Mathematical Methods of Operations Research

, Volume 66, Issue 3, pp 373–407 | Cite as

Biconvex sets and optimization with biconvex functions: a survey and extensions

Original Article


The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a function \(f:X\times Y\to{\mathbb{R}}\) is called biconvex, if f(x,y) is convex in y for fixed xX, and f(x,y) is convex in x for fixed yY. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives some extensions. In particular, we focus on biconvex minimization problems and survey methods and algorithms for the constrained as well as for the unconstrained case. Furthermore, we state new theoretical results for the maximum of a biconvex function over biconvex sets.


Biconvex functions Biconvex sets Biconvex optimization Biconcave optimization Non-convex optimization Generalized convexity 


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  1. Al-Khayyal F (1990) Jointly constrained bilinear programs and related problems: an overview. Comput Math Appl 19(11):53–62MATHCrossRefMathSciNetGoogle Scholar
  2. Al-Khayyal F, Falk J (1983) Jointly constrained biconvex programming. Math Oper Res 8(2):273–286MATHMathSciNetGoogle Scholar
  3. Audet C, Hansen P, Jaumard B, Savard G (2000) A branch and cut algorithm for non-convex quadratically constrained quadratic programming. Math Program Ser A 87(1):131–152MATHMathSciNetGoogle Scholar
  4. Aumann R, Hart S (1986) Bi-convexity and bi-martingales. Isr J Math 54(2):159–180MATHMathSciNetGoogle Scholar
  5. Barmish B (1994) New Tools for Robustness of Linear Systems. Maxwell, Macmillan International, New YorkMATHGoogle Scholar
  6. Barmish B, Floudas C, Hollot C, Tempo R (1995) A global programming solution to some open robustness problems including matrix polytope stability. In: Proceedings of the American Control Conference, Seattle, Washington, pp 3871–3877Google Scholar
  7. Bazaraa M, Sherali H, Shetty C (1993) Nonlinear programming—theory and algorithms, 2nd edn. Wiley, New YorkMATHGoogle Scholar
  8. Benders J (1962) Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4:238–252MATHCrossRefMathSciNetGoogle Scholar
  9. Besl P, McKay N (1992) A method for registration of 3-D shapes. IEEE Trans Pattern Anal Mach Intell 14:239–256CrossRefGoogle Scholar
  10. Borwein J (1986) Partially monotone operators and the generic differentiability of convex-concave and biconvex mappings. Isr J Math 54(1):42–50MATHGoogle Scholar
  11. Burkholder D (1981) A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann Probab 9(6):997–1011MATHMathSciNetGoogle Scholar
  12. Burkholder D (1986) Lecture Notes in Mathematics. Probability and Analysis (Varenna, 1985), vol 1206. Chapter Martingales and Fourier analysis in Banach spaces, pp 61–108. Springer, HeidelbergGoogle Scholar
  13. Cooper L (1963) Location-allocation problems. Oper Res 11:331–343MATHGoogle Scholar
  14. Cooper L (1964) Heuristic methods for location-allocation problems. SIAM Rev 6:37–53CrossRefMathSciNetGoogle Scholar
  15. de Leeuw J (1994) Block relaxation algorithms in statistics. In: Bock H, Lenski W, Richter M (eds) Information systems and data analysis. Springer, Heidelberg, pp 308–325Google Scholar
  16. Falk J, Soland R (1969) An algorithm for separable nonconvex programming problems. Manage Sci 15(9):550–569MathSciNetMATHGoogle Scholar
  17. Floudas C (1995) Nonlinear and mixed integer optimization: fundamentals and applications. Oxford Press, New YorkMATHGoogle Scholar
  18. Floudas C (2000) Deterministic global optimization, 1st edn. Kluwer, DordrechtGoogle Scholar
  19. Floudas C, Visweswaran V (1990) A global optimization algorithm (GOP) for certain classes of nonconvex NLPs: I. Theory. Comput Chem Eng 14(12):1397–1417CrossRefGoogle Scholar
  20. Floudas C, Visweswaran V (1993) A primal-relaxed dual global optimization approach. J Optim Theory Appl 78(2):187–225MATHCrossRefMathSciNetGoogle Scholar
  21. Gao Y, Xu C (2002) An outer approximation method for solving the biconcave programs with separable linear constraints. Math Appl 15(3):42–46MATHMathSciNetGoogle Scholar
  22. Gelbaum B, Olmsted J (2003) Counterexamples in analysis. Dover, Mineola, New YorkMATHGoogle Scholar
