Abstract
In this paper, a full Nesterov–Todd step infeasible interior-point method for solving semidefinite optimization problems based on a new kernel function is analyzed. In each iteration, the algorithm involves a feasibility step and several centrality steps. The centrality step is focused on Nesterov–Todd search directions, while we used a kernel function with trigonometric barrier term to induce the feasibility step. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.
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Bai, Y., El Ghami, M., Roos, C.: A comparative study of kernel functions for primal–dual interior-point algorithms in linear optimization. SIAM J. Optim. 1, 101–128 (2004)
Bhatia, R.: Matrix Analysis. Springer, Berlin (1997)
De Klerk, E.: Aspects of Semidefinite Programming: Interior Point Methods and Selected Applications. Kluwer Academic Publisher, Dordrecht (2002)
Kheirfam, B., Mahdavi-Amiri, N.: A full Nesterov–Todd step infeasible interior-point algorithm for symmetric cone linear complementarity problem. Bull. Iran. Math. Soc. 40(3), 541–564 (2014)
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)
Kheirfam, B.: A generic interior-point algorithm for monotone symmetric cone linear complementarity problems based on a new kernel function. J. Math. Model. Algorithms Oper. Res. 13(4), 471–491 (2014)
Kheirfam, B.: A full-Newton step infeasible interior-point algorithm for linear complementarity problems based on a kernel function. Algor. Oper. Res. 7(1), 103–110 (2013)
Kheirfam, B.: A full Nesterov–Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function. Numer. Algebra Control Optim. 3(4), 601–614 (2013)
Kheirfam, B., Haghighi, M.: A full-Newton step infeasible interior-point method for linear optimization based on a trigonometric kernel function. Optimization 65(4), 841–857 (2015)
Kheirfam, B., Moslemi, M.: A polynomial-time algorithm for linear optimization based on a new kernel function with trigonometric barrier term. Yugosl. J. Oper. Res. 25(2), 233–250 (2015)
Kojima, M., Shida, M., Shindoh, S.: Local convergence of predictor–corrector infeasible-interior-point method for SDPs and SDLCPs. Math. Program. 80(2), 129–160 (1998)
Liu, Z., Sun, W., Tian, F.: A full-Newton step infeasible interior-point algorithm for linear programming based on a kernel function. Appl. Math. Optim. 60, 237–251 (2009)
Mansouri, H., Roos, C.: Simplified \(O(nL)\) infeasible interior-point algorithm for linear optimization using full-Newton step. Optim. Methods Softw. 22(3), 519–530 (2007)
Mansouri, H., Roos, C.: A new full-Newton step \(O(n)\) infeasible interior point algorithm for semidefinite optimization. Numer. Algorithms 52(2), 225–255 (2007)
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.: 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.: New complexity analysis of the primal–dual method for semidefinite optimization based on the NT-direction. J. Optim. Theory Appl. 109(2), 327–343 (2001)
Potra, F.A., Sheng, R.: On homogeneous interior-point algorithms for semidefinite programming. Optim. Methods Softw. 9(1–3), 161–184 (1998)
Potra, F.A., Sheng, R.: A superlinearly convergent primal–dual infeasible-interior-point algorithm for semidefinite programming. SIAM J. Optim. 8(4), 1007–1028 (1998)
Roos, C., Terlaky, T., Vial, J.-P.: Theory and Algorithms for Linear Optimization: An Interior-Point Approach, 2nd edn. Springer, Berlin (2006)
Roos, C.: A full-Newton step \(O(n)\) infeasible interior-point algorithm for linear optimization. SIAM J. Optim. 16(4), 1110–1136 (2006)
Wang, G.Q., Bai, Y.Q.: A new full Nesterov–Todd step primal–dual path-following interior-point algorithm for symmetric optimization. J. Math. Anal. Appl. 154(3), 966–985 (2012)
Wang, G.Q., Bai, Y.Q.: A new primal–dual path-following interio-point algorithm for semidefinite optimization. J. Math. Anal. Appl. 353(1), 339–349 (2009)
Wang, G.Q., Zhu, D.T.: A unified kernel function approach to primal-dual interior-point algorithms for convex quadratic SDO. Numer. Algorithms 57(4), 537–558 (2011)
Wolkowicz, H., Saigal, R., Vadenberghe, L.: Handbook of Semidefinite Programming, Theory, Algorithm, and Applications. Kluwer Academic Publishers, Dordrecht (2000)
Zhang, Y.: On extending some primal–dual interior-point algorithms from linear programming to semidefinite programming. SIAM J. Optim. 8, 365–386 (1998)
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Moslemi, M., Kheirfam, B. Complexity analysis of infeasible interior-point method for semidefinite optimization based on a new trigonometric kernel function. Optim Lett 13, 127–145 (2019). https://doi.org/10.1007/s11590-018-1257-7
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DOI: https://doi.org/10.1007/s11590-018-1257-7