Abstract
Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithms for solving symmetric optimization problems. In this paper we present a new kernel function for which interior point method yields iteration bounds \({\mathcal {O}}(\sqrt{r}\log r\log \frac{r}{\epsilon })\) and \({\mathcal {O}}(\sqrt{r}\log \frac{r}{\epsilon })\) for large-and small-update methods, respectively, which matches currently the best known bounds for such methods.
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Bai, Y.Q., El Ghami, M., Roos, C.: 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., 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 (2008)
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., 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., El Ghami, M., Roos, C.: A new efficient large-update primal-dual interior-point method based on a finite barrier. SIAM J. Optim. 13(3), 766–782 (2003)
El Ghami, M., Guennoun, Z.A., Bouali, S., Steihaug, T.: Interior-point methods for linear optimization based on a kernel function with trigonmetric barrier term. J. Comput. Appl. Math. 236(15), 3613–3623 (2012)
Faybusovich, L.: A Jordan-algebraic approach to potential-reduction algorithms. Math. Z. 239(1), 117–129 (2002)
Faybusovich, L.: Linear systems in Jordan algebras and primal-dual interior-point algorithms. J. Comput. Appl. Math. 86, 149–175 (1997)
Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1994)
Güler, O.: Barrier functions in interior-point methods. Math. Oper. Res. 21(4), 860–885 (1996)
Kheirfam, B.: Primal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier term. Numer. Algorithms 61(4), 659–680 (2012)
Korányi, A.: Monotone functions on formally real Jordan algebras. Math. Ann. 269(1), 73–76 (1984)
Lesaja, G., Roos, C.: Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones. J. Optim. Theory Appl. 150(3), 444–474 (2011)
Monteiro, R.D.C., Zhang, Y.: A unified analysis for a class of path-following primal-dual interior-point algorithms for semidefinite programming. Math. Program. 8, 281–299 (1998)
Nesterov, Y.E., Todd, M.J.: Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22(1), 1–42 (1997)
Nesterov, Y.E., Nemirovskii, A.S.: Interior Point Polynomial Algorithms in Convex Programming. in: SIAM Stud. Appl. Math., vol. 13, SIAM, Philadelphia (1994)
Peng, J., Roos, C., Terlaky, T.: Self-regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton (2002)
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)
Peng, J., Roos, C., Terlaky, T.: A new class of polynomial primal-dual methods for linear and semidefinite optimization. Eur. J. Oper. Res. 143, 234–256 (2002)
Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization. An Interior-Point Approach. Wiley, Chichester (1997)
Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior-point algorithms to symmetric cones. Math. Program. 96(3), 409–438 (2003)
Vieira, M.V.C.: Jordan algebraic approach to symmetric optimization. Ph.D thesis, Delft University of Technology (2007)
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Kheirfam, B. A primal-dual interior-point algorithm for symmetric optimization based on a new kernel function with trigonometric barrier term yielding the best known iteration bounds. Afr. Mat. 28, 389–406 (2017). https://doi.org/10.1007/s13370-016-0455-7
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DOI: https://doi.org/10.1007/s13370-016-0455-7