  23. Geng Z, Huang L (2000a) Robust stability of systems with both parametric and dynamic uncertainties. Syst Control Lett 39:87–96MATHCrossRefMathSciNetGoogle Scholar
  24. Geng Z, Huang L (2000b) Robust stability of the systems with mixed uncertainties under the IQC descriptions. Int J Control 73(9):776–786MATHCrossRefMathSciNetGoogle Scholar
  25. Geoffrion A (1972) Generalized Benders decomposition. J Optim Theory Appl 10(4):237–260MATHCrossRefMathSciNetGoogle Scholar
  26. Goh K, Turan L, Safonov M, Papavassilopoulos G, Ly J (1994) Biaffine matrix inequality properties and computational methods. In: Proceedings of the American Control Conference, Baltimore, Maryland, pp 850–855Google Scholar
  27. Goh K, Safonov M, Papavassilopoulos G (1995) Global optimization for the biaffine matrix inequality problem. J Global Optim 7:365–380MATHCrossRefMathSciNetGoogle Scholar
  28. Hodgson M, Rosing K, Shmulevitz (1993) A review of location-allocation applications literature. Stud Locat Anal 5:3–29Google Scholar
  29. Horst R, Thoai N (1996) Decomposition approach for the global minimization of biconcave functions over polytopes. J Optim Theory Appl 88(3):561–583MATHCrossRefMathSciNetGoogle Scholar
  30. Horst R, Tuy H (1990) Global optimization, deterministic approaches. Springer, BerlinMATHGoogle Scholar
  31. Huang S, Batta R, Klamroth K, Nagi R (2005) K-Connection location problem in a plane. Ann Oper Res 136:193–209MATHCrossRefMathSciNetGoogle Scholar
  32. Jouak M, Thibault L (1985) Directional derivatives and almost everywhere differentiability of biconvex and concave-convex operators. Math Scand 57:215–224MATHMathSciNetGoogle Scholar
  33. Lee J (1993) On Burkholder’s biconvex-function characterisation of Hilbert spaces. In: Proceedings of the American Mathematical Society, vol 118, pp 555–559Google Scholar
  34. Luenberger D (1989) Linear and nonlinear programming, 2nd edn. Addison-Wesley, ReadingGoogle Scholar
  35. Meyer R (1976) Sufficient conditions for the convergence of monotonic mathematical programming algorithms. J Comput Syst Sci 12:108–121MATHGoogle Scholar
  36. Ostrowski A (1966) Solution of equations and systems of equations, 2nd edn. Academic, New YorkMATHGoogle Scholar
  37. Plastria F (1995) Continuous location problems. In: Drezner Z (ed) Facility location, pp 225–262. Springer Series in Operations ResearchGoogle Scholar
  38. Rockafellar R (1997) Convex analysis, 1st edn. Princeton University Press, PrinctonMATHGoogle Scholar
  39. Sherali H, Alameddine A, Glickman T (1994) Biconvex models and algorithms for risk management problems. Am J Math Manage Sci 14(3–4):197–228MATHMathSciNetGoogle Scholar
  40. Sherali H, Alameddine A, Glickman T (1995) Biconvex models and algorithms for risk management problems. Oper Res Manage Sci 35(4):405–408Google Scholar
  41. Thibault L (1984) Continuity of measurable convex and biconvex operators. In: Proceedings of the American Mathematical Society, vol 90, pp 281–284Google Scholar
  42. Tuyen H, Muu L (2001) Biconvex programming approach to optimization over the weakly efficient set of a multiple objective affine fractional problem. Oper Res Lett 28:81–92MATHCrossRefMathSciNetGoogle Scholar
  43. Visweswaran V, Floudas C (1990) A global optimization algorithm (GOP) for certain classes of nonconvex NLPs: II. Application of theory and test problems. Comput Chem Eng 14(12):1419–1434CrossRefGoogle Scholar
  44. Visweswaran V, Floudas C (1993) New properties and computational improvement of the GOP algorithm for problems with quadratic objective function and constraints. J Global Optim 3(3):439–462MATHCrossRefMathSciNetGoogle Scholar
  45. Wendell R, Hurter A Jr (1976) Minimization of non-separable objective function subject to disjoint constraints. Oper Res 24(4):643–657MATHMathSciNetGoogle Scholar
  46. Zangwill W (1969) Convergence conditions for nonlinear programming algorithms. Manage Sci 16(1):1–13MathSciNetCrossRefMATHGoogle Scholar
  47. Zitová B, Flusser J (2003) Image registration methods: a survey. Image Vis Comput 21:977–1000CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Jochen Gorski
    • 1
  • Frank Pfeuffer
    • 1
  • Kathrin Klamroth
    • 1
  1. 1.Institute for Applied MathematicsFriedrich-Alexander-University Erlangen-NurembergErlangenGermany

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