Abstract
An interior point algorithm is proposed for linearly constrained convex programming following a parameterized central path, which is a generalization of the central path and requires weaker convergence conditions. The convergence and polynomial-time complexity of the proposed algorithm are proved under the assumption that the Hessian of the objective function is locally Lipschitz continuous. In addition, an initialization strategy is proposed and some numerical results are provided to show the efficiency and attractiveness of the proposed algorithm.
Similar content being viewed by others
References
Alder, I., Monteiro, R.D.: Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Math. Program. 50(1), 29–51 (1991)
Andrei, N.: Predictor-corrector interior-point methods for linear constrained optimization. Stud. Inform. Control. 7(2), 155–177 (1998)
Andri, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10(1), 147–181 (2008)
Burke, J.V., Xu, S.: The global linear convergence of a noninterior path-following algorithm for linear complementarity problems. Math. Oper. Res. 23(3), 719–734 (1998)
Burke, J.V., Xu, S.: A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem. Math. Program. 87(1), 113–130 (2000)
Burke, J.V., Xu, S.: Complexity of a noninterior path-following method for the linear complementarity problem. J. Optim. Theory Appl. 112(1), 53–76 (2002)
Chen, B., Chen, X.: A global and local superlinear continuation-smoothing method for \({P}_0\) and \({R}_0\) NCP or monotone NCP. SIAM J. Optim. 9(3), 624–645 (1999)
Chen, B., Chen, X.: A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints. Comput Optimi Appl. 17(2–3), 131–158 (2000)
Chen, B., Harker, P.T.: A non-interior-point continuation method for linear complementarity problems. SIAM J. Matrix Anal. Appl. 14(4), 1168–1190 (1993)
Chen, B., Xiu, N.: A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions. SIAM J. Optim. 9(3), 605–623 (1999)
Chen, X., Tseng, P.: Non-interior continuation methods for solving semidefinite complementarity problems. Math. Program. 95(3), 431–474 (2003)
Chubanov, S.: A polynomial-time descent method for separable convex optimization problems with linear constraints. SIAM J. Optim. 26(1), 856–889 (2016)
Den Hertog, D., Roos, C., Terlaky, T.: On the classical logarithmic barrier function method for a class of smooth convex programming problems. J. Optim. Theory Appl. 73(1), 1–25 (1992)
Dikin, I.I.: Iterative solution of problems of linear and quadratic programming. Dokl. Akad. Nauk SSSR 174, 747–748 (1967)
Drummond, L.G., Svaiter, B.F.: On well definedness of the central path. J. Optim. Theory Appl. 102(2), 223–237 (1999)
Grossmann, C.: Asymptotic analysis of a path-following barrier method for linearly constrained convex problems. Optimization 45(1–4), 69–87 (1999)
Grossmann, C.: Penalty/Barrier path-following in linearly constrained optimization. Discuss. Math. Differ. Incl. Control Optim. 20(1), 7–26 (2000)
Kanzow, C.: Some noninterior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17(4), 851–868 (1996)
Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)
Kojima, M., Megiddo, N., Noma, T.: Homotopy continuation methods for nonlinear complementarity problems. Math. Oper. Res. 16(4), 754–774 (1991)
Kojima, M., Mizuno, S., Noma, T.: Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems. Math. Oper. Res. 15(4), 662–675 (1990)
Kojima, M., Mizuno, S., Yoshise, A.: A polynomial-time algorithm for a class of linear complementarity problems. Math. Program. 44(1–3), 1–26 (1989)
Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming. In: Progress in mathematical programming (Pacific Grove, CA, 1987), pp. 29–47. Springer, New York (1989)
Kojima, M., Mizuno, S., Yoshise, A.: An \( O (\sqrt{n} {L}) \) iteration potential reduction algorithm for linear complementarity problems. Math. Program. 50(1–3), 331–342 (1991)
Kortanek, K.O., Potra, F., Ye, Y.: On some efficient interior point methods for nonlinear convex programming. Linear Alg. Appl. 152, 169–189 (1991)
Kortanek, K.O., Zhu, J.: A polynomial barrier algorithm for linearly constrained convex programming problems. Math. Oper. Res. 18(1), 116–127 (1993)
Kortanek, K.O., Zhu, J.: On controlling the parameter in the logarithmic barrier term for convex programming problems. J. Optim. Theory Appl. 84(1), 117–143 (1995)
Liao, L.-Z.: A study of the dual affine scaling continuous trajectories for linear programming. J. Optim. Theory Appl. 163(2), 548–568 (2014)
Mclinden, L.: An analogue of Moreau’s proximation theorem, with application to the nonlinear complementarity problem. Pac. J. Math. 88(1), 101–161 (1980)
Megiddo, N.: Pathways to the optimal set in linear programming. In: Progress in mathematical programming (Pacific Grove, CA, 1987), pp. 131–158. Springer, New York (1989)
Mehrotra, S., Sun, J.: An algorithm for convex quadratic programming that requires \({O}(n^{3.5}{L})\) arithmetic operations. Math. Oper. Res. 15(2), 342–363 (1990)
Mehrotra, S., Sun, J.: An interior point algorithm for solving smooth convex programs based on Newton’s method. In: Mathematical developments arising from linear programming (Brunswick, ME, 1988), Contemp. Math., vol. 114, pp. 265–284. Amer. Math. Soc., Providence, RI (1990)
Mehrotra, S., Sun, J.: A method of analytic centers for quadratically constrained convex quadratic programs. SIAM J. Numer. Anal. 28(2), 529–544 (1991)
