Abstract
Kernel functions play an important role in the design and analysis of primal-dual interior-point algorithms. They are not only used for determining the search directions but also for measuring the distance between the given iterate and the μ-center for the algorithms. In this paper we present a unified kernel function approach to primal-dual interior-point algorithms for convex quadratic semidefinite optimization based on the Nesterov and Todd symmetrization scheme. The iteration bounds for large- and small-update methods obtained are analogous to the linear optimization case. Moreover, this unifies the analysis for linear, convex quadratic and semidefinite optimizations.
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Amini, K., Haseli, A.: A new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methods. ANZIAM J. 49(2), 259–270 (2007)
Alfakih, A.Y., Khandani, A., Wolkowicz, H.: Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12(1–3), 13–30 (1999)
Bai, Y.Q., Guo, J., Roos, C.: A new kernel function yielding the best known iteration bounds for primal-dual interior-point algorithms. Acta Math. Sin., Engl. Ser. 25(12), 2169–2178 (2009)
Bai, Y.Q., Lesaja, G., Roos, C., Wang, G.Q., El Ghami, M.: A class of large-update and small-update primal-dual interior-point algorithms for linear optimization. J. Optim. Theory Appl. 138(3), 341–359 (2008)
Bai, Y.Q., Roos, C.: A primal-dual interior-point method based on a new kernel function with linear growth rate. In: Proceedings of Industrial Optimization Symposium and Optimization Day. Australia (2002)
Bai, Y.Q., Roos, C.: A polynomial-time algorithm for linear optimization based a new simple kernel function. Optim. Methods Softw. 18(6), 631–646 (2003)
Bai, Y.Q., Roos, C., El Ghami, M.: A primal-dual interior-point method for linear optimization based a new proximity function. Optim. Methods Softw. 17(6), 985–1008 (2002)
Bai, Y.Q., Roos, C., El Ghami, M.: A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM J. Optim. 15(1), 101–128 (2004)
Bai, Y.Q., Wang, G.Q., Roos, C.: Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions. Nonlinear Anal. 70(10), 3584–3602 (2009)
El Ghami, M., Bai, Y.Q., Roos, C.: Kernel-function based algorithms for semidefinite optimization. RAIRO Oper. Res. 43, 189–199 (2009)
EL Ghami, M., Ivanov, I.D., Roos, C., Steihag, T.: A polynomial-time algorithm for LO based on generalized logarithmic barrier functions. Int. J. Appl. Math. 21(1), 99–115 (2008)
Horn, R.A., Charles, R.J.: Matrix Analysis. Cambridge University Press, UK (1986)
de Klerk, E.: Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications. Kluwer Academic Publishers, Dordrecht (2002)
Kojima, M., Shindoh, S., Hara, S.: Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM J. Optim. 7(1), 86–125 (1997)
Liu, Z.Y., Sun, W.Y.: An infeasible interior-point algorithm with full-Newton step for linear optimization. Numer. Algorithms 46(2), 173–188 (2007)
Nie, J.W., Yuan, Y.X.: A potential reduction algorithm for an extended SDP problem. Sci. China, Ser. A 43(1), 35–46 (2000)
Nie, J.W., Yuan, Y.X.: A predictor-corrector algorithm for QSDP combining Dikin-type and Newton centering steps. Ann. Oper. Res. 103, 115–133 (2001)
Nesterov, Y.E., Todd, M.J.: Self-scaled barries and interior-point methods for convex programming. Math. Oper. Res. 22(1), 1–42 (1997)
Nesterov, Y.E., Todd, M.J.: Primal-dual interior-point methods for self-scaled cones. SIAM J. Optim. 8(2), 324–364 (1998)
Peng, J., Roos, C., Terlaky, T.: New complexity analysis of the primal-dual Newton method for linear optimization. Ann. Oper. Res. 99, 23–39 (2000)
Peng, J., Roos, C., Terlaky, T.: Self-regular functions and new search directions for linear and semidefinite optimization. Math. Program. 93(1), 129–171 (2002)
Qi, H., Sun, D.: A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM J. Matrix Anal. Appl. 28(2), 360–385 (2006)
Roos, C., Terlaky, T., Vial, J.Ph.: Theory and Algorithms for Linear Optimization. An Interior-Point Approach. Wiley, Chichester (1997)
Toh, K.C.: An inexact primal-dual path following algorithm for convex quadratic SDP. Math. Program. 112(1), 221–254 (2008)
Toh, K.C., Tütüncü, R.H., Todd, M.J.: Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems. Pac. J. Optim. 3(1), 135–164 (2007)
Wang, G.Q., Bai, Y.Q., Roos, C.: Primal-dual interior-point algorithms for semidefinite optimization based on a simple kernel function. J. Math. Model. Algorithms 4(4), 409-433 (2005)
Wang, G.Q., Bai, Y.Q.: Primal-dual interior-point algorithms for convex quadratic semidefinite optimization. Nonlinear Anal. 71(7–8), 3389–3402 (2009)
Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming, Theory, Algorithms, and Applications. Kluwer Academic Publishers, Dordrecht (2000)
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Wang, G., Zhu, D. A unified kernel function approach to primal-dual interior-point algorithms for convex quadratic SDO. Numer Algor 57, 537–558 (2011). https://doi.org/10.1007/s11075-010-9444-3
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DOI: https://doi.org/10.1007/s11075-010-9444-3
Keywords
- Convex quadratic semidefinite optimization
- Interior-point algorithm
- Kernel function
- Large- and small-update methods
- Iteration bound