Skip to main content
SpringerLink
Log in

A Global Optimization Method, QBB, for Twice-Differentiable Nonconvex Optimization Problem

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

A global optimization method, QBB, for twice-differentiable NLPs (Non-Linear Programming) is developed to operate within a branch-and-bound framework and require the construction of a relaxed convex problem on the basis of the quadratic lower bounding functions for the generic nonconvex structures. Within an exhaustive simplicial division of the constrained region, the rigorous quadratic underestimation function is constructed for the generic nonconvex function structure by virtue of the maximal eigenvalue analysis of the interval Hessian matrix. Each valid lower bound of the NLP problem with the division progress is computed by the convex programming of the relaxed optimization problem obtained by preserving the convex or linear terms, replacing the concave term with linear convex envelope, underestimating the special terms and the generic terms by using their customized tight convex lower bounding functions or the valid quadratic lower bounding functions, respectively. The standard convergence properties of the QBB algorithm for nonconvex global optimization problems are guaranteed. The preliminary computation studies are presented in order to evaluate the algorithmic efficiency of the proposed QBB approach.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • C.S. Adjiman S. Dallwig C.A. Floudas A. Neumaier (1998a) ArticleTitleA global optimization method, αBB, for general twice-differentiable constrained NLPs - I. Theoretical advances Computers and Chemical Engineering 22 1137–1158

    Google Scholar 

  • C.S. Adjiman I.P. Androulakis C.A. Floudas (1998b) ArticleTitleA global optimization method, αBB, for general twice-differentiable constrained NLPs - II. Implementation and computational results Computers Chemical Engineering 22 1159–1179

    Google Scholar 

  • F.A. Al-Khayyal J.E. Falk (1983) ArticleTitleJointly constrained biconvex programming Mathematics of Operational and Research 8 273–286

    Google Scholar 

  • A.R. Colville (1970) A comparative study of nonlinear programming codes H.W. Kuhn (Eds) Princeton Symposium on Mathematical Programming Princeton Univ. Press New Jersy

    Google Scholar 

  • C.A. Floudas A. Aggarwal A.R. Ciric (1989) ArticleTitleGlobal optimum search for non-convex NLP and MINLP problems Computers and Chemical Engineering 13 1117–1132

    Google Scholar 

  • C.A. Floudas V. Visweswaran (1990) ArticleTitleA global optimization algorithm (GOP) for certain classes of nonconvex NLPs: I. Theory Computers and Chemical Engineering 14 1397–1417 Occurrence Handle10.1016/0098-1354(90)80020-C

    Article  Google Scholar 

  • C.A. Floudas P.M. Pardalos (1990) A Collection of Test Problems for Constrained Global Optimization Algorithms Springer-Verlag Berlin

    Google Scholar 

  • C.A. Floudas V. Visweswaran (1993) ArticleTitleA primal-relaxed dual global optimization approaches Journal of Optimization Theory and Application 78 187–225

    Google Scholar 

  • C.A. Floudas (2000) Deterministic Global Optimization: Theory, Methods and Applications Kluwer Academic Publishers Dordrecht

    Google Scholar 

  • S. Gerschgorin (1931) ArticleTitleUber die abgrenzung der eigenwerte einer matrix Izv.Akademii Nauk SSSR, Ser.fiziku-material 6 749–754

    Google Scholar 

  • I.E. Grossmann (Eds) (1996) Global Optimization in Engineering Design, Kluwer book series in nonconvex optimization and its applications Kluwer Academic Publishers Dordrecht

    Google Scholar 

  • E.R. Hansen (1992) Global Optimization using Interval Analysis Marcel Dekker New York

    Google Scholar 

  • R. Horst H. Tuy (1990) Global Optimization : Deterministic Approaches Springer-Verlag Berlin

    Google Scholar 

  • R. Horst P.M. Pardalos (1995) Handbook of Global Optimization Kluwer Academic Publishers New York

    Google Scholar 

  • R. Horst P.M. Pardalos N.V. Thoai (1995) Introduction to Global Optimization Kluwer Academic Publishers Dordrecht, The Netherlands

    Google Scholar 

  • R.B. Kearfott (1996) Rigorous Global Search: Continuous Problems Kluwer Academic Publishers Dordrecht, The Netherlands

    Google Scholar 

  • Knuppel O. (1993). Programmer’s Runtime Optimized Fast Interval Library. Technische Informatik III, Technische Universitat Hamburg-Harburg.

  • H. Konno P.T. Thach H. Tuy (1997) Optimization on Low Rank Nonconvex Structures, Nonconvex Optimization and its Applications Kluwer Academic Publishers Dordrecht, The Netherlands

    Google Scholar 

  • Lasdon L. (2000). LSGRG Version 3.0 Release Notes. MSIS Department, College of Business Administration, the University of Texas at Austin.

