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doi: 10.3934/jimo.2018190

A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

1. 

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

2. 

School of Mathematical Sciences, University of Chinