Advertisement

D.C. Optimization: Theory, Methods and Algorithms

  • Hoang Tuy
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 2)

Abstract

Optimization problems involving d.c. functions (differences of convex functions) and d.c. sets (differences of convex sets) occur quite frequently in operations research,economics, engineering design and other fields. We present a review of the theory, methods and algorithms for this class of global optimization problems which have been elaborated in recent years

Key words:

d.c. functions d.c. sets duality reverse convex programming d.c. optimization continuous optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A.D. Alexandrov: 1949, ‘On surfaces which may be represented by a difference of convex functions’ (in Russian), Izvestiya Akademii Nauk ICazakhskoj SSR, Seria FizikoMatematicheskikh 3, 3-20.Google Scholar
  2. 2.
    A.D. Alexandrov: 1950, ‘On surfaces which may be represented by differences of convex functions’ (in Russian), Doklady Akademii Nauk SSR, 72, 613-616.Google Scholar
  3. 3.
    F.A. Al-Khayyal: 1989, ‘Jointly Constrained Bilinear Programs and Related Problems. An Overview’, Computers and Mathematics with Applications, 19, 53-62.MathSciNetGoogle Scholar
  4. 4.
    F.A. Al-Khayyal and J.E. Falk: 1983, ‘Jointly Constrained Biconvex Programming’, Mathematics of Operations Research, 8, 273-286.MathSciNetzbMATHGoogle Scholar
  5. 5.
    F.A. Al-Khayyal, R. Horst and P.M. Pardalos: 1989, ‘Global Optimization of Concave Functions Subject to Separable Quadratic Constraints. An Application in Nonlinear Bilevel Programming’, to appear in Annals of Operations Research.Google Scholar
  6. 6.
    E. Asplund: 1973, ‘Differentiability of the metric projection in finite dimensional Euclidean space’, Proceedings of the American Mathematical Society, 38, 218-219.MathSciNetzbMATHGoogle Scholar
  7. 7.
    M. Avriel and A.C. Williams: 1970, ‘Complementary Geometric Programming’, SIAM Journal on Applied Mathematics, 19, 125-141.MathSciNetzbMATHGoogle Scholar
  8. 8.
    E. Balas, J.M. Tama and J. Tind: 1989, ‘Sequential convexification in reverse convex and disjunctive programming’, Mathematical Programming, 44, 337-351.MathSciNetzbMATHGoogle Scholar
  9. 9.
    P.P. Barisal and S.E. Jacobsen: 1975, ‘An algorithm for optimizing network flow capacity under economies of scale’, Journal of Optimization Theory and Applications,15,565-586.MathSciNetGoogle Scholar
  10. 10.
    R. Benacer: 1986, ‘Contribution à l’étude des algorithmes de l’optimisation nonconvexe et nondifférentiable’, Thèse, Université Scientifique, Technologique et Médicale de Grenoble.Google Scholar
  11. 11.
    K. Ben Nahia: 1986, ‘Autour de la biconvexité’, Thèse, Université de Toulouse.Google Scholar
  12. 12.
    S. Ben Saeed and S.E. Jacobsen: 1990, ‘A Level Set Algorithm for a Class of Reverse Convex Programs, Annals of Operations Research, 25, 19-42.MathSciNetGoogle Scholar
  13. 13.
    H.P. Benson: 1989, ‘On the Structure and Properties of a Linear Multilevel Programming Problem’, Journal of Optimization Theory and Applications, 60, 353-373.MathSciNetzbMATHGoogle Scholar
  14. 14.
    H.P. Benson: 1991, ‘An All-Linear Programming Relaxation Algorithm for Optimization over the Efficient Set’, Journal of Global Optimization, 1, 83-104.MathSciNetzbMATHGoogle Scholar
  15. 15.
    H.P. Benson: 1992, ‘An Algorithm for Optimizing over the Weakly-Efficient Set’, European J. of Operations Research, 25, 192-199.MathSciNetGoogle Scholar
  16. 16.
    H.P. Benson: 1993, ‘A Bisection-Extreme Point Search Algorithm for Optimizing over the Efficient Set in the Linear Dependence Case’, Journal of Global Optimization, 3, 95-111.MathSciNetzbMATHGoogle Scholar
  17. 17.
    L. Bittner: 1970, ‘Some representation theorems for functions and sets and their applications to nonlinear programming’, Numerische Mathematik, 16, 32-51.MathSciNetzbMATHGoogle Scholar
  18. 18.
    V.P. Bulatov: 1977, ‘Embedding methods in optimization problems’, Nauka, Novosibirsk (in Russian)Google Scholar
  19. 19.
    V.P. Bulatov: 1990, ‘Methods for solving multiextremal problems (global search)’, Annals of Operations Research, 25, 253-278.MathSciNetzbMATHGoogle Scholar
  20. 20.
    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.MathSciNetzbMATHGoogle Scholar
  21. 21.
    P.-C. Chen, P. Hansen, B. Jaumard and H. Tuy: 1992, ‘Weber’s Problem with Attraction and Repulsion’, Journal of Regional Science, 32, 467-486.Google Scholar
  22. 22.
    P.-C. Chen, P. Hansen, B. Jaumard and H. Tuy: 1992, ‘Solution of the Multifacility Weber and Conditional Weber Problems by D.C. Programming’, Cahier du GERAD G-92-35, Ecole Polytechnique, Montréal.Google Scholar
  23. 23.
