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April  2013, 9(2): 391-409. doi: 10.3934/jimo.2013.9.391

## A penalty-free method for equality constrained optimization

 1 School of Mathematics Science, Soochow University, Suzhou, 215006, China, China, China

Received  September 2011 Revised  January 2013 Published  February 2013

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.
Citation: Zhongwen Chen, Songqiang Qiu, Yujie Jiao. A penalty-free method for equality constrained optimization. Journal of Industrial and Management Optimization, 2013, 9 (2) : 391-409. doi: 10.3934/jimo.2013.9.391
##### References:
 [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.

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##### References:
 [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.
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