Quadratic Programming Models

Part of the International Series in Operations Research & Management Science book series (ISOR, volume 137)


Quadratic programming (QP) deals with a special class of mathematical programs in which a quadratic function of the decision variables is required to be optimized (i.e., either minimized or maximized) subject to linear equality and/or inequality constraints.

Let x = (x 1, , x n ) T denote the column vector of decision variables. In mathematical programming, it is standard practice to handle a problem requiring the maximization of a function f(x) subject to some constraints by minimizing − f(x) subject to the same constraints. Both problems have the same set of optimum solutions. Because of this, we restrict our discussion to minimization problems.

A quadratic function of decision variables x is a function of the form
$$Q(x) = \sum \limits _{i=1}^{n} \sum \limits _{j=i}^{n}{q}_{ ij}{x}_{i}{x}_{j}\ \ + \sum \limits _{j=1}^{n}{c}_{ j}{x}_{j}\ \ + {c}_{0}.$$


Quadratic Program Linear Complementarity Problem Convex Quadratic Program Nonconvex Quadratic Program General Nonlinear Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Abadie J, Carpentier J (1969) Generalization of the wolfe reduced gradient method to the case of nonlinear constraints. In: Fletcher R (ed) Optimization. Academic Press, NYGoogle Scholar
  2. Avi-Itzak (1994) High-accuracy correlation-based pattern recognition. Ph.D. thesis, EE, Stanford University, Stanford, CA; Dantzig, Thappa (1997) vol 1Google Scholar
  3. Brooke A, Kendrick D, Meeraus A (1988) GAMS: a user’s guide. Scientific Press, San FranciscoGoogle Scholar
  4. Conn AR, Gould NIM, Toint PL (2000) Trust-region methods. MPS-SIAM Series on OptimizationGoogle Scholar
  5. Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. Academic Press, NYGoogle Scholar
  6. Crum RL, Nye DL (1981) A network model of insurance company cash flow management. Math Program Stud 15:86–101CrossRefGoogle Scholar
  7. Dennis JB (1959) Mathematical programming and electrical networks. Wiley, NYGoogle Scholar
  8. Dennis JE Jr, Schnabel RB (1983) Numerical methods for unconstrained optimization and nonlinear equations. Prentice Hall, NJGoogle Scholar
  9. Eldersveld SK (1991) Large scale sequential quadratic programming, SOL91. Department of OR, Stanford University, CAGoogle Scholar
  10. Fang SC, Puthenpura S (1993) Linear optimization and extensions: theory and algorithms. Prentice Hall, NJGoogle Scholar
  11. Fletcher R (1987) Practical methods of optimization, 2nd edn. Wiley, NYGoogle Scholar
  12. Fourer R, Gay DM, Kernighan BW (1993) AMPL: a modeling language for mathematical programming. Scientific Press, San FranciscoGoogle Scholar
  13. Frank M, Wolfe P (1956) An algorithm for quadratic programming. Nav Res Logist Q 3:95–110CrossRefGoogle Scholar
  14. Glassey CR (1978) A quadratic network optimization model for equilibrium single commodity trade flows. Math Program 14:98–107CrossRefGoogle Scholar
  15. Han SP (1976) Superlinearly convergent variable metric algorithms for general nonlinear programming problems. Math Program 11:263–282CrossRefGoogle Scholar
  16. Hestenes MR, Stiefel E (1952) Method of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49:409–436CrossRefGoogle Scholar
  17. IBM (1990) OSL- Optimization subroutine library guide and reference. IBM Corp, NYGoogle Scholar
  18. Kojima M, Megiddo N, Noma T, Yoshise A (1991) A unified approach to interior point algorithms for linear complementarity problems, Lecture Notes in Computer Science 538. Springer, NYCrossRefGoogle Scholar
  19. Lemke CE (1965) Bimatrix equilibrium points and mathematical programming. Manag Sci 11:681–689CrossRefGoogle Scholar
  20. Markovitz HM (1959) Portfolio selection: efficient diversification of investments. Wiley, NYGoogle Scholar
  21. Mulvey JM (1987) Nonlinear network models in finance. Adv Math Program Finan Plann 1:253–271Google Scholar
  22. Murtagh BA, Saunders MA (1987) MINOS 5.4 user’s guide, SOL 83-20R. Department of OR, Stanford University, CAGoogle Scholar
  23. Murty KG (1972) On the number of solutions of the complementarity problem and spanning properties of complementary cones. Lin Algebra Appl 5:65–108CrossRefGoogle Scholar
  24. Murty KG (2008a) Forecasting for supply chain and portfolio management, Chap 3. In: Neogy SK, Bapat RB, Das AK, Parthasarathy T (eds) Mathematical programming and game theory for decision making, vol 1, pp 231–255. World Scientific, SingaporeCrossRefGoogle Scholar
  25. Murty KG (2008b) A new practically efficient IPM for convex quadratic programming, Chap 3. In: Neogy SK, Bapat RB, Das AK, Parthasarathy T (eds) Mathematical programming and game theory for decision making, vol 1, pp 21–31. World Scientific, SingaporeCrossRefGoogle Scholar
  26. Murty KG, Kabadi SN (1987) Some NP-complete problems in quadratic and nonlinear programming. Math Program 39:117–129CrossRefGoogle Scholar
  27. Powell MJD (1978) Algorithms for nonlinear constraints that use Lagrangian functions. Math Program 14:224–248CrossRefGoogle Scholar
  28. Theil H, van de Panne C (1961) Quadratic programming as an extension of conventional quadratic maximization. Manag Sci 7:1–20CrossRefGoogle Scholar
  29. Vavasis SA (1992) Local minima for indefinite quadratic knapsack problems. Math Program 54:127–153CrossRefGoogle Scholar
  30. White FC (1983) Trade-off in growth and stability in state taxes. Natl Tax J 36:103–114Google Scholar
  31. Wilson RB (1963) A simplicial algorithm for convex programming, Ph.D. dissertation, School of Business Administration, HarvardGoogle Scholar
  32. Wolfe P (1959) The simplex method for quadratic programming. Econometrica 27:382–398CrossRefGoogle Scholar
  33. Wood AJ (1984) Power generation, operation, and control. Wiley, NYGoogle Scholar
  34. Ye Y (1991) Interior point algorithms for quadratic programming. In: Kumar S (ed) Recent developments in mathematical programming, pp 237–261. Gordon and Breach, PAGoogle Scholar
  35. Zhou JL, Tits AL (1992) User’s guide to FSQP Version 3.1. SRC TR-92-107r2, Institute for Systems Research, University of Maryland, College ParkGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. Industrial and Operations EngineeringUniversity of MichiganAnn ArborUSA
  2. 2.Systems Engineering DepartmentKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

Personalised recommendations