Advertisement

Journal of Global Optimization

, Volume 6, Issue 1, pp 87–105 | Cite as

A method for solving d.c. programming problems. Application to fuel mixture nonconvex optimization problem

  • Thai Quynh Phong
  • Pham Dinh Tao
  • Thi Hoai Le An
Article

Abstract

We present a numerical method for solving the d.c. programming problem
$$c^* = \min \{ \langle c,x\rangle s.t. f_i (x) \leqslant 0, i = 1,...,m, x \in D\} $$
wherefi, i=1,...,m are d.c. (difference of two convex functions) and D is a convex set in ℝn. An (ɛ, η)-solutionx(ɛ, η) satisfying
$$x(\varepsilon ,\eta ) \in D, \langle c,x(\varepsilon ,\eta )\rangle \leqslant c^* + \varepsilon , f_i (x(\varepsilon ,\eta )) \leqslant \eta , i = 1,...,m,$$
can be found after a finite number of iterations. This algorithm combines an outer approximation procedure for solving a system of d.c. inequalities with a simple general scheme for minimizing a linear function over a compact set. As an application we discuss the numerical solution of a fuel mixture problem (encountered in the oil industry).

Key words

Nonlinear programming global optimization d.c. programming outer approximation system of d.c. inequalities (ɛ, η)-solution fuel mixture problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Benacer (1986),Contribution à l'étude des algorithmes de l'optimisation non convexe et non differentiable. Thèse de doctorat en math, appl., Université J. Fourier, Grenoble, France.Google Scholar
  2. 2.
    M.C. Bohringer and S.E. Jacobsen (1983), Two general algorithms for solving linear programs with an additional reverse convex constraint. InLectures Notes in Control and Inform. Sc, number 59 in Sys. Mod. and Optim., Copenhagen. 11th of IFIP Working Conference.Google Scholar
  3. 3.
    J. Chaarani (1989),Etude d'une classe d'algorithmes d'optimisation non convexe. Implementation et Applications. Thèse de doctorat, Univ. Joseph Fourier, Grenoble, France.Google Scholar
  4. 4.
    P.C. Chen, P. Hansen, and B. Jaumard (1991), On-line and off-line vertex enumeration by adjacency lists.Operations Research Letters 10, 403–409.Google Scholar
  5. 5.
    A. Chine (1991),Algorithmes robustes en optimisation non convexe. Codes et Simulations numeriques en grande dimension. Thèse de doctorat, Univ. Joseph Fourier, Grenoble, France.Google Scholar
  6. 6.
    J.E. Falk and K.L. Hoffman (1976), A successive underestimating method for concave programming problems.Mathematics of Operations Research 1, 251–259.Google Scholar
  7. 7.
    C.A. Floudas and V. Visweswaran (1990), A global optimization algorithm for certain classes of nonconvex NLPs -I. Theory.Computers and Chemical Engineering 14, 1397, 1990.Google Scholar
  8. 8.
    J. Fülöp (1990), A finite cutting plane method for solving linear programs with an additional reverse convex constraint.European J. Oper. Research 44, 395–409.Google Scholar
  9. 9.
    F. Giannessi, L. Jurina, and G. Maier (1979), Optimal excavation profile for a pipeline freely resting on the sea-floor.Engineering Structures 1, 81–91.Google Scholar
  10. 10.
    R.J. Hillestad (1975), Optimization problems subject to a budget constraint with economies of scale.Operations Research 23, 1091–1098.Google Scholar
  11. 11.
    R.J. Hillestad and Jacobsen S.E. (1980), Linear programs with an additional reverse convex constraint.Applied Mathematics and Optimization 6, 257–269.Google Scholar
  12. 12.
    R.J. Hillestad and Jacobsen S.E. (1980), Reverse convex programming.Applied Mathematics and Optimization 6, 63–78.Google Scholar
  13. 13.
    J.B. Hiriart-Urruty (1985), Generalized differentiability, duality and optimization for problems dealing with difference of two convex functions. InLectures Notes in Economics and Mathematical Systems 256, 37–69. Springer-Verlag, Berlin.Google Scholar
  14. 14.
    R. Horst (1988), Deterministic global optimization: Recent advances and new fields of application.Naval Res. Log. Quar. 37, 433–471.Google Scholar
  15. 15.
    R. Horst and Thoai N.V. (1988), Branch and bound methods for solving systems of equations and inequalities.J. of Optimization Theory and Applications 134, 426–430.Google Scholar
  16. 16.
    R. Horst, Thoai N.V., and H. Benson (1991), Concave minimization via conical partitions and polyhedral outer approximation.Mathematical Programming 50, 259–274.Google Scholar
  17. 17.
