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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 & 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.  doi: 10.1007/s10107-006-0077-1.  Google Scholar

[2]

R. H. Bielschowsky and F. A. M. Gomes, Dynamic control of infeasibility in equality constrained optimization,, SIAM J. Optim., 19 (2008), 1299.  doi: 10.1137/070679557.  Google Scholar

[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.   Google Scholar

[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, (1997).   Google Scholar

[5]

R. H. Byrd, F. E. Curtis and J. Nocedal, An inexact SQP method for equality constrained optimization,, SIAM J Optim., 19 (2008), 351.  doi: 10.1137/060674004.  Google Scholar

[6]

R. H. Byrd, F. E. Curtis and J. Nocedal, An inexact Newton method for nonconvex equality constrained optimization,, Math. Prog., 122 (2010), 273.  doi: 10.1007/s10107-008-0248-3.  Google Scholar

[7]

Z. W. Chen, A penalty-free-type nonmonotone trust-region method for nonlinear constrained optimization,, Appl. Math. and Comput., 173 (2006), 1014.  doi: 10.1016/j.amc.2005.04.031.  Google Scholar

[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.   Google Scholar

[9]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,", MPS-SIAM Ser. Optim., (2000).  doi: 10.1137/1.9780898719857.  Google Scholar

[10]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles,, Math. Prog. Serial A., 91 (2002), 201.  doi: 10.1007/s101070100263.  Google Scholar

[11]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Math. Prog. Ser. A, 91 (2002), 239.  doi: 10.1007/s101070100244.  Google Scholar

[12]

R. Fletcher, S. Leyffer and Ph. L. Toint, On the global convergence of a filter-SQP algorithm,, SIAM J. Optim., 13 (2002), 44.  doi: 10.1137/S105262340038081X.  Google Scholar

[13]

R. Fletcher, S. Leyffer and Ph. L. Toint, "A Brief History of Filter Methods,", Optimization Online, (2006).   Google Scholar

[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.  doi: 10.1007/s10107-008-0244-7.  Google Scholar

[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.  doi: 10.1137/080739884.  Google Scholar

[16]

S. Qiu and Z. Chen, A new penalty-free-type algorithm that based on trust region techniques,, Appl. Math. Comput., 218 (2012), 11089.  doi: 10.1016/j.amc.2012.04.065.  Google Scholar

[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.  doi: 10.1007/s10107-002-0343-9.  Google Scholar

[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.  doi: 10.1007/s10107-003-0477-4.  Google Scholar

[19]

A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Local convergence,, SIAM J. Optim., 16 (2005), 32.  doi: 10.1137/S1052623403426544.  Google Scholar

[20]

H. Yamashita, "A Globally Convergent Quasi-Newton Method for Equality Constrained Optimization that Does Not Use a Penalty Function,", Technical Report, (1979).   Google Scholar

[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, (2003).   Google Scholar

[22]

C. Zoppke-Donaldson, "A Tolerance Tube Approach to Sequential Quadratic Programming with Applications,", Ph.D Thesis, (1995).   Google Scholar

show all references

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.  doi: 10.1007/s10107-006-0077-1.  Google Scholar

[2]

R. H. Bielschowsky and F. A. M. Gomes, Dynamic control of infeasibility in equality constrained optimization,, SIAM J. Optim., 19 (2008), 1299.  doi: 10.1137/070679557.  Google Scholar

[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.   Google Scholar

[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, (1997).   Google Scholar

[5]

R. H. Byrd, F. E. Curtis and J. Nocedal, An inexact SQP method for equality constrained optimization,, SIAM J Optim., 19 (2008), 351.  doi: 10.1137/060674004.  Google Scholar

[6]

R. H. Byrd, F. E. Curtis and J. Nocedal, An inexact Newton method for nonconvex equality constrained optimization,, Math. Prog., 122 (2010), 273.  doi: 10.1007/s10107-008-0248-3.  Google Scholar

[7]

Z. W. Chen, A penalty-free-type nonmonotone trust-region method for nonlinear constrained optimization,, Appl. Math. and Comput., 173 (2006), 1014.  doi: 10.1016/j.amc.2005.04.031.  Google Scholar

[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.   Google Scholar

[9]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,", MPS-SIAM Ser. Optim., (2000).  doi: 10.1137/1.9780898719857.  Google Scholar

[10]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles,, Math. Prog. Serial A., 91 (2002), 201.  doi: 10.1007/s101070100263.  Google Scholar

[11]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Math. Prog. Ser. A, 91 (2002), 239.  doi: 10.1007/s101070100244.  Google Scholar

[12]

R. Fletcher, S. Leyffer and Ph. L. Toint, On the global convergence of a filter-SQP algorithm,, SIAM J. Optim., 13 (2002), 44.  doi: 10.1137/S105262340038081X.  Google Scholar

[13]

R. Fletcher, S. Leyffer and Ph. L. Toint, "A Brief History of Filter Methods,", Optimization Online, (2006).   Google Scholar

[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.  doi: 10.1007/s10107-008-0244-7.  Google Scholar

[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.  doi: 10.1137/080739884.  Google Scholar

[16]

S. Qiu and Z. Chen, A new penalty-free-type algorithm that based on trust region techniques,, Appl. Math. Comput., 218 (2012), 11089.  doi: 10.1016/j.amc.2012.04.065.  Google Scholar

[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.  doi: 10.1007/s10107-002-0343-9.  Google Scholar

[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.  doi: 10.1007/s10107-003-0477-4.  Google Scholar

[19]

A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Local convergence,, SIAM J. Optim., 16 (2005), 32.  doi: 10.1137/S1052623403426544.  Google Scholar

[20]

H. Yamashita, "A Globally Convergent Quasi-Newton Method for Equality Constrained Optimization that Does Not Use a Penalty Function,", Technical Report, (1979).   Google Scholar

[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, (2003).   Google Scholar

[22]

C. Zoppke-Donaldson, "A Tolerance Tube Approach to Sequential Quadratic Programming with Applications,", Ph.D Thesis, (1995).   Google Scholar

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