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Dynamic updates of the barrier parameter in primal-dual methods for nonlinear programming. (English) Zbl 1180.90305
Summary: We introduce a framework in which updating rules for the barrier parameter in primal-dual interior-point methods become dynamic. The original primal-dual system is augmented to incorporate explicitly an updating function. A Newton step for the augmented system gives a primal-dual Newton step and also a step in the barrier parameter. Based on local information and a line search, the decrease of the barrier parameter is automatically adjusted. We analyze local convergence properties, report numerical experiments on a standard collection of nonlinear problems and compare our results to a state-of-the-art interior-point implementation. In many instances, the adaptive algorithm reduces the number of iterations and of function evaluations. Its design guarantees a better fit between the magnitudes of the primal-dual residual and of the barrier parameter along the iterations.

##### MSC:
 90C30 Nonlinear programming 90C51 Interior-point methods
##### Software:
CUTEr; Ipopt; COPS; SifDec; KELLEY
Full Text:
##### References:
 [1] Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic, London (1982) · Zbl 0572.90067 [2] Byrd, R.H., Liu, G., Nocedal, J.: On the local behavior of an interior point method for nonlinear programming. In: Griffiths, D.F., Higham, D.J. (eds.) Numerical Analysis 1997, pp. 37–56. Addison Wesley Longman (1997). Proceedings of the Dundee 1997 Conference on Numerical Analysis · Zbl 0902.65021 [3] Dolan, E., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. Ser. B 91, 201–213 (2002) · Zbl 1049.90004 [4] Dolan, E.D., Moré, J.J., Munson, T.S.: Benchmarking optimization software with COPS 3.0. Technical Report ANL/MCS-273, Argonne National Laboratory (2004) [5] Dussault, J.-P.: Numerical stability and efficiency of penalty algorithms. SIAM J. Numer. Anal. 32(1), 296–317 (1995) · Zbl 0816.65039 [6] El-Bakry, A.S., Tapia, R.A., Tsuchiya, T., Zhang, Y.: On the formulation and theory of newton interior point methods for nonlinear programming. J. Optim. Theory Appl. 89(3), 507–541 (1996) · Zbl 0851.90115 [7] Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, Chichester (1968). Reprinted as Classics in Applied Mathematics. SIAM, Philadelphia (1990) · Zbl 0193.18805 [8] Gay, D.M., Overton, M.L., Wright, M.H.: A primal-dual interior method for nonconvex nonlinear programming. In: Yuan, Y. (ed.) Advances in Nonlinear Programming, pp. 31–56. Kluwer Academic, Dordrecht (1998) · Zbl 0908.90236 [9] Gould, N.I.M., Orban, D., Sartenaer, A., Toint, P.L.: Superlinear convergence of primal-dual interior point algorithms for nonlinear programming. SIAM J. Optim. 11(4), 974–1002 (2001) · Zbl 1003.65066 [10] Gould, N.I.M., Orban, D., Sartenaer, A., Toint, P.L.: Componentwise fast convergence in the solution of full-rank systems of nonlinear equation. Math. Program. Ser. B 92(3), 481–508 (2002) · Zbl 1012.65046 [11] Gould, N.I.M., Orban, D., Toint, P.L.: $$\mathsf{CUTEr}$$ and $$\mathsf{SifDec}$$ , a constrained and unconstrained testing environment, revisited. Trans. ACM Math. Softw. 29(4), 373–394 (2003) · Zbl 1068.90526 [12] Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics, vol. 16. SIAM, Belmont (1995) · Zbl 0832.65046 [13] Mehrotra, S.: On the implementation of a primal-dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992) · Zbl 0773.90047 [14] Nocedal, J., Wächter, A., Waltz, R.A.: Adaptive barrier strategies for nonlinear interior methods. Technical Report, Northwestern University, IL, USA (2005) · Zbl 1176.49036 [15] Vanderbei, R.J., Shanno, D.F.: An interior point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13(2), 231–252 (1999) · Zbl 1040.90564 [16] Vicente, L.N., Wright, S.J.: Local convergence of a primal-dual method for degenerate nonlinear programming. Comput. Optim. Appl. 22(3), 311–328 (2002) · Zbl 1039.90093 [17] Wächter, A., Biegler, L.T.: Failure of global convergence for a class of interior point methods for nonlinear programming. Math. Program. Ser. B 88(3), 565–574 (2000) · Zbl 0963.65063 [18] Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. Ser. A 106(1), 25–57 (2006) · Zbl 1134.90542 [19] Yabe, H., Yamashita, H.: Q-superlinear convergence of primal-dual interior point quasi-Newton methods for constrained optimization. J. Oper. Res. Soc. Jpn. 40(3), 415–436 (1997) · Zbl 0914.90246 [20] Yamashita, H., Yabe, H.: Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization. Math. Program. Ser. A 75(3), 377–397 (1996) · Zbl 0874.90175 [21] Yamashita, H., Yabe, H., Tanabe, T.: A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization. Math. Program. Ser. A 102(1), 111–151 (2005) · Zbl 1062.90036
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