Methods for Linearly Constrained Problems

  • H. A. Eiselt
  • Carl-Louis Sandblom
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 282)


In this and the following two chapters, several algorithms for solving nonlinear constrained optimization problems will be described. First the most special of all constrained nonlinear programming problems is considered, namely the quadratic programming problem for which the objective function is convex and quadratic and the constraints are linear. In the second section of the chapter methods for the more general problem of optimizing a differentiable convex function subject to linear constraints are discussed. Although every convex quadratic programming problem could be solved also by these more general methods, it is generally preferable to employ quadratic programming methods when possible. As a general principle it is advisable to use more specialized techniques for more specialized problems. Consequently, for a given problem one should select a method (covered in the previous, this, or the next two chapters) from a box as high up in Table 6.1 as possible. The third section considers problems in which the objective function is quadratic, but concave, a difficult case.


  1. Aganagić M (1984) Newton’s method for linear complementarity problems. Mathematical Programming 28: 349-362MathSciNetCrossRefGoogle Scholar
  2. Balinski ML, Cottle RW (eds.) (1978) Complementarity and fixed point problems. Mathematical Programming Study 7, North Holland, AmsterdamzbMATHGoogle Scholar
  3. Bazaraa MS, Sherali HD, Shetty CM (2013) Nonlinear programming: theory and algorithms. (3rd ed.) Wiley, New YorkzbMATHGoogle Scholar
  4. Beale EML (1955) On minimizing a convex function subject to linear inequalities. Journal of the Royal Statistical Society 17B: 173-184MathSciNetzbMATHGoogle Scholar
  5. Beale EML (1959) On quadratic programming. Naval Research Logistics Quarterly 6: 227-243MathSciNetCrossRefGoogle Scholar
  6. Ben-Israel A, Greville T (2003) Generalized inverses. Springer-Verlag, Berlin-Heidelberg-New YorkzbMATHGoogle Scholar
  7. Bertsekas DP (2016) Nonlinear programming. (3rd ed.) Athena Scientific, Belmont, MAzbMATHGoogle Scholar
  8. Charnes A, Cooper WW (1961) Management models and industrial applications of linear programming. Wiley, New YorkGoogle Scholar
  9. Cottle RW, Dantzig GB (1968) Complementary pivot theory of mathematical programming. Linear Algebra and its Applications 1: 103-125MathSciNetCrossRefGoogle Scholar
  10. Cottle RW, Pang JS (1978) On solving linear complementarity problems as linear programs. Mathematical Programming Study 7: 88-107MathSciNetCrossRefGoogle Scholar
  11. Cottle RW, Pang JS, Stone RE (2009) The linear complementarity problem. Classics in Applied Mathematics, SIAM, PhiladelphiaCrossRefGoogle Scholar
  12. Cottle RW, Thapa MN (2017) Linear and nonlinear optimization. Springer-Verlag, Berlin-Heidelberg-New YorkCrossRefGoogle Scholar
  13. Eaves BC (1971) The linear complementarity problem. Management Science 17: 612-634MathSciNetCrossRefGoogle Scholar
  14. Eaves BC (1978) Computing stationary points. Mathematical Programming Study 7: 1-14MathSciNetCrossRefGoogle Scholar
  15. Eiselt HA, Sandblom C-L (2007) Linear programming and its applications. Springer-Verlag, Berlin-HeidelbergzbMATHGoogle Scholar
  16. Floudas CA (2000) Deterministic global optimization. Theory, methods, and applications. Springer Science + Business Media, DordrechtCrossRefGoogle Scholar
  17. Floudas CA, Visweswaran V (1995) Quadratic optimization. In: Horst R, Pardalos PM (eds.) Handbook of global optimization. Kluwer, Boston, MAzbMATHGoogle Scholar
  18. Frank M, Wolfe P (1956) An algorithm for quadratic programming. Naval Research Logistics Quarterly 3: 95-110MathSciNetCrossRefGoogle Scholar
  19. Holloway CA (1974) An extension of the Frank and Wolfe method of feasible directions. Mathematical Programming 6: 14-27MathSciNetCrossRefGoogle Scholar
  20. Horst R, Pardalos PM, Thoai NV (2000) Introduction to global optimization, vol. 2. Kluwer, Dordrecht, The NetherlandsCrossRefGoogle Scholar
  21. Karamardian S (1969a) The nonlinear complementarity problem with applications, part I. Journal of Optimization Theory and Applications 4: 87-98MathSciNetCrossRefGoogle Scholar
  22. Karamardian S (1969b) The nonlinear complementarity Problem with Applications, part II. Journal of Optimization Theory and Applications 4: 167-181MathSciNetCrossRefGoogle Scholar
  23. Karamardian S (1972) The complementarity problem. Mathematical Programming 2: 107-129MathSciNetCrossRefGoogle Scholar
  24. Künzi HP, Krelle W, Oettli W (1966) Nonlinear programming. Blaisdell Publ. Co., MAGoogle Scholar
  25. Lemke CE (1968) On complementary pivot theory. pp. 95-114 in Dantzig GB, Veinott AF (eds.) Mathematics of the Decision Sciences, Part 1. American Mathematical Society, Providence, RIGoogle Scholar
  26. Luenberger DL, Ye Y (2008) Linear and nonlinear programming. (3rd ed.) Springer-Verlag, Berlin-Heidelberg-New YorkCrossRefGoogle Scholar
  27. Mangasarian OL (1976) Linear complementarity problems solvable by a single linear program. Mathematical Programming 10: 263-270MathSciNetCrossRefGoogle Scholar
  28. Moore EH (1935) General analysis, part I. Memoranda of the American Philosophical Society, Vol. IGoogle Scholar
  29. Penrose R (1955) A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society 51: 406-413MathSciNetCrossRefGoogle Scholar
  30. Powell MJD (1981) An example of cycling in a feasible point algorithm. Mathematical Programming 20: 353-357MathSciNetCrossRefGoogle Scholar
  31. Rosen JB (1960) The gradient projection method for non-linear programming: part I, linear constraints. Journal of SIAM 8: 181-217CrossRefGoogle Scholar
  32. Rosen JB (1961) The gradient projection method for non-linear programming: part II. Journal of SIAM 9: 514-532zbMATHGoogle Scholar
  33. Talman D, Van der Heyden L (1981) Algorithms for the linear complementarity problem which allow an arbitrary starting point. Cowles Foundation Discussion Paper No. 600. Yale University, New Haven, CTGoogle Scholar
  34. Topkis DM, Veinott AF (1967) On the convergence of some feasible direction algorithms for nonlinear programming. SIAM Journal on Control 5/2: 268-279MathSciNetCrossRefGoogle Scholar
  35. Wolfe P (1959) The simplex method for quadratic programming. Econometrica 27: 382-398MathSciNetCrossRefGoogle Scholar
  36. Wolfe P (1970) Convergence theory in nonlinear programming. pp. 1-36 in Abadie J (ed.) Integer and nonlinear programming. North Holland, AmsterdamGoogle Scholar
  37. Zoutendijk G (1960) Methods of feasible directions. Elsevier, New YorkGoogle Scholar

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Authors and Affiliations

  • H. A. Eiselt
    • 1
  • Carl-Louis Sandblom
    • 2
  1. 1.Faculty of ManagementUniversity of New BrunswickFrederictonCanada
  2. 2.Department of Industrial EngineeringDalhousie UniversityHalifaxCanada

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