zbMATH — the first resource for mathematics

A line search filter-SQP method with Lagrangian function for nonlinear inequality constrained optimization. (English) Zbl 1370.49031
Summary: In this paper, we propose a line search filter technique in association with Sequential Quadratic Programming (SQP) for solving the nonlinear inequality constrained optimization. The Lagrangian function value instead of the objective function value is used in the filter together with an appropriate infeasibility measure. The search direction which is generated by solving the quadratic programming is decomposed into its normal space and tangential space vectors. Under some reasonable conditions, the global convergence is established for every possible choice of the starting point. By using the Lagrangian function value in the filter, it is shown that the algorithm does not suffer from the Maratos effect without a second order correction, so that local superlinear convergence rate is achieved. Numerical results show that the proposed algorithm is efficient.

MSC:
 49M37 Numerical methods based on nonlinear programming 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming 90C55 Methods of successive quadratic programming type
Software:
ipfilter; SNOPT; TFOCS
Full Text:
References:
 [1] Becker, SR; Candés, EJ; Grant, MC, Templates for convex cone problems with applications to sparse signal recovery, Math. Prog. Comp., 3, 165-218, (2011) · Zbl 1257.90042 [2] Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS-SIAM Series on Optimization, SIAM, Philadelphia (2000) · Zbl 0958.65071 [3] Chin, CM; Rashid, AHA; Nor, KM, Global and local convergence of a filter line search method for nonlinear programming, Optim. Methods Softw., 22, 365-390, (2007) · Zbl 1193.90192 [4] Fletcher, R, A sequential linear constraint programming algorithm for NLP, SIAM J. Optim., 22, 772-794, (2012) · Zbl 1258.65061 [5] Fletcher, R; Gould, NIM; Leyffer, S; Toint, PhL; Wächter, A, Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming, SIAM J. Optim., 13, 635-659, (2002) · Zbl 1038.90076 [6] Fletcher, R; Leyffer, S, Nonlinear programming without a penalty function, Math. Program., 91, 239-269, (2002) · Zbl 1049.90088 [7] Fletcher, R; Leyffer, S; Toint, PL, On the global convergence of a filter-SQP algorithm, SIAM J. Optim., 13, 44-59, (2002) · Zbl 1029.65063 [8] Gonzaga, CC; Karas, EW; Vanti, M, A globally convergent filter method for nonlinear programming, SIAM J. Optim., 14, 646-669, (2003) · Zbl 1079.90129 [9] Gould, NIM; Loh, Y; Robinson, DP, A filter method with unified step computation for nonlinear optimization, SIAM J. Optim., 24, 175-209, (2014) · Zbl 1301.49070 [10] Gill, PhE; Murray, W; Saunders, MA, SNOPT: an SQP algorithm for large-scale constrained optimization, SIAM Rev., 47, 99-131, (2005) · Zbl 1210.90176 [11] Hock, W., Schittkowski, K.: Test examples for nonlinear programming codes. In: Lecture Notes in Economics and Mathematics System, vol. 187. Springer, Berlin (1981) · Zbl 0452.90038 [12] Milzarek, A; Ulbrich, M, A semismooth Newton method with multidimensional filter globalization for $$l_1$$-optimization, SIAM J. Optim., 24, 298-333, (2014) · Zbl 1295.49022 [13] NEOS server: http://neos.mcs.anl.gov/neos/solvers/nco:SNOPT/GAMS.html (2006) · Zbl 1038.90076 [14] Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999) · Zbl 0930.65067 [15] Shen, C; Leyffer, S; Fletcher, R, A nomonotone filter method for nonlinear optimization, Comput. Optim. Appl., 52, 583-607, (2012) · Zbl 1259.90140 [16] Shen, C; Xue, W; Pu, D, Global convergence of a tri-dimensional filter SQP based on the line search method, Appl. Numer. Math., 59, 235-250, (2009) · Zbl 1155.90023 [17] Ulbrich, S, On the superlinear local convergence of a filter-SQP method, Math. Program., 100, 217-245, (2004) · Zbl 1146.90525 [18] Ulbrich, M; Ulbrich, S; Vicente, LN, A globally convergent primal-dual interior-point filter method for nonlinear programming, Math. Program., 100, 379-410, (2004) · Zbl 1070.90110 [19] Wächter, A; Biegler, LT, Line search filter method for nonlinear programming: motivation and global convergence, SIAM J. Comput., 16, 1-31, (2005) · Zbl 1114.90128 [20] Wächter, A; Biegler, LT, Line search filter method for nonlinear programming: local convergence, SIAM J. Optim., 16, 32-48, (2005) · Zbl 1115.90056 [21] Wen, Z; Yin, W; Zhang, Y, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Math. Prog. Comput., 4, 333-361, (2012) · Zbl 1271.65083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.