Abstract
Quadratic programming (QP) is a family of methods, techniques, and algorithms that can be used to minimize quadratic objective functions subject to linear constraints. QP shares many combinatorial features with linear programming (LP) and it is often used as the basis of constrained nonlinear programming. An important branch of QP is convex QP where the objective function is a convex quadratic function.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
F. Alizadeh, “Combinational optimization with interior point methods and semidefinite matrices,” Ph.D. dissertation, University of Minnesota, Oct. 1991.
Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming. Philadelphia, PA: SIAM, 1994.
L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Review, vol. 38, pp. 49–95, Mar. 1996.
C. Helmberg, F. Rendl, R. Vanderbei, and H. Wolkowicz, “An interior-point method for semidefinite programming,” SIAM J. Optim., vol. 6, pp. 342–361, 1996.
M. Kojima, S. Shindoh, and S. Hara, “Interior-point methods for the monotone linear complementarity problem in symmetric matrices,” SIAM J. Optim., vol. 7, pp. 86–125, Nov. 1997.
Y. Nesterov and M. Todd, “Primal-dual interior-point method for self-scaled cones,” SIAM J. Optim., vol. 8, pp. 324–364, 1998.
F. Alizadeh, J. A. Haeberly, and M. L. Overton, “Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results,” SIAM J. Optim., vol. 8, pp. 746–768, 1998.
R. Monteiro, “Polynomial convergence of primal-dual algorithm for semidefinite programming based on Monteiro and Zhang family of directions,” SIAM J. Optim., vol. 8, pp. 797–812, 1998.
G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed. Baltimore, MD: Johns Hopkins University Press, 2013.
R. Fletcher, Practical Methods of Optimization, 2nd ed. New York: Wiley, 1987.
P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization. New York: Academic Press, 1981.
D. Goldfarb and A. Idnani, “A numerically stable dual method for solving strictly convex quadratic programs,” Math. Programming, vol. 27, pp. 1–33, 1983.
C. L. Lawson and R. J. Hanson, Solving Least Squares Problems. Englewood Cliffs, NJ: Prentice-Hall, 1974.
R. D. C. Monteiro and I. Adler, “Interior path following primal-dual algorithms, Part II: Convex quadratic programming,” Math. Programming, vol. 44, pp. 45–66, 1989.
R. D. C. Monteiro and I. Adler, “Interior path following primal-dual algorithms, Part I: Linear programming,” Math. Programming, vol. 44, pp. 27–41, 1989.
Y. Ye, Interior Point Algorithms: Theory and Analysis. New York: Wiley, 1997.
S. J. Wright, Primal-Dual Interior-Point Methods. Philadelphia, PA: SIAM, 1997.
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994.
A. S. Lewis and M. L. Overton, “Eigenvalue optimization,” Acta Numerica, vol. 5, pp. 149–190, 1996.
M. Shida, S. Shindoh, and M. Kojima, “Existence and uniqueness of search directions in interior-point algorithms for the SDP and the monotone SDLCP,” SIAM J. Optim., vol. 8, pp. 387–398, 1998.
S. Mehrotra, “On the implementation of a primal-dual interior point method,” SIAM J. Optim., vol. 2, pp. 575–601, 1992.
M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone programming,” Linear Algebra and Its Applications, vol. 284, pp. 193–228, Nov. 1998.
R. D. C. Monteiro and T. Tsuchiya, “Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions,” Math. Programming, vol. 88, pp. 61–83, 2000.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2021 Springer Science+Business Media, LLC, part of Springer Nature
About this chapter
Cite this chapter
Antoniou, A., Lu, WS. (2021). Quadratic, Semidefinite, and Second-Order Cone Programming. In: Practical Optimization. Texts in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-0843-2_13
Download citation
DOI: https://doi.org/10.1007/978-1-0716-0843-2_13
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-0716-0841-8
Online ISBN: 978-1-0716-0843-2
eBook Packages: Computer ScienceComputer Science (R0)