  • C.D. Maranas C.A. Floudas (1992) ArticleTitleA global optimization approach for Lennard-Jones microclusters Journal of Chemical Physics 97 7667–7678 Occurrence Handle10.1063/1.463486

    Article  Google Scholar 

  • C.D. Maranas C.A. Floudas (1994) ArticleTitleGlobal minimum potential-energy conformations of small molecules Journal of Global Optimization 4 135–170 Occurrence Handle10.1007/BF01096720

    Article  Google Scholar 

  • G.P. McCormick (1976) ArticleTitleComputability of global solutions to factorable nonconvex programs: Part I – Convex underestimating problems Mathematical Programming 10 147–175 Occurrence Handle10.1007/BF01580665

    Article  Google Scholar 

  • C.M. McDonald C.A. Floudas (1994) ArticleTitleDecomposition based and branch and bound global optimization approaches for the phase equilibrium problems Journal of Global Optimization 5 205–251 Occurrence Handle10.1007/BF01096454

    Article  Google Scholar 

  • A. Neumaier (1990) Interval Methods for Systems of Equations Cambridge Univ. Press Cambridge, England

    Google Scholar 

  • A. Neumaier (1996) ArticleTitleSecond-order sufficient optimality conditions for local and global nonlinear programming Journal of Global Optimization 9 141–151 Occurrence Handle10.1007/BF00121660

    Article  Google Scholar 

  • G. Parthasarathy M.M. El-Halwagi (2000) ArticleTitleOptimum mass integration strategies for condensation and allocation of multicomponent VOCs Chemical Engineering Science 55 881–895 Occurrence Handle10.1016/S0009-2509(99)00382-6

    Article  Google Scholar 

  • R.T. Rockafellar (1972) Convex Analysis Princeton Univ. Press Princeton

    Google Scholar 

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

    Article  Google Scholar 

  • H.S. Ryoo N.V. Sahinidis (1996) ArticleTitleA branch-and-reduce approach to global optimization Journal Global Optimization 8 IssueID2 107–138

    Google Scholar 

  • H.S. Ryoo N.V. Sahinidis (2003) ArticleTitleGlobal optimization of multiplicative programs Journal of Global Optimization 26 IssueID4 387–418 Occurrence Handle10.1023/A:1024700901538

    Article  Google Scholar 

  • N.V. Sahinidis (1996) ArticleTitleBARON: A general purpose global optimization software package Journal of Global Optimization 8 201–205 Occurrence Handle10.1007/BF00138693

    Article  Google Scholar 

  • E.M.B. Smith C.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 

  • S. Smith L. Lasdon (1992) ArticleTitleSolving large sparse nonlinear programs using GRG ORSA Journal on Computing 4 IssueID1 2–15

    Google Scholar 

  • G. Stephanopoulos A.W. Westerberg (1975) ArticleTitleThe use of Hestenes’ method of multipliers to resolve dual gaps in engineering system optimization Journal of Optimization Theory and Application 15 285

    Google Scholar 

  • H. Tuy (1998) Convex Analysis and Global Optimization Kluwer Academic Publishers Dordrecht

    Google Scholar 

  • M. Tawarmalani N.V. Sahinidis (2001) ArticleTitleSemidefinite relaxation of fractional programs via novel convexification techniques Journal of Global Optimization 20 137–158 Occurrence Handle10.1023/A:1011233805045

    Article  Google Scholar 

  • M. Tawarmalani N.V. Sahinidis (2002) ArticleTitleConvex extensions and envelopes of lower semi-continuous functions Mathematical Programming 93 IssueID2 247–263 Occurrence Handle10.1007/s10107-002-0308-z

    Article  Google Scholar 

  • M. Tawarmalani N.V. Sahinidis (2004) ArticleTitleGlobal optimization of mixed-integer nonlinear programs: A theoretical and computational study Mathematical Programming 99 IssueID3 563–591 Occurrence Handle10.1007/s10107-003-0467-6

    Article  Google Scholar 

  • S. Vavasis (1991) Nonlinear Optimization: Complexity Issues Oxford University Press New York, N.Y.

    Google Scholar 

  • D.J. Wales H.A. Scheraga (1999) ArticleTitleGlobal optimization of clusters, crystals, and Biomolecules Science 285 1368–1372 Occurrence Handle10.1126/science.285.5432.1368

    Article  Google Scholar 

  • A.W. Westerberg J.V. Shah (1978) ArticleTitleAssuring a global optimum by the use of an upper bound on the lower (dual) bound Computers and Chemical Engineering 2 83

    Google Scholar 

  • Y. Zhu Z. Xu (1999) ArticleTitleCalculation of liquid-liquid equilibrium based on the global stability analysis for ternary mixtures by using a novel branch and bound algorithm: Application to UNIQUAC equation Industrial and Engineering Chemistry Research 38 3549–3556 Occurrence Handle10.1021/ie980476f

    Article  Google Scholar 

  • Y. Zhu K. Inoue (2001) ArticleTitleCalculation of chemical and phase equilibrium based on stability analysis by QBB algorithm: Application to NRTL equation Chemical Engineering Science 56 6915–6931

    Google Scholar 

  • Y. Zhu T. Kuno (2003) ArticleTitleGlobal optimization of nonconvex MINLP by a hybrid branch-and-bound and revised general Benders decomposition method Industrial and Engineering Chemical Research 42 528–539

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Information Sciences and Electronics, University of Tsukuba, Ibaraki, 305-8573, Japan

    Yushan Zhu & Takahito Kuno

  2. Department of Chemical Engineering, Tsinghua University, Beijing, 100084, China

    Yushan Zhu

Authors
  1. Yushan Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takahito Kuno
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yushan Zhu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhu, Y., Kuno, T. A Global Optimization Method, QBB, for Twice-Differentiable Nonconvex Optimization Problem. J Glob Optim 33, 435–464 (2005). https://doi.org/10.1007/s10898-005-0936-y

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-005-0936-y

Keywords

Search

Navigation