    R. Horst, Thoai, N.V., and J. de Vries (1988), On finding new vertices and redudant constraints in cutting plane algorithms for global optimization.Operations Research Letters 7(2), 85–90.Google Scholar
  18. 18.
    R. Horst, Thoai N.V., and H. Tuy (1987), Outer approximation by polyhedral convex sets.Operations Research Spektrum 9, 153–159.Google Scholar
  19. 19.
    R. Horst, T.Q. Phong, and N.V. Thoai (1990), On solving general reverse convex programming problems by a sequence of linear programs and linear searches.Annals of Operations Research 25, 1–18.Google Scholar
  20. 20.
    R. Horst, T.Q. Phong, N.V. Thoai, and J. de Vries (1991), On solving a d.c. programming problem by a sequence of linear programs.J. of Global Optimization 1, 183–203.Google Scholar
  21. 21.
    R. Horst and H. Tuy (1993),Global Optimization: Deterministic Approaches. Springer-Verlag, Berlin New York, 2 editition.Google Scholar
  22. 22.
    T.H. Matheis and D.S Rubin (1980), A survey and comparison of methods for finding all vertices of convex polyhedral sets.Mathematics of Operations Research 5, 167–185.Google Scholar
  23. 23.
    L.D. Muu (1985), Convergent algorithm for solving linear programs with an additional reverse convex constraint.Kybernetica 21, 428–435.Google Scholar
  24. 24.
    N.D. Nghia and N.D. Hieu (1986), A method for solving reverse convex programming problems.Acta Mathematica Vietnamica 11(2), 241–252.Google Scholar
  25. 25.
    V.H. Nguyen and J.J. Strodiot (1992), Computing a global optimal solution to a design centering problem.Mathematical Programming 53, 111–123.Google Scholar
  26. 26.
    V.H. Nguyen, J.J. Strodiot, and N.V. Thoai (1985), On an optimum shape design problem. Tech. Report 85/5, Department of Mathematics, Univ. of Namur.Google Scholar
  27. 27.
    S. Sen and H.D. Sherali (1987), Nondifferentiable reverse convex programs and facial convexity cuts via disjunctive characterization.Mathematical Programming 37, 169–183.Google Scholar
  28. 28.
    Pham D. Tao (1991) Un algorithme pour la résolution du programme linéaire général.RAIRO-Recherche opérationnelle 25, 183–201.Google Scholar
  29. 29.
    Pham D. Tao and El Bernoussi (1988), Duality in d.c. (difference of convex functions) optimization. Subgradient methods. In K.H. Hoffmannet al., eds,Trends in Mathematical Optimization, volume 84 ofInternational Series of Numerische Mathematik. Birkhauser.Google Scholar
  30. 30.
    Pham D. Tao and El Bernoussi (1989), Numerical method for solving a class of global nonconvex optimization problems.International Series of Numerische Mathematik 87, 97–132.Google Scholar
  31. 31.
    P.T. Thach (1993), D.c. sets, d.c. functions and nonlinear equations.Mathematical programming 58, 415–428.Google Scholar
  32. 32.
    T.V. Thieu, B.T. Tam, and V.T. Ban (1985), An outer approxmation method for globally minimizing a concave function over a compact convex set.Acta Mathematica Vietnamica 2.Google Scholar
  33. 33.
    N.V. Thoai (1988), A modified version of Tuy's method for solving d.c. programming problems.Optimization 19, 665–674.Google Scholar
  34. 34.
    N.V. Thuong and H. Tuy (1984), A finite algorithm for solving linear programs with an additional reverse convex constraint. In V. Demyannov and H. Pallaschlke, editors,Lecture Notes in Economics and Mathematical Systems 225, 291–302. Springer.Google Scholar
  35. 35.
    H. Tuy (1983), On outer approximation methods for solving concave minimization problems.Acta Mathematica Vietnamica 8, 3–34.Google Scholar
  36. 36.
    H. Tuy (1986), A general deterministic approach to global optimization via d.c. programming. In J.B. Hiriart-Urruty, editor,Fermats Days 1985: Mathematics for Optimization, 137–162. North-Holland, Amsterdam.Google Scholar
  37. 37.
    H. Tuy (1987), Global minimization of a difference of two convex functions.Mathematical Programming Study 30, 150–182.Google Scholar
  38. 38.
    L. Vigidal and S. Director (1982), A design centering algorithm for nonconvex regions of acceptability.IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 13–24.Google Scholar
  39. 39.
    A.B. Zalessky (1980), Nonconvexity of feasible domains and optimization of management decisions.Ekonomika i Mat. Metody. Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Thai Quynh Phong
    • 1
  • Pham Dinh Tao
    • 1
  • Thi Hoai Le An
    • 1
  1. 1.LMI-INSA RouenCNRS, URA 1378Mt St Aignan CedexFrance

Personalised recommendations