    J.P. Crouzeix: 1981, ‘ Duality framework in quasiconvex programming’, in Generalized Convexity in Optimization and Economics, eds. S. Schaible andW.T. Ziemba, Academic Press, 207-226.Google Scholar
  24. 24.
    L.C.W. Dixon and G.P. Szegö (eds.): 1975, Towards Global Optimization, North-Holland, AmsterdamGoogle Scholar
  25. 25.
    L.C.W. Dixon and G.P. Szegö (eds.): 1978, Towards Global Optimization 2, North-Holland, Amsterdam.zbMATHGoogle Scholar
  26. 26.
    P.C. Duong: 1987, ‘Finding the global extremum of a polynomial function’, in Essays in Nonlinear Analysis and Optimization Problems, Institute of Mathematics, Hanoi, 111-120.Google Scholar
  27. 27.
    B.C. Eaves and W. I. Zangwill: 1971, ‘ Generalized cutting plane algorithms’, SIAM Journal on Control, 9; 529-542.MathSciNetGoogle Scholar
  28. 28.
    R. Ellaia: 1984, Contribution it l’analyse et l’optimisation de différences de fonctions convexes, Thèse de trosième cycle, Université de Toulouse.Google Scholar
  29. 29.
    Y.G. Evtushenko: 1971, ‘Numerical methods for finding the global extremum of a function’, USSR Computational Mathematics and Mathematical Physics, 11, 38-54.Google Scholar
  30. 30.
    J.E. Falk: 1973, ‘A Linear Max-Min Problem’, Mathematical Programming, 5, 169-188.MathSciNetzbMATHGoogle Scholar
  31. 31.
    J.E. Falk: 1973, ‘Conditions for Global Optimality in Nonlinear Programming’, Operations Research,21, 337-340.MathSciNetzbMATHGoogle Scholar
  32. 32.
    J.E. Falk: 1974, ‘Sharper Bounds on Nonconvex Programs’, Operations Research, 21, 337-340.MathSciNetGoogle Scholar
  33. 33.
    J.E. Falk and S.W. Palocsay: 1991, ‘Optimizing the Sum of Linear Fractional Functions’, in Recent Advances in Global Optimization, eds. C. Floudas and P. Pardalos, Princeton University Press, 221-258.Google Scholar
  34. 34.
    J. E. Falk and R.M. Soland: 1969, ‘An algorithm for separable nonconvex programming problems’, Management Science, 15, 550-569.MathSciNetzbMATHGoogle Scholar
  35. 35.
    C.A. Floudas and P.M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Problems, Lecture Notes in Computer Science 455 (eds. G. Goos and J. Hartmanis), Springer-Verlag, Berlin.Google Scholar
  36. 36.
    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.Google Scholar
  37. 37.
    F. Forgo: 1975, ‘The solution of a special quadratic problem’ (in Hungarain), Szigma, 53-59.Google Scholar
  38. 38.
    F. Forgo: 1988, Nonconvex Programming, Akadémiai Kiadó, Budapest.Google Scholar
  39. 39.
    J. Fülöp: 1990, ‘A Finite Cutting Plane Method for Solving Linear Programs with an Additional Reverse Convex Constraint’, European Journal of Operations Research, 44, 395-409.zbMATHGoogle Scholar
  40. 40.
    J. Fülöp: 1992, ‘An outer approximation method for solving canonical d.c. problems’, in Operations Research 91, eds. P: Gritzmann, R. Tettich, R. Horst and E. Sachs, PhysicaVerlag, Heidelberg, 95-98.Google Scholar
  41. 41.
    R. Gabasov and F.M. Kirillova: 1980, Linear Programming Methods, Part 3 (Special Problems) (in Russian), Minsk.Google Scholar
  42. 42.
    R.P. Ge and Y.F. Qin: 1987, ‘A Class of Filled Functions for Finding Global Minimizers of a Function of Several Variables’, Journal of Optimization Theory and Applications, 54, 241-252.MathSciNetzbMATHGoogle Scholar
  43. 43.
    A. Grock, L. Vidigal and S. Director: 1985, ‘A new global optimization method for electronic circuit design’, IEEE Transactions on Circuits and Systems, 32, 160-179.Google Scholar
  44. 44.
    G.M. Guisewite and P.M. Pardalos: 1992, ‘A Polynomial Time Solvable Concave Network Flow Problem’, Network, 23, 143-148.MathSciNetGoogle Scholar
  45. 45.
    T.R. Gurlitz: 1985, ‘Algorithms for Reverse Convex Programs’, Ph. D. Dissertation, UCLA.Google Scholar
  46. 46.
    T.R. Gurlitz and S.E. Jacobsen: 1991, ‘On the use of cuts in reverse convex programs’, Journal of Optimization Theory and Applications,68, 257-273.MathSciNetzbMATHGoogle Scholar
  47. 47.
    H. Günzel, R. Hirabayashi, H. Th. Jongen, and S, Shindoh: 1992, ‘On the topological complexity of dc-sets’, Preprint 39, Departement of Mathematics, Rheinisch-Westfälische Technische Hocschule, Aachen.Google Scholar
  48. 48.
    M. Hamami : 1982, ‘Finitely Convergent Tuy-Type Algorithms for Concave Minimization’, Ph.D. Dissertation, UCLA.Google Scholar
  49. 49.
