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2012, 2(1): 19-29. doi: 10.3934/naco.2012.2.19

## Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints

 1 Department of Applied Mathematics, Beijing University of Technology, Beijing 100124 2 Department of Applied Mathematics, Hebei University of Technology, Tianjin 300401

Received  March 2011 Revised  May 2011 Published  March 2012

A sequential quadratic programming (SQP) algorithm is presented for solving nonlinear optimization with overdetermined constraints. In each iteration, the quadratic programming (QP) subproblem is always feasible even if the gradients of constraints are always linearly dependent and the overdetermined constraints may be inconsistent. The $\ell_2$ exact penalty function is selected as the merit function. Under suitable assumptions on gradients of constraints, we prove that the algorithm will terminate at an approximate KKT point of the problem, or there is a limit point which is either a point satisfying the overdetermined system of equations or a stationary point for minimizing the $\ell_2$ norm of the residual of the overdetermined system of equations.
Citation: Chunlin Hao, Xinwei Liu. Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 19-29. doi: 10.3934/naco.2012.2.19
##### 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. Program., 111 (2008), 5-32. doi: doi:10.1007/s10107-006-0077-1. [2] N. Arora and L. T. Biegler, A trust region SQP algorithm for equality constrained parameter estimation with simple parameter bounds, Comput. Optim. Appl., 28 (2004), 51-86. doi: doi:10.1023/B:COAP.0000018879.40214.11. [3] J. T. Betts, Very low-thrust trajectory optimization using a direct SQP method, J. Comput. Appl. Math., 120 (2000), 27-40. doi: doi:10.1016/S0377-0427(00)00301-0. [4] J. V. Burke, A sequential quadratic programming method for potentially infeasible mathematical programs, J. Math. Anal. Appl., 139 (1989), 319-351. doi: doi:10.1016/0022-247X(89)90111-X. [5] J. V. Burke and S. P. Han, A robust sequential quadratic programming method, Math. Program., 43 (1989), 277-303. doi: doi:10.1007/BF01582294. [6] 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: doi:10.1137/060674004. [7] R. H. Byrd, M. Marazzi and J. Nocedal, On the convergence of Newton iterations to non- \break stationary points, Math. Program., 99 (2004), 127-148. doi: doi:10.1007/s10107-003-0376-8. [8] A. R. Conn, N. Gould and Ph. L. Toint, "Trust-Region Methods," SIAM, Philadelphia, USA, 2000. doi: doi:10.1137/1.9780898719857. [9] H. Dai, "Matrix Theory," Science Press, 2001. [10] R. Fletcher, "Practical Methods for Optimization. Vol. 2: Constrained Optimization," John Wiley and Sons, Chichester, UK, 1981. [11] R. Fletcher, N. Gould, S. Leyffer, Ph. L. Toint and A. Wächter, Global convergence of a trust-region SQP-filter allgorithm for general nonlinear programming, SIAM J. Optim., 13 (2002), 635-659. doi: doi:10.1137/S1052623499357258. [12] R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Math. Program., 91 (2002), 239-269. doi: doi:10.1007/s101070100244. [13] G. H. Golub and C. F. V. Loan, "Matrix Computation," Third Edition, The Johns Hopkins University Press, 1996. [14] S. P. Han, A globally convergent method for nonlinear programming, J. Optim. Theory Appl., 22 (1977), 297-309. doi: doi:10.1007/BF00932858. [15] M. Heinkenschloss and L. N. Vicente, Analysis of inexact trust-region SQP algorithms, SIAM J. Optim., 12 (2002), 283-302. doi: doi:10.1137/S1052623499361543. [16] X. W. Liu, Global convergence on an active set SQP for inequality constrained optimization, J. Comput. Appl. Math., 180 (2005), 201-211. doi: doi:10.1016/j.cam.2004.10.012. [17] X. W. Liu and J. Sun, A robust primal-dual interior point algorithm for nonlinear programs, SIAM J. Optim., 14 (2004), 1163-1186. doi: doi:10.1137/S1052623402400641. [18] X. W. Liu and Y. X. Yuan, A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties, Math. Program., 125 (2010), 163-193. doi: doi:10.1007/s10107-009-0272-y. [19] X. W. Liu and Y. X. 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. [20] W. Murray, Sequential quadratic programming methods for large-scale problems, J. Comput. Optim. Appl., 7 (1997), 127-142. doi: doi:10.1023/A:1008671829454. [21] J. Nocedal and S. Wright, "Numerical Optimization," Springer-Verlag New York, Inc., 1999. doi: doi:10.1007/b98874. [22] M. J. D. Powell, A fast algorithm for nonlinearly constrained optimization calculations, in "Proceedings 1977 Dundee Biennial Conference on Numerical Analysis"(ed. G.A. Watson), Springer-Verlag, Berlin, 1978, 144-157. [23] P. Spellucci, An SQP method for general nonlinear programs using only equality constrained subproblems, Math. Program., 82 (1998), 413-448. doi: doi:10.1007/BF01580078. [24] G. W. Walster and E. R. Hansen, Computing interval parameter bounds from fallible measurements using overdetermined (Tall) systems of nonlinear equations, Lecture Notes in "Computer Science, Global Optimization and Constraint Satisfaction"(eds. C. Blieket al.), COCOS 2002, LNCS 2861, 171-177, Springer Berlin Heidelberg, 2003. [25] S. J. Wright, Modifying SQP for degenerate problems, SIAM J. Optim., 13 (2002), 470-497. doi: doi:10.1137/S1052623498333731. [26] Y. X. Yuan, On the convergence of a new trust region algorithm, Numer. Math., 70 (1995), 515-539. doi: doi:10.1007/s002110050133.

