem Industrial and Engineering Chemistry Research 38 1956–1972 Occurrence Handle10.1021/ie980666q

    Article  Google Scholar 

  2. Adjiman, C. (1998), Global Optimization Techniques for Process Systems Engineering. Ph.D. thesis, Princeton University.

  3. C. Adjiman S. Dallwig C. Floudas A. Neumaier (1998) ArticleTitleA global optimization method, α BB, for general twice-differentiable constrained NLPs: I. Theoretical advances Computers and Chemical Engineering 22 IssueID9 1137–1158 Occurrence Handle10.1016/S0098-1354(98)00027-1

    Article  Google Scholar 

  4. A. Aho J. Hopcroft J. Ullman (1983) Data Structures and Algorithms Addison-Wesley Reading, MA

    Google Scholar 

  5. F. Al-Khayyal J. Falk (1983) ArticleTitleJointly constrained biconvex programming Mathematics of Operations Research 8 IssueID2 273–286

    Google Scholar 

  6. Epperly, T. (1995), Global optimization of nonconvex nonlinear programs using parallel branch and bound. Ph.D. thesis, University of Winsconsin, Madison.

  7. T. Epperly E. Pistikopoulos (1997) ArticleTitleA reduced space branch and bound algorithm for global optimization Journal of Global Optimization 11 287–311 Occurrence Handle10.1023/A:1008212418949

    Article  Google Scholar 

  8. Gill, P. (1999), User's Guide for SNOPT 5.3. Systems Optimization Laboratory, Department of EESOR, Stanford University, California.

  9. P. Kesavan P. Barton (2000) ArticleTitleDecomposition algorithms for nonconvex mixed-integer nonlinear programs AIChE Symposium Series 96 IssueID323 458–461

    Google Scholar 

  10. B. Korte J. Vygen (2000) Combinatorial Optimization, Theory and Algorithms Springer-Verlag Berlin

    Google Scholar 

  11. L. Liberti (2004) ArticleTitleReduction constraints for the global optimization of NLPs International Transactions in Operations Research 11 33–41 Occurrence Handle10.1111/j.1475-3995.2004.00438.x

    Article  Google Scholar 

  12. Liberti, L., Tsiakis, P., Keeping, B. and Pantelides, C. (2001),ooOPS , 1.24 ed., Centre for Process Systems Engineering, Chemical Engineering Department, Imperial College, London, UK.

  13. G. McCormick (1976) ArticleTitleComputability of global solutions to factorable nonconvex programs: Part I–Convex underestimating problems Mathematical Programming 10 146–175 Occurrence Handle10.1007/BF01580665

    Article  Google Scholar 

  14. H.S. Ryoo N.V. Sahinidis (1995) ArticleTitleGlobal optimization of nonconvex NLPs and MINLPs with applications in process design Computers and Chemical Engineering 19 IssueID5 551–566 Occurrence Handle10.1016/0098-1354(94)00097-8

    Article  Google Scholar 

  15. H. Sherali A. Alameddine (1992) ArticleTitleA new Reformulation–Linearization Technique for bilinear programming problems Journal of Global Optimization 2 379–410 Occurrence Handle10.1007/BF00122429

    Article  Google Scholar 

  16. H. Sherali J. Smith W. Adams (2000) ArticleTitleReduced first-level representations via the reformulation–linearization technique: Results, counterexamples and computations. Discrete Applied Mathematics 101 247–267 Occurrence Handle10.1016/S0166-218X(99)00225-5

    Article  Google Scholar 

  17. Smith, E. (1996), On the optimal design of continuous processes. Ph.D. thesis, Imperial College of Science, Technology and Medicine, University of London.

  18. E. Smith C. Pantelides (1997) ArticleTitleGlobal optimisation of nonconvex MINLPs Computers and Chemical Engineering 21 S791–S796

    Google Scholar 

  19. E. Smith C. Pantelides (1999) ArticleTitleA symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs Computers and Chemical Engineering 23 457–478 Occurrence Handle10.1016/S0098-1354(98)00286-5

    Article  Google Scholar 

  20. R. Vaidyanathan M. El-Halwagi (1996) Global optimization of nonconvex MINLPs by interval analysis I. Grossmann (Eds) Global Optimization in Engineering Design Kluwer Academic Publishers Dordrecht 175–193

    Google Scholar 

  21. L. Wolsey (1998) Integer Programming Wiley New York

    Google Scholar 

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  1. Politecnico di Milano, DEI, P.zza L. da Vinci 32, I-20133, Milano, Italy

    Leo Liberti

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  1. Leo Liberti
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Liberti, L. Linearity Embedded in Nonconvex Programs. J Glob Optim 33, 157–196 (2005). https://doi.org/10.1007/s10898-004-0864-2

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  • DOI: https://doi.org/10.1007/s10898-004-0864-2

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