    P. Hartman: 1959, ‘On Functions Representable as a Difference of Convex Functions’, Pacific Journal of Mathematics, 9, 707-713.MathSciNetzbMATHGoogle Scholar
  50. 50.
    R.J. Hillestad: 1975, ‘Optimization Problems Subject to a Budget Constraint with Economies of Scale’, Operations Research, 23, 1091-1098.MathSciNetzbMATHGoogle Scholar
  51. 51.
    R.J. Hillestad and S.E. Jacobsen: 1980, ‘Reverse Convex Programming’, Applied Mathemattics and Optimization, 6, 63-78MathSciNetzbMATHGoogle Scholar
  52. 52.
    R.J. Hillestad and S.E. Jacobsen: 1980, ‘Linear Programs with an Additional Reverse Convex Constraint’, Applied Mathematics and Optimization, 6, 257-269.MathSciNetzbMATHGoogle Scholar
  53. 53.
    J.-B. Hiriart-Urruty: 1985 ‘Generalized Differentiability, Duality and Optimization for Problems Dealing with Differences of Convex Functions’, Lecture Notes in Economics and Mathematical Systems, 256, 37-69, Springer-Verlag, Berlin.Google Scholar
  54. 54.
    J.-B. Hiriart-Urruty: 1989, ‘From convex optimization to nonconvex optimization, Part 1: Necessary and sufficient conditions for global optimality’, in Nonsmooth optimization and related topics, eds. F.H. Clarke, V.F. Demyanov, F. Giannessi, Plenum, New York.Google Scholar
  55. 55.
    J.-B, Hiriart-Urruty: 1989, ‘Conditions nécessaires et suffisantes d’optimalité globale en optimisation de différences de deux fonctions convexes’, C.R. Acad. Sci. Paris, 309, Série I, 459-462.Google Scholar
  56. 56.
    K. Holmberg and H. Tuy: 1993, ‘A production -transportation problem with stochastic demands and concave production cost’, Preprint„ Department of Mathematics, Linköping University.Google Scholar
  57. 57.
    R. Horst: 1988, ‘Deterministic Global Optimization with Partition Sets whose Feasibility is not Known. Application to Concave Minimization, Reverse Convex Constraints, D.C. Programming and Lipschitz Optimization’, Journal of Optimization Theory and Application, 58, 11-37.Google Scholar
  58. 58.
    R. Horst: 1990, ‘Deterministic Global Optimization: Some Recent Advances and New Fields of Application’, Naval Research Logistics, 37, 433-471.MathSciNetzbMATHGoogle Scholar
  59. 59.
    R. Horst, T.Q. Phong and N.V. Thoai: 1990, ‘On Solving General Reverse Convex Programming Problems by a Sequence of Linear Programs and Line Searches’, Annals of Operations Research, 25, 1-18.MathSciNetzbMATHGoogle Scholar
  60. 60.
    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’, Journal of Global Optimization, 1, 183-203.Google Scholar
  61. 61.
    R. Horst and N.V. Thoai: 1989, ‘Modification, Implementation and Comparison of Three Algorithms for Globally Solving Linearly Constrained Concave Minimization Problems’, Computing, 42, 271-289.MathSciNetGoogle Scholar
  62. 62.
    R. Horst, N.V. Thoai and H.P. Benson: 1991, ‘Concave minimization via conical partitions and polyhedral outer approximation’, Mathematical Programming, 50, 259-274.MathSciNetzbMATHGoogle Scholar
  63. 63.
    R. Horst, N.V. Thoai and J. de Vries: 1988, ‘On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization’, Operations Research Letters, 7, 85-90.MathSciNetzbMATHGoogle Scholar
  64. 64.
    R. Horst, N.V. Thoai and J. de Vries: 1992, ‘On Geometry and Convergence of a Class of Simplicial Covers’, Optimization, 25, 53-64.MathSciNetzbMATHGoogle Scholar
  65. 65.
    R. Horst and H. Tuy: 1993, Global Optimization (Deterministic Approaches), second edition, Springer-Verlag, Berlin New York.Google Scholar
  66. 66.
    B. Klinz and H. Tuy: 1993, ‘Minimum Concave-Cost Network Flow Problems with a Single Nonlinear Arc Cost’, in Network Optimization Problems, eds. P. Pardalos and Dingzhu Du, World Scientific, 125-143.Google Scholar
  67. 67.
    H. Konno: 1988, ‘Minimum concave cost production sytem: a further generalization of multi-echelon model’, Mathematical Programming, 41, 185-193.MathSciNetzbMATHGoogle Scholar
  68. 68.
    H. Konno and T. Kuno: 1989, ‘Linear Multiplicative Programming ‘, Preprint IHSS 89-13, Tokyo Institute of Technology, to appear in Mathematical Programming, Series A.Google Scholar
  69. 69.
    H. Konno and T. Kuno: 1990, ‘Generalized Linear Multiplicative and Fractional Programming’, Annals of Operations Research,25, 147-162.MathSciNetzbMATHGoogle Scholar
  70. 70.
    H. Konno and Y. Yajima: 1991, ‘Minimizing and Maximizing the Product of Linear Fractional Functions’, in Recent Advances in Global Optimization, eds. C. Floudas and P. Pardalos, Princeton University Press.Google Scholar
  71. 71.
    H. Konno, Y. Yajima and T. Matsui: 1991, ‘Parametric Simplex Algorithms for Solving a Special Class of Nonconvex Minimization Problems. Journal of Global Optimization, 1, 65-82.MathSciNetGoogle Scholar
  72. 72.