Mizuno, S.: Polynomiality of the Kojima-Megiddo-Mizuno infeasible interior point algorithm for linear programming. Tech. Rep., Operations Research and Industrial Engineering, Cornell University (1992)
Mizuno, S., Todd, M.J., Ye, Y.: On adaptive-step primal-dual interior-point algorithms for linear programming. Math. Oper. Res. 18(4), 964–981 (1993)
Monteiro, R.D.: Convergence and boundary behavior of the projective scaling trajectories for linear programming. Math. Oper. Res. 16(4), 842–858 (1991)
Monteiro, R.D.: On the continuous trajectories for a potential reduction algorithm for linear programming. Math. Oper. Res. 17(1), 225–253 (1992)
Monteiro, R.D.: A globally convergent primal-dual interior point algorithm for convex programming. Math. Program. 64(1–3), 123–147 (1994)
Monteiro, R.D., Adler, I.: Interior path following primal-dual algorithms. Part I: Linear programming. Math. Program. 44(1–3), 27–41 (1989)
Monteiro, R.D., Adler, I.: Interior path following primal-dual algorithms. Part II: Convex quadratic programming. Math. Program. 44(1–3), 43–66 (1989)
Monteiro, R.D., Adler, I.: An extension of Karmarkar type algorithm to a class of convex separable programming problems with global linear rate of convergence. Math. Oper. Res. 15(3), 408–422 (1990)
Monteiro, R.D., Tsuchiya, T., Wang, Y.: A simplified global convergence proof of the affine scaling algorithm. Ann. Oper. Res. 46(2), 443–482 (1993)
Monteiro, R.D., Zou, F.: On the existence and convergence of the central path for convex programming and some duality results. Comput. Optim. Appl. 10(1), 51–77 (1998)
Necoara, I., Suykens, J.: Interior-point Lagrangian decomposition method for separable convex optimization. J. Optim. Theory Appl. 143(3), 567–588 (2009)
Potra, F., Ye, Y.: A quadratically convergent polynomial algorithm for solving entropy optimization problems. SIAM J. Optim. 3(4), 843–860 (1993)
Qi, L., Sun, D.: Improving the convergence of non-interior point algorithms for nonlinear complementarity problems. Math. Comput. 69(229), 283–304 (2000)
Qian, X., Liao, L.-Z.: Generalized affine scaling trajectory analysis for linearly constrained convex programming. In: International Symposium on Neural Networks, pp. 139–147. Springer (2018)
Qian, X., Liao, L.-Z., Sun, J.: Analysis of some interior point continuous trajectories for convex programming. Optimization 66(4), 589–608 (2017)
Qian, X., Liao, L.-Z., Sun, J.: A strategy of global convergence for the affine scaling algorithm for convex semidefinite programming. Math. Program. 179(1), 1–19 (2020)
Qian, X., Liao, L.-Z., Sun, J., Zhu, H.: The convergent generalized central paths for linearly constrained convex programming. SIAM J. Optim. 28(2), 1183–1204 (2018)
Sheu, R.L.: A generalized interior-point barrier function approach for smooth convex programming with linear constraints. J. Inform. Optim. Sci. 20(2), 187–202 (1999)
Shi, Y.: A combination of potential reduction steps and steepest descent steps for solving convex programming problems. Numer. Linear Algebra Appl. 9(3), 195–203 (2002)
Shi, Y.: A projected-steepest-descent potential-reduction algorithm for convex programming problems. Numer. Linear Algebra Appl. 11(10), 883–893 (2004)
Sonnevend, G., Stoer, J., Zhao, G.: On the complexity of following the central path of linear programs by linear extrapolation II. Math. Program. 52(1–3), 527–553 (1991)
Sun, J.: A convergence analysis for a convex version of Dikin’s algorithm. Ann. Oper. Res. 62(1), 357–374 (1996)
Sun, J., Zhu, J., Zhao, G.: A predictor-corrector algorithm for a class of nonlinear saddle point problems. SIAM J. Control. Optim. 35(2), 532–551 (1997)
Todd, M.J., Ye, Y.: A centered projective algorithm for linear programming. Math. Oper. Res. 15(3), 508–529 (1990)
Tseng, P., Luo, Z.Q.: On the convergence of the affine-scaling algorithm. Math. Program. 56(1–3), 301–319 (1992)
Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)
Wolfe, P.: A duality theorem for non-linear programming. Quart. Appl. Math. 19(3), 239–244 (1961)
Xu, S., Burke, J.V.: A polynomial time interior-point path-following algorithm for LCP based on Chen-Harker-Kanzow smoothing techniques. Math. Program. 86(1), 91–103 (1999)
Ye, Y.: An \({O} (n^3{L})\) potential reduction algorithm for linear programming. Math. Program. 50(1–3), 239–258 (1991)
Ye, Y., Anstreicher, K.: On quadratic and \( {O}(\sqrt{n}{L}) \) convergence of a predictor-corrector algorithm for LCP. Math. Program. 62(1–3), 537–551 (1993)
Ye, Y., Güler, O., Tapia, R.A., Zhang, Y.: A quadratically convergent \({O}(\sqrt{n}{L})\)-iteration algorithm for linear programming. Math. Program. 59(1–3), 151–162 (1993)
Zhu, J.: A path following algorithm for a class of convex programming problems. Z. Oper. Res. 36(4), 359–377 (1992)
Acknowledgements
The authors would like to thank the Associate Editor and one anonymous referee for their constructive comments and suggestions on the earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work of L.-Z. Liao was supported in part by grants from General Research Fund (GRF) of Hong Kong. The work of J. Sun was partially supported by Australia Council Research under grant DP160102918.
Rights and permissions
About this article
Cite this article
Hou, L., Qian, X., Liao, LZ. et al. An Interior Point Parameterized Central Path Following Algorithm for Linearly Constrained Convex Programming. J Sci Comput 90, 95 (2022). https://doi.org/10.1007/s10915-022-01765-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-022-01765-3