Article Contents
Article Contents

# A penalty-free method for equality constrained optimization

• A penalty-free method is introduced for solving nonlinear programming with nonlinear equality constraints. This method does not use any penalty function, nor a filter. It uses trust region technique to compute trial steps. By comparing the measures of feasibility and optimality, the algorithm either tries to reduce the value of objective function by solving a normal subproblem and a tangential subproblem or tries to improve feasibility by solving a normal subproblem only. In order to guarantee global convergence, the measure of constraint violation in each iteration is required not to exceed a progressively decreasing limit. Under usual assumptions, we prove that the given algorithm is globally convergent to first order stationary points. Preliminary numerical results on CUTEr problems are reported.
Mathematics Subject Classification: Primary: 65K05; Secondary: 90C26, 90C30.

 Citation:

•  [1] R. Andreani, E. G. Birgin, J. M. Martinez and M. L. Schuverdt, Augmented Lagrangian methods under the constant positive linear dependence constraint qualification, Math. Prog. Ser. B, 111 (2008), 5-32.doi: 10.1007/s10107-006-0077-1. [2] R. H. Bielschowsky and F. A. M. Gomes, Dynamic control of infeasibility in equality constrained optimization, SIAM J. Optim., 19 (2008), 1299-1325.doi: 10.1137/070679557. [3] I. Bongartz, A. R. Conn, N. I. M. Gould and Ph. L. Toint, CUTE: Constrained and unconstrained testing enviroment, ACM Tran. Math. Software, 21 (1995), 123-160. [4] I. Bongartz, A. R. Conn, N. I. M. Gould, M. A. Saunders and Ph. L. Toint, "A Numerical Comparison between the LANCELOT and MINOS Packages for Large-Scale Constrained Optimization: The Complete Numerical Results," Report 97/14, Departement de Mathematique, Faculties Universitaires de Namur, 1997. [5] R. H. Byrd, F. E. Curtis and J. Nocedal, An inexact SQP method for equality constrained optimization, SIAM J Optim., 19 (2008), 351-369.doi: 10.1137/060674004. [6] R. H. Byrd, F. E. Curtis and J. Nocedal, An inexact Newton method for nonconvex equality constrained optimization, Math. Prog., 122 (2010), 273-299.doi: 10.1007/s10107-008-0248-3. [7] Z. W. Chen, A penalty-free-type nonmonotone trust-region method for nonlinear constrained optimization, Appl. Math. and Comput., 173 (2006), 1014-1046.doi: 10.1016/j.amc.2005.04.031. [8] C. M. Chin and R. Fletcher, On the global convergence of an SLP-filter algorithm that takes EQP steps, Math. Prog. Ser. A, 96 (2003), 161-177. [9] A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods," MPS-SIAM Ser. Optim., SIAM, Philadelphia, PA, Mathematical Programming Society (MPS), Philadelphia, PA, 2000.doi: 10.1137/1.9780898719857. [10] E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Prog. Serial A., 91 (2002), 201-213.doi: 10.1007/s101070100263. [11] R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Math. Prog. Ser. A, 91 (2002), 239-269.doi: 10.1007/s101070100244. [12] R. Fletcher, S. Leyffer and Ph. L. Toint, On the global convergence of a filter-SQP algorithm, SIAM J. Optim., 13 (2002), 44-59.doi: 10.1137/S105262340038081X. [13] R. Fletcher, S. Leyffer and Ph. L. Toint, "A Brief History of Filter Methods," Optimization Online, September 26, 2006. [14] N. I. M. Gould and Ph. L. Toint, Nonlinear programming without a penalty function or a filter, Math. Prog. Ser. A, 122 (2010), 155-196.doi: 10.1007/s10107-008-0244-7. [15] X. Liu and Y. Yuan, A sequential quadratic programming method without a penalty function or a filter for nonlinear equality constrained optimization, SIAM J. Optim., 21 (2011), 545-571.doi: 10.1137/080739884. [16] S. Qiu and Z. Chen, A new penalty-free-type algorithm that based on trust region techniques, Appl. Math. Comput., 218 (2012), 11089-11099.doi: 10.1016/j.amc.2012.04.065. [17] M. Ulbrich and S. Ulbrich, Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function, Math. Prog. Ser. B, 95 (2003), 103-135.doi: 10.1007/s10107-002-0343-9. [18] M. Ulbrich, S. Ulbrich and L. N. Vicente, A globally convergent primal-dual interior-point filter method for nonlinear programming, Math. Prog. Ser. A, 100 (2004), 379-410.doi: 10.1007/s10107-003-0477-4. [19] A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Local convergence, SIAM J. Optim., 16 (2005), 32-48.doi: 10.1137/S1052623403426544. [20] H. Yamashita, "A Globally Convergent Quasi-Newton Method for Equality Constrained Optimization that Does Not Use a Penalty Function," Technical Report, Mathematical System Inc., Tokyo, Japan, June 1979 (revised September 1982). [21] H. Yamashita and H. Yabe, "A Globally and Superlinearly Convergent Trust-Region SQP Method Without a Penalty Function for Nonlinearly Constrained Optimization," Technical Report, Mathematical System Inc., Tokyo, Japan, September 2003 (revised July 2007). [22] C. Zoppke-Donaldson, "A Tolerance Tube Approach to Sequential Quadratic Programming with Applications," Ph.D Thesis, Department of Mathematics and Computer Science, University of Dundee, Dundee, Scotland, UK, 1995.