    T. Kuno and H. Konno: 1991, ‘A Parametric Successive Underestimation Method for Convex Multiplicative Programming Problems’, Journal of Global Optimization, 1, 267-286 .MathSciNetzbMATHGoogle Scholar
  73. 73.
    T. Kuno, H. Konno and Y. Yamamoto: 1991, ‘A Parametric Successive Underestimation Method for Convex Programming Problems with an Additional Convex Multiplicative Constraint’, Preprint IHSS 90-23, Tokyo Institute of Technology, to appear in Journal of the Operations Research Society of Japan.Google Scholar
  74. 74.
    T. Kuno, Y. Yajima and H. Konno: 1991, ‘An Outer Approximation Method for Minimizing the Product of p Convex Functions on a Convex Set’, Preprint IHSS 91-33, Tokyo Institute of Technology.Google Scholar
  75. 75.
    E.M. Landis: 1951, On functions representable as the difference of two convex functions’, Dokl. Akad. Nauk SSSR, 80, 9-11.MathSciNetzbMATHGoogle Scholar
  76. 76.
    A. L. Levy and A. Montalvo: 1988, ‘The Tunneling Algorithm for the Global Minimization of Functions’, SIAM Journal Sci. Stat. Comp., 6, 15-29MathSciNetGoogle Scholar
  77. 77.
    J.E. Martinez-Legaz: 1988, ‘Quasiconvex duality theory by generalized conjugation methods’, Optimization, 19, 603-652.MathSciNetzbMATHGoogle Scholar
  78. 78.
    G.P. McCormik: 1972, ‘Attempts to Calculate Global Solutions of Problems that may have Local Minima’, in Numerical Methods for Nonlinear Optimization. ed. F. Lootsma, Academic Press, London New York, 209-221.Google Scholar
  79. 79.
    G.P. McCormik: 1976, ‘Computability of global solutions to factorable nonconvex programs: Part I- Convex Underestimating Problems’, Mathematical Programming, 10, 147-175.MathSciNetGoogle Scholar
  80. 80.
    G.P. McCormik: 1980, ‘Locating an Isolated Minimizer of a Constrained Nonconvex Program’, Mathematics of Operations Research, 5, 435-443.MathSciNetGoogle Scholar
  81. 81.
    G.P. McCormik: 1988, ‘Nonlinear Programming: Theory, Algorithms and Applications,’ John Wiley and Sons, New York.Google Scholar
  82. 82.
    D. Melzer: 1986, ‘On the expressibility of piecewise-linear continuous functionsss as the difference of two piecewise-linear convex functions,’Mathematical Programming Study 29, 118-134.MathSciNetGoogle Scholar
  83. 83.
    R. Meyer: 1970, ‘The Validity of a Family of Optimization Methods’, SIAM Journal on Control, 8, 41-54.zbMATHGoogle Scholar
  84. 84.
    K.G. Murty and S.N. Kabadi: 1987, ‘Some NP-complete problems in quadratic and nonlinear programming’, Mathematical Programming, 39, 117-130.MathSciNetzbMATHGoogle Scholar
  85. 85.
    L.D. Muu: 1985, ‘A Convergent Algorithm for Solving Linear Programs with an Additional Reverse Convex Constraint’, Kybernetika, 21, 428-435.MathSciNetzbMATHGoogle Scholar
  86. 86.
    L.D. Muu: 1990, ‘A Method for Solving Convex Programs with an Additional Convex-Concave Constraint’, Preprint, Institute of Mathematics, Hanoi.Google Scholar
  87. 87.
    L.D. Muu: 1991, ‘Convex-Concave Programming’, Preprint, Institute of Mathematics, Hanoi.Google Scholar
  88. 88.
    L.D. Muu and W. Oettli: 1991, ‘Method for minimizing a convex concave function over a convex set’, Journal of Optimization Theory and Applications, 70, 377-384.MathSciNetzbMATHGoogle Scholar
  89. 89.
    L-D. Muu: 1993, ‘Methods for Optimizing a Linear Function over the Efficient Set’, Preprint, Institute of Mathematics, Hanoi.Google Scholar
  90. 90.
    L.D. Muu and B.T. Tam: 1992, ‘Minimizing the Sum of a Convex Function and the Product of Two Affine Functions over a Convex Set’, Optimization, 24, 57-62.MathSciNetzbMATHGoogle Scholar
  91. 91.
    L.D. Muu and B.T. Tam: 1992, ‘Efficient Methods for Solving certain Bilinear Programming Problems’, Preprint, Institute of Mathematics, Hanoi, Vietnam.Google Scholar
  92. 92.
    N.D. Nghia and N.D. Hieu: 1986, ‘A Method for Solving Reverse Convex Programming Problems’, Acta Mathematica Vietnamica, 11, 241-252.MathSciNetzbMATHGoogle Scholar
  93. 93.
    V.H. Nguyen and J.J. Strodiot: 1992, ‘Computing a global optimal solution to a design centering problem’, Mathematical Programming, 53, 111-123.MathSciNetzbMATHGoogle Scholar
  94. 94.
    P.M. Pardalos: 1988, ‘Enumerative Techniques for Solving some Nonconvex Global Optimization Problems’, Operations Research Spektrum, 10, 29-35MathSciNetzbMATHGoogle Scholar
  95. 95.
