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Quadratic, Semidefinite, and Second-Order Cone Programming

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Practical Optimization

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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.

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References

  1. F. Alizadeh, “Combinational optimization with interior point methods and semidefinite matrices,” Ph.D. dissertation, University of Minnesota, Oct. 1991.

    Google Scholar 

  2. Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming.   Philadelphia, PA: SIAM, 1994.

    Google Scholar 

  3. L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Review, vol. 38, pp. 49–95, Mar. 1996.

    Google Scholar 

  4. 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.

    Google Scholar 

  5. 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.

    Google Scholar 

  6. Y. Nesterov and M. Todd, “Primal-dual interior-point method for self-scaled cones,” SIAM J. Optim., vol. 8, pp. 324–364, 1998.

    Google Scholar 

  7. 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.

    Google Scholar 

  8. 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.

    Google Scholar 

  9. G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed.   Baltimore, MD: Johns Hopkins University Press, 2013.

    Google Scholar 

  10. R. Fletcher, Practical Methods of Optimization, 2nd ed.   New York: Wiley, 1987.

    Google Scholar 

  11. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization.   New York: Academic Press, 1981.

    Google Scholar 

  12. D. Goldfarb and A. Idnani, “A numerically stable dual method for solving strictly convex quadratic programs,” Math. Programming, vol. 27, pp. 1–33, 1983.

    Google Scholar 

  13. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems.   Englewood Cliffs, NJ: Prentice-Hall, 1974.

    Google Scholar 

  14. 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.

    Google Scholar 

  15. 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.

    Google Scholar 

  16. Y. Ye, Interior Point Algorithms: Theory and Analysis.   New York: Wiley, 1997.

    Google Scholar 

  17. S. J. Wright, Primal-Dual Interior-Point Methods.   Philadelphia, PA: SIAM, 1997.

    Google Scholar 

  18. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory.   Philadelphia, PA: SIAM, 1994.

    Google Scholar 

  19. A. S. Lewis and M. L. Overton, “Eigenvalue optimization,” Acta Numerica, vol. 5, pp. 149–190, 1996.

    Google Scholar 

  20. 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.

    Google Scholar 

  21. S. Mehrotra, “On the implementation of a primal-dual interior point method,” SIAM J. Optim., vol. 2, pp. 575–601, 1992.

    Google Scholar 

  22. 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.

    Google Scholar 

  23. 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.

    Google Scholar 

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Correspondence to Andreas Antoniou .

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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

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  • DOI: https://doi.org/10.1007/978-1-0716-0843-2_13

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