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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. Program., 111 (2008), 5-32. doi: doi:10.1007/s10107-006-0077-1. [2] N. Arora and L. T. Biegler, A trust region SQP algorithm for equality constrained parameter estimation with simple parameter bounds, Comput. Optim. Appl., 28 (2004), 51-86. doi: doi:10.1023/B:COAP.0000018879.40214.11. [3] J. T. Betts, Very low-thrust trajectory optimization using a direct SQP method, J. Comput. Appl. Math., 120 (2000), 27-40. doi: doi:10.1016/S0377-0427(00)00301-0. [4] J. V. Burke, A sequential quadratic programming method for potentially infeasible mathematical programs, J. Math. Anal. Appl., 139 (1989), 319-351. doi: doi:10.1016/0022-247X(89)90111-X. [5] J. V. Burke and S. P. Han, A robust sequential quadratic programming method, Math. Program., 43 (1989), 277-303. doi: doi:10.1007/BF01582294. [6] 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: doi:10.1137/060674004. [7] R. H. Byrd, M. Marazzi and J. Nocedal, On the convergence of Newton iterations to non- \break stationary points, Math. Program., 99 (2004), 127-148. doi: doi:10.1007/s10107-003-0376-8. [8] A. R. Conn, N. Gould and Ph. L. Toint, "Trust-Region Methods," SIAM, Philadelphia, USA, 2000. doi: doi:10.1137/1.9780898719857. [9] H. Dai, "Matrix Theory," Science Press, 2001. [10] R. Fletcher, "Practical Methods for Optimization. Vol. 2: Constrained Optimization," John Wiley and Sons, Chichester, UK, 1981. [11] R. Fletcher, N. Gould, S. Leyffer, Ph. L. Toint and A. Wächter, Global convergence of a trust-region SQP-filter allgorithm for general nonlinear programming, SIAM J. Optim., 13 (2002), 635-659. doi: doi:10.1137/S1052623499357258. [12] R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Math. Program., 91 (2002), 239-269. doi: doi:10.1007/s101070100244. [13] G. H. Golub and C. F. V. Loan, "Matrix Computation," Third Edition, The Johns Hopkins University Press, 1996. [14] S. P. Han, A globally convergent method for nonlinear programming, J. Optim. Theory Appl., 22 (1977), 297-309. doi: doi:10.1007/BF00932858. [15] M. Heinkenschloss and L. N. Vicente, Analysis of inexact trust-region SQP algorithms, SIAM J. Optim., 12 (2002), 283-302. doi: doi:10.1137/S1052623499361543. [16] X. W. Liu, Global convergence on an active set SQP for inequality constrained optimization, J. Comput. Appl. Math., 180 (2005), 201-211. doi: doi:10.1016/j.cam.2004.10.012. [17] X. W. Liu and J. Sun, A robust primal-dual interior point algorithm for nonlinear programs, SIAM J. Optim., 14 (2004), 1163-1186. doi: doi:10.1137/S1052623402400641. [18] X. W. Liu and Y. X. Yuan, A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties, Math. Program., 125 (2010), 163-193. doi: doi:10.1007/s10107-009-0272-y. [19] X. W. Liu and Y. X. 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. [20] W. Murray, Sequential quadratic programming methods for large-scale problems, J. Comput. Optim. Appl., 7 (1997), 127-142. doi: doi:10.1023/A:1008671829454. [21] J. Nocedal and S. Wright, "Numerical Optimization," Springer-Verlag New York, Inc., 1999. doi: doi:10.1007/b98874. [22] M. J. D. Powell, A fast algorithm for nonlinearly constrained optimization calculations, in "Proceedings 1977 Dundee Biennial Conference on Numerical Analysis"(ed. G.A. Watson), Springer-Verlag, Berlin, 1978, 144-157. [23] P. Spellucci, An SQP method for general nonlinear programs using only equality constrained subproblems, Math. Program., 82 (1998), 413-448. doi: doi:10.1007/BF01580078. [24] G. W. Walster and E. R. Hansen, Computing interval parameter bounds from fallible measurements using overdetermined (Tall) systems of nonlinear equations, Lecture Notes in "Computer Science, Global Optimization and Constraint Satisfaction"(eds. C. Blieket al.), COCOS 2002, LNCS 2861, 171-177, Springer Berlin Heidelberg, 2003. [25] S. J. Wright, Modifying SQP for degenerate problems, SIAM J. Optim., 13 (2002), 470-497. doi: doi:10.1137/S1052623498333731. [26] Y. X. Yuan, On the convergence of a new trust region algorithm, Numer. Math., 70 (1995), 515-539. doi: doi:10.1007/s002110050133.
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