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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. [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. [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. [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). [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. [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. [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. [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. [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. [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. [11] R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Math. Prog. Ser. A, 91 (2002), 239. 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. doi: 10.1137/S105262340038081X. [13] R. Fletcher, S. Leyffer and Ph. L. Toint, "A Brief History of Filter Methods,", Optimization Online, (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. 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. 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. 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. 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. 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. 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, (1979). [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). [22] C. Zoppke-Donaldson, "A Tolerance Tube Approach to Sequential Quadratic Programming with Applications,", Ph.D Thesis, (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. 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. 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. [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). [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. [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. [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. [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. [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. [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. [11] R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Math. Prog. Ser. A, 91 (2002), 239. 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. doi: 10.1137/S105262340038081X. [13] R. Fletcher, S. Leyffer and Ph. L. Toint, "A Brief History of Filter Methods,", Optimization Online, (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. 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. 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. 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. 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. 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. 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, (1979). [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). [22] C. Zoppke-Donaldson, "A Tolerance Tube Approach to Sequential Quadratic Programming with Applications,", Ph.D Thesis, (1995).
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