    P.M. Pardalos and J.B. Rosen: 1987, ‘Constrained Global Optimization: Algorithms and Applications’, Lecture Notes in Computer Science, 268, Springer-Verlag, Berlin.Google Scholar
  96. 96.
    P.M. Pardalos and A.T. Phillips: 1991, ‘Global Optimization of Fractional Programs’, Journal of Global Optimization, 1, 173-182.MathSciNetzbMATHGoogle Scholar
  97. 97.
    P.M. Pardalos and S.A. Vavasis: 1991, ‘Quadratic Programming with One Negative Eigen-value is NP-hard’, Journal of Global Optimization, 21, 843-855.Google Scholar
  98. 98.
    J.-P. Penot and M. L. Bougeard: 1988,’Approximation and decomposition properties of some classes of locally d.c. functions’, Mathematical Programming, 41, 195-227.Google Scholar
  99. 99.
    J.P. Penot and M. Volle: 1990, ‘On quasiconvex duality’, Mathematics of Operations Research, 14, 597-625.MathSciNetGoogle Scholar
  100. 100.
    U. Pferschy and H. Tuy: 1993, ‘Linear Programs with an Additional Rank Two Reverse Convex Constraint’, Journal of Global Optimization, forthcoming.Google Scholar
  101. 101.
    D.T. Pham and S. El Bernoussi: 1988, ‘Duality in D.C. (Difference of Convex Functions)Optimization. Subgradient Methods’, International Series of Numerical Mathematics, 84.Google Scholar
  102. 102.
    D.T. Pham and S. El Bernoussi: 1989, ‘Numerical Method for Solving a Class of Global Nonconvex Optimization Problems, International Series of Numerical Mathematics, 87, 97132Google Scholar
  103. 103.
    J. Pinter, J. Szabo and L. Somlyody: 1986, ‘Multiextremal Optimization for Calibrating Water Resources Models’, Environmental Sofware, 1, 98-105.Google Scholar
  104. 104.
    J. Pinter: 1990, ‘Solving Nonlinear Equations Systems via Global Partition and Search’, Computing, 43, 309-323.MathSciNetzbMATHGoogle Scholar
  105. 105.
    H. Ratschek and J. Rokne: 1988, New Computer Methods for Global Optimization, Horwood, Chichester.Google Scholar
  106. 106.
    H. Ratschek and R.L. Voller: 1991, ‘ What Can Interval Analysis Do for Global Optimization ?’ Journal of Global Optimization, 1, 11-130.Google Scholar
  107. 107.
    R.T. Rockafellar: 1970, Convex Analysis, Princeton University Press, Princeton.Google Scholar
  108. 108.
    J.B. Rosen: 1966, ‘Iterative Solution of Nonlinear Optimal Control Problems’, SIAM Journal on Control, 4, 223-244.zbMATHGoogle Scholar
  109. 109.
    S. Sen and H.D. Sherali: 1987, ‘Nondifferentiable reverse convex programs and facial convexity cuts via a disjunctive characterization’, Mathematical Programming, 37, 169-183.MathSciNetzbMATHGoogle Scholar
  110. 110.
    A. Shapiro: 1983, ‘On functions representable as a difference of two convex functions in inequality constrained optimization’, Research Report, University of South Africa.Google Scholar
  111. 111.
    A. Shapiro and Y. Yomdin: 1981, ‘On functions repesentable as a difference of two convex functions and necessary conditions in constrained optimization’, Preprint, Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheve, Israel.Google Scholar
  112. 112.
    H.D. Sherali and A. Alameddine: 1991, ‘A New Reformulation-Linearization Technique for Bilinear Programming Problems’, Journal of Global Optimization, 2, 379-411.MathSciNetGoogle Scholar
  113. 113.
    H.D. Sherali and C.H. Tuncbilek: 1991, ‘A Global Optimization Algorithm for Polynomial Programming Problems Using a Reformulation- Linearization Tchnique’, Journal of Global Optimization, 2, 101-112.MathSciNetGoogle Scholar
  114. 114.
    N.Z. Shor and S.I. Stetsenko: 1989, Quadratic extremal problems and nondiffereniiable optimization (in Russian), Kiev.Google Scholar
  115. 115.
    N.Z. Shor: 1990, ‘Dual quadratic estimates in polynomial and boolean programming’, Annals of Operations Research, 25, 163-168.MathSciNetzbMATHGoogle Scholar
  116. 116.
    N.Z. Shor: 1991, ‘Dual Estimates in Multiextremal Problems’, Journal of Global Optimization, 2, 411-418.MathSciNetGoogle Scholar
  117. 117.
    I. Singer: 1980, ‘Minimization of Continuous Convex Functionals on Complements of Convex Sets’, Mathematische Operationsforschung and Statistik, Series Optimization, 11, 221-234.MathSciNetzbMATHGoogle Scholar
  118. 118.
    I. Singer: 1986, ‘A General Theory of Dual Optimization Problems’, Journal of Mathematical Analysis and Applications, 116, 77-130.MathSciNetzbMATHGoogle Scholar
  119. 119.
    I. Singer: 1987, ‘Generalization of Convex Supremization Duality’, in Nonlinear and Convex Analysis, eds. B.L. Lin and S. Simons, Lecture Notes in Pure and Applied Mathematics 107, M. Desher, NewYork.Google Scholar
  120. 120.
    R.M. Soland: 1971, ‘An Algorithm for Separable Nonconvex Programming Problems II: Nonconvex Constraints’, Management Science, 17, 759-773.MathSciNetzbMATHGoogle Scholar
  121. 121.
    A. C. Strekalovski: 1987, ‘On the global extremum problem’ (in Russian), Soviet Doklady, 292, 1062-1066.Google Scholar
  122. 122.
    A. C. Strekalovski: 1990, ‘On problems of global extremum in nonconvex extremal problems’ (in Russian), Izvestya Vuzov, ser. Matematika, No 8, 74-80.Google Scholar
  123. 123.
    A. C. Strekalovski: 1991, ‘On conditions for global extremum in a nonconvex minimization problem’, (in Russian), Izvestya Vuzov, ser. Matematika, No 2, 94-96.Google Scholar
  124. 124.
    R.G. Strongin: 1978, ‘Numerical Methods for Multiextremal Problems with Nonconvex Constraints’ (in Russian), Nauka, Moscow.Google Scholar
  125. 125.
    R.G. Strongin: 1984, ‘Numerical Methods for Multiextremal Nonlinear Programming Problems with Nonconvex Constraints’, in Nondifferentiable Optimization: Motivations and Applications, eds. V. F. Demyanov and D.Pallaschke, Proceedings IIASA Workshop, Sopron 1984, IIASA, Laxenburg.Google Scholar
  126. 126.
    R.G. Strongin: 1992, ‘Algorithms for Multi-Extremal Mathematical Programming Problems Employing the Set of Joint Space-Filling Curves’, Journal of Global Optimization, 2, 357-378.MathSciNetzbMATHGoogle Scholar
  127. 127.
    T. Tanaka, P.T. Thach and S. Suzuki: 1991, ‘Methods for an Optimal Ordering Policy for Jointly Replenished Products’, Journal of Operations Research Society of Japan, 34, 109.MathSciNetzbMATHGoogle Scholar
  128. 128.
    P.T. Thach: 1985, ‘Convex Programs with Several Additional Reverse Convex Constraints’, Acta Mathematica Vietnamica; 10, 35-37.MathSciNetzbMATHGoogle Scholar
  129. 129.
    P.T. Thach: 1988, ‘The design centering problem as a d.c. programming problem’, Mathematical Programming, 41, 229-248.Google Scholar
  130. 130.
    P.T. Thach: 1990, ‘A decomposition method for the minconcave cost flow problem with a staircase structure’, Japan Journal of Applied Mathematics, 7, 103-120.MathSciNetzbMATHGoogle Scholar
  131. 131.
    P.T. Thach: 1991, ‘Quasiconjugates of functions, duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint, and applications’, Journal of Mathematical Analysis and Applications, 159, 299-322.MathSciNetzbMATHGoogle Scholar
  132. 132.
    P.T. Thach: 1991, ‘A nonconvex duality with zero gap and applications, Preprint, Department of Mathematics, Trier University. To appear in SIAM Journal on Optimization.Google Scholar
  133. 133.
    P.T. Thach: 1991, ‘Global optimality criterions and a duality with a zero gap in nonconvex optimization problems, Preprint, Department of Mathematics, Trier University, to appear in SIAM Journal on Mathematical Analysis.Google Scholar
  134. 134.
    P.T. Thach: 1992, ‘New partitioning method for a class of nonconvex optimization problems’, Mathematics of Operations Research, 17, 43-69.MathSciNetzbMATHGoogle Scholar
  135. 135.
    P.T. Thach: 1993, ‘D.c. sets, d.c. functions and nonlinear equations’, Mathematical Programming, 58, 415-428.Google Scholar
  136. 136.
    P.T. Thach, R.E. Burkard and W. Oettli: 1991, ‘Mathematical programs with a two dimensional reverse convex constraint’, Journal of Global Optimization, 1, 145-154.MathSciNetGoogle Scholar
  137. 137.
    P.T. Thach and H. Konno: 1993, ‘On the degree and separability of nonconvexity and applications to optimization problems’, Preprint, IHSS 92-52, Tokyo Institute of Technology.Google Scholar
  138. 138.
    P.T. Thach, H. Konno and D. Yokota: 1993, ‘A dual approach to a minimization on the set of Pareto-optimal solutions’, Preprint, IHSS, Tokyo Institute of Technology.Google Scholar
  139. 139.
    P.T. Thach and H. Tuy: 1990, ‘The relief indicator method for constrained global optimization, Naval Research Logistics, 37, 473-497MathSciNetzbMATHGoogle Scholar
  140. 140.
    P.T. Thach and H. Tuy: 1990, ‘Dual outer approximation methods for concave programs and reverse convex programs’, Preprint, IHSS, Tokyo Institute of Technology (submitted)Google Scholar
  141. 141.
    N.V. Thoai: 1988, ‘A Modified Version of Tuy’s Method for Solving D.C. Programming Problems’, Optimization, 19, 665-674.Google Scholar
  142. 142.
    N.V. Thoai: 1991, ‘A Global Optimization Approach for Solving the Convex Multiplicative Programming Problem’, Journal of Global Optimization, 4, 341-358.Google Scholar
  143. 143.
    N.V. Thuong and H. Tuy: 1984, ‘A Finite Algorithm for Solving Linear Programs with and Additional Reverse Convex Constraint’, in Nondifferentiable Optimization, eds. V. Demyannov and H. Pallaschlke, Lecture Notes in Econonmics and Mathematical Systems, 225, Springer, 291-302.Google Scholar
  144. 144.
    J. F. Toland: 1978, ‘Duality in nonconvex optimization’, Journal of Mathematical Analysis and Applications, 66, 399-415.MathSciNetzbMATHGoogle Scholar
  145. 145.
    H. Tuy: 1964, ‘Concave programming under linear constraints’, Soviet Mathematics, 5, 1437-1440.Google Scholar
  146. 146.
    H. Tuy: 1983, ‘On outer approximation methods for solving concave minimization problems’, Acta Mathematica Vietnamica, 8,3-34.zbMATHGoogle Scholar
  147. 147.
    H. Tuy: 1986, ‘A general deterministic approach to global optimization via d.c. programming’, in J.B. Hiriart-Urruty ed., Fermat Days 1985: Mathematics for Optimization, North-Holland, Amsterdam, 137-162.Google Scholar
  148. 148.
    H. Tuy: 1987, ‘Global Minimization of a Difference of Two Convex Functions’, Mathematical Programming Study, 30, 150-182zbMATHGoogle Scholar
  149. 149.
    H. Tuy: 1987, ‘Convex Programs with an Additional Reverse Convex Constraint’, Journal of Optimization Theory and Applications, 52, 463-486.MathSciNetzbMATHGoogle Scholar
  150. 150.
    H. Tuy: 1990, ‘On polyhedral annexation method for concave minimization’, in Functional Analysis, Optimization and Mathematical Economics, eds. Lev J. Leifman and J.B. Rosen, Oxford University Press, 248-260.Google Scholar
  151. 151.
    H. Tuy: 1991, ‘Normal conical algorithm for concave minimization over polytopes’, Mathematical Programming, 51, 229-245.MathSciNetzbMATHGoogle Scholar
  152. 152.
    H. Tuy: 1991, ‘Computing fixed points by global optimization methods’, in Fixed Point Theory and Applications, eds. MA Thera and Baillon, Longman Scientific and Technical, 231-244.Google Scholar
  153. 153.
    Theory and Applications, eds. MA Thera and Baillon, Longman Scientific and Technical, 231-244.Google Scholar
  154. 154.
    H. Thy: 1991, ‘Effect of the Subdivision Strategy on Convergence and Efficiency of Some Global Optimization Algorithms’, Journal of Global Optimization , 1, 23-36.Google Scholar
  155. 155.
    H. Tuy: 1991, ‘Polyhedral Annexation, Dualization and Dimension Reduction Technique in Global Optimization’, Journal of Global Optimization, 1, 229-244.MathSciNetzbMATHGoogle Scholar
  156. 156.
    H. Thy: 1992, ‘The Complementary Convex Structure in Global Optimization’, Journal of Global Optimization„ 2, 21-40.Google Scholar
  157. 157.
    H. Thy: 1992, ‘On Nonconvex Optimization Problems with Separated Nonconvex Variables’, Journal of Global Optimization, 2, 133-144.Google Scholar
  158. 158.
    H. Thy: 1992, Introduction to Global Optimization, Ph.D. Course (manuscript), Department of Mathematics, Linköping University.Google Scholar
  159. 159.
    H. Tuy and F.A. Al-Khayyal: 1988, ‘Concave minimization by piecewise affine approximation’, Preprint, Institute of Mathematics, Hanoi.Google Scholar
  160. 160.
    H. Thy and F.A. Al-Khayyal: 1992, ‘Global Optimization of a Nonconvex Single Facility Location Problem by Sequential Unconstrained Convex Minimization’, Journal of Global Optimization, 2, 61-71.Google Scholar
  161. 161.
    H. Thy, N.D. Dan and S. Ghannadan: 1992, ‘Strongly Polynomial Time Algorithm for Certain Concave Minimization Problems on Networks’, Operations Research Letters, 14, 99-109.Google Scholar
  162. 162.
    H. Thy, S. Ghannadan, A. Migdalas and P. Värbrand: 1993, ‘Strongly Polynomial Algorithm for a Production-Transportation Problem with Concave Production Cost’, Optimization, 27, 205-207.MathSciNetGoogle Scholar
  163. 163.
    H. Thy, S. Ghannadan, A. Migdalas and P. Värbrand: 1992, ‘Strongly Polynomial Algorithm for Two Special Minimum Concave Cost Network Flow Problems’, Preprint, Department of Mathematics, Linköping University.Google Scholar
  164. 164.
    H. Tuy, S. Ghannadan, A. Migdalas and P. Värbrand: 1992b, ‘The Minimum Concave Cost Flow Problem with Fixed Numbers of Nonlinear Arc Costs and Sources’, Preprint, Department of Mathematics, Linköping University.Google Scholar
  165. 165.
    H. Thy and R. Horst: 1988, ‘Convergence and restart in branch and bound algorithms for global optimization. Application to concave minimization and d.c. optimization problems’, Mathematical Programming, 42, 161-184.Google Scholar
  166. 166.
    H. Thy and R. Horst: 1991, ‘The Geometric Complementarity Problem and Transcending Stationarity Problem in Global OPtimization’, DIMACS Series in Discrete Mathematics and Computer Science, Vol. 4, Applied Geometry and Discrete Mathematics, The Victor Klee Festschrift, 341-353.Google Scholar
  167. 167.
    H. Thy, V. Khachaturov and S. Utkin: 1987,’A Class of Exhaustive Cone Splitting Procedures in Conical Algorithms for Concave Minimization’, Optimization, 18, 791-807.Google Scholar
  168. 168.
    H. Thy, A. Migdalas and P. Värbrand: 1993, ‘A Global Optimization Approach for the Linear Two-Level Program’, Journal of Global Optimization, 3,1-23.Google Scholar
  169. 169.
    H. Thy, A. Migdalas and P. Värbrand: 1993, ‘A Quasiconcave Minimization Method for Solving Linear Two Level Programs’, to appear in Journal of Global Optimization.Google Scholar
  170. 170.
    H. Tuy and W. Oettli: 1991, ‘On necessary and sufficient conditions for global optimality’, Preprint, Institute of Mathematics, Hanoi.Google Scholar
  171. 171.
    H. Tuy and N.V. Thuong: 1988, ‘On the Global Minimization of a Convex Function Under General Nonconvex Constraints’, Applied Mathematics and Optimization, 18, 119-142.MathSciNetzbMATHGoogle Scholar
  172. 172.
    H. Thy and B.T. Tam: 1992, ‘An efficient solution method for rank two quasiconcave minimization problems’ Optimization, 24, 43-56.Google Scholar
  173. 173.
    H. Thy, B.T. Tam and N.D. Dan: 1994, ‘Minimizing the sum of a convex function and a specially structured nonconvex function’, Optimization, 28, 237-248.MathSciNetGoogle Scholar
  174. 174.
    H. Thy and B.T. Tam: 1993, ‘Polyhedral annexation method for minimizing a rank k quasiconcave function over a polyhedron’, Preprint, Institute of Mathematics, Hanoi.Google Scholar
  175. 175.
    H. Thy and P.T. Thach, ‘The Relief Indicator Method as a New Approach to Constrained Global Optimization’, in System Modelling and Optimization, Proceedings 14th IFIP Conference, Leipzig, Lecture Notes in Control Information Sciences, 143, 219-233.Google Scholar
  176. 176.
    U. Ueing: 1972, ‘A Combinatorial Method to Compute a Global Solution of Certain Non-convex Optimization Problems’, in Numerical Methods for Nonlinear Optimization, ed. F.A. Lootsma, Academic Press, New York, 223-230.Google Scholar
  177. 177.
    S. Utkin, V. Khachaturov and H. Thy: 1988, ‘A new exhaustive procedure for concave minimization’ (in Russian), USSR Computational Mathematics and Mathematical Physics, 7, 992-999.Google Scholar
  178. 178.
    L. Vidigal 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
  179. 179.
    V: Visweswaran and C.A. Floudas: 1990, ‘A Global Optimization Algorithm for Certain Classes of Nonconvex NLPs–II. Application and Test Problems’, Computers and Chemical Engineering, 14, 1419.Google Scholar
  180. 180.
    V: Visweswaran and C.A. Floudas: 1992, ‘Unconstrained and Constrained Global Optimization of Polynomial Functions in One Variable’, Journal of Global Optimization, 2, 73-100.Google Scholar
  181. 181.
    M. Volle: 1988, ‘Concave duality: Applications to problems dealing with difference of functions’, Mathematical Programming, 41, 261-278.MathSciNetzbMATHGoogle Scholar
  182. 182.
    U.-P. Wen and S: T Hsu: 1991, ‘Linear Bilevel Programming Problems – A Review’, J. Opl Rea. Soc., 42, 125-133.zbMATHGoogle Scholar
  183. 183.
    D.R. Wingo: 1985, ‘Globally Minimizing Polynomials without Evaluating Derivatives’, International Journal of Computer Mathematics, 17, 287.zbMATHGoogle Scholar
  184. 184.
    Y. Yajima and H. Konno: 1992, ‘Efficient Algorithms for Solving Rank Two ans Rank Three Bilinear Programming Problems’, Journal of Global Optimization, 1, 155-171.Google Scholar
  185. 185.
    A.B. Zaleesky: 1990, ‘Nonconvexity of feasible domains and optimization of management decisions’, (in Russian), Ekonomika i Matematitcheskie Methody, 16, 1069-1081.Google Scholar
  186. 186.
    V.A. Zalgaller: 1963, ‘On the representation of functions of two variables as a difference of convex functions’ (in Russian), Vestnik Leningradskogo Universiteta 1, 44-45.MathSciNetGoogle Scholar
  187. 187.
    W.I. Zangwill: 1968, ‘Minimum Concave Cost Flows in Certain Networks’, Management Science, 14, 429-450.MathSciNetzbMATHGoogle Scholar
  188. 188.
    V.S. Zhirov: 1985, ‘Searching for a Global Extremum of a Polynomial on a Parallelepiped’, USSR Computational Mathematics and Mathematical Physics, 25, 163-180.MathSciNetGoogle Scholar
  189. 189.
    S.I. Zukhovitzki, R.A. Polyak and M.E. Primak: 1968, ‘On a Class of Concave Programming Problems’ (in Russian),Ekonomika i Matematitcheskie Methody, 4, 431-443.Google Scholar
  190. 190.
    P.B. Zwart:1974, ‘Global Maximization of a Convex Function with Linear Inequality Constraints’, Operations Research, 22, 602-609.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Hoang Tuy
    • 1
  1. 1.Institute of MathematicsBo Ho HanoiVietnam

Personalised recommendations