Mathematical Programming

, Volume 31, Issue 2, pp 153–191 | Cite as

CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems

  • Arne Drud


The paper presents CONOPT, an optimization system for static and dynamic large-scale nonlinearly constrained optimization problems. The system is based on the GRG algorithm. All computations involving the Jacobian of the constraints use sparse-matrix algorithms from linear programming, modified to deal with the nonlinearity and to take maximum advantage of the periodic structure in dynamic models. The paper presents the main features of the system, espcially the inversion routines and their data structures, the dynamic setting of tolerances in Newton’s algorithm, and the user features in the overal packaging. The difficulties with implementing a practical GRG algorithm are described in detail. Computational experience with some medium to large models is presented, idicating the viability of CONOPT for certain real-life problems, particularly those involving almost as many constraints as variables.

Key words

Large-scale Systems Dynamic Models Optimization Nonlinear Programming Generalized Reduced Gradient Nonlinear Constraints Sparse Matrix Techniques Dynamic Tolerances 


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  1. [1]
    J. Abadie, “Optimization problems with coupled blocks”,Economic Cybernetics Studies and Research (1970b).Google Scholar
  2. [2]
    J. Abadie, “Application of the GRG algorithm to optimal control problems”, in: J. Abadie, ed.,Nonlinear and integer programming (North-Holland, Amseterdam, 1972) pp. 191–211.Google Scholar
  3. [3]
    J. Abadie and J. Carpentier, “Generalization of the Wolfe reduced gradient method to the cases of nonlinear constraints”, in R. Fletcher, ed.,Optimization (Academic Press, New York, 1969), pp. 37–47.Google Scholar
  4. [4]
    P.O. Beck and L.S. Lasdon, “Scaling nonlinear programs”,Operations Research Letters 1 (1981) 6–9.zbMATHCrossRefGoogle Scholar
  5. [5]
    J. Bisschop and A. Meeraus, “On the development of a general algebraic modeling system in a strategic planning environment”,Mathematical Programming Study 20 (1982) 1–29.Google Scholar
  6. [6]
    T. Cauchois, “The world coffee model”, M.Sc. Diss.,Massachusetts Institute of Technology (Cambridge, MA, 1980).Google Scholar
  7. [7]
    C.F. Coleman and J.J. More. “Estimation of sparse jacobian matrices and graph coloring problems”,SIAM Journal of Numerical Analysis (1983) 187–209.Google Scholar
  8. [8]
    A.R. Colville, “A comparative study of nonlinear programming codes”, in: H.W. Kuhn, ed.,Proceedings of the princeton Symposium on Mathematical Programming (Princeton University Press, 1970).Google Scholar
  9. [9]
    A.R. Curtis, M.J.D. Powell and J.K. Reid, “On the estimation of sparse jacobian matrices”,Journal of the Institute of Mathematics and its Applications 13 (1974) 117–119.zbMATHGoogle Scholar
  10. [10]
    R.S. Dembo and S. Sahi, “A globally convergent framework for linearly constrained nonlinear optimization”, Working Paper B69, Yale School of Organization and Management, Yale University (New Haven, CT, 1983).Google Scholar
  11. [11]
    A. Drud, “Optimization in large partly nonlinear systems”, in: J. Cea, ed.,Optimization techniques. Modeling and optimization in the service of Man, Part 2, Lecture notes in computer science, Vol. 41 (Springer-Verlag, Berlin, Heidelberg, New York, 1976) 312–329.Google Scholar
  12. [12]
    A. Drud and A. Meeraus, “CONOPT—A system for large-scale dynamic optimization—User’s guide”, Technical note 16, Development Research Center, World Bank (Washington, DC, 1980).Google Scholar
  13. [13]
    R. Fourer, “Solving staircase linear programs by the simplex method, Part UI: Inversion”,Mathematical Programming 23 (1983) 274–313.CrossRefMathSciNetGoogle Scholar
  14. [14]
    C.B. Garcia and W.I. Zangwill,Pathways to solutions, fixed points, and equilibria (Prentice-Hall, NJ, 1983).Google Scholar
  15. [15]
    A. Gelb, “Oil rent and development strategies: A model for Indonesia”, Development Research Department, World Bank (Washington, DC, 1983).Google Scholar
  16. [16]
    P.M.J. Harris, “Pivot selection methods of the devex LP code”,Mathematical Programming 5 (1973) 1–28.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    E. Hellerman and D. Rarick, “Reinversion with the preassigned pivot procedure”,Mathematical Programming 1 (1971) 195–216.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    E. Hellerman and D. Rarick, “The partitioned preassigned pivot procedure”, in: D.J. Rose and R.A. Willoughby, eds.,Sparse matrices and their applications (Plenum Press, New York, 1972) pp. 67–76.Google Scholar
  19. [19]
    J.E. Kalan, “Aspects of large-scale in-core linear programming”, in:Proceedings of ACM conference, Chicago, 1971, pp. 304–313.Google Scholar
  20. [20]
    L.S. Lasdon, A.D. Waren, A. Jain and M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming”,ACM Transactions on Mathematical Software 4 (1978) 34–50.zbMATHCrossRefGoogle Scholar
  21. [21]
    L.S. Lasdon and N.H. Kim, “SLP User’s Guide”, Department of General Business, School of Business Administration, University of Texas, (Austin, Texas, 1983).Google Scholar
  22. [22]
    J.B. Mantell and L.S. Lasdon, “A GRG algorithm for econometric control problems”,Annals of Economic and Social Measurement 6 (1978) 581–597.Google Scholar
  23. [23]
    B.A. Murtagh and M.A. Saunder, “Large-scale linearly constrained optimization”,Mathematical Programming 14 (1978) 41–72.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    B.A. Murtagh and M.A. Saunders, “MINOS/AUGMENTED user’s manual”, Report SOL 80-14 (1980), Department of Operations Research, Stanford University, Stanford, CA.Google Scholar
  25. [25]
    B.A. Murtagh and M.A. Saunders, “A projected larrangian algorithm and its implementation for sparse nonlinear constraints”,Mathematical Programming Study 16 (1982) 84–117.zbMATHMathSciNetGoogle Scholar
  26. [26]
    B.A. Murtagh and M.A. Sauders, MINOS 5.0 User’s Guide”, Report SOL 83-20 (1983). Department of Operations Research, Stanford University, Stanford, CA.Google Scholar
  27. [27]
    R.S. Pindyck, “Gains to producers from the cartelization of exhaustible resources”,Review of Economics and Statistics 60 (1978) 238–251.CrossRefGoogle Scholar
  28. [28]
    M.J.D. Powell, “A hybrid method for nonlinear equations”, and “AFortran subrotine for solving systems of nonlinear algebraic equations”, in: P. Rabinowitz, ed.,Numerical methods for nonlinear algebraic equations (Gordon and Breach, London, 1970).Google Scholar
  29. [29]
    K. Schittkowski,Nonlinear programming codes. Lecture Note in Economics and Mathematical Systems, vol. 183 (Springer-Verlag, Berlin, Heidelberg, New York, 1980).zbMATHGoogle Scholar
  30. [30]
    “APEX III Reference Manual Version 1.2”, CDC Manual 76070000.Google Scholar
  31. [31]
    “Mathematical Programming System-Extended (MPSX), and Generalized Upper Bounding (GUB)”, IBM manual SH20-0968-1.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1985

Authors and Affiliations

  • Arne Drud
    • 1
  1. 1.Development Research DepartmentThe World BankWashington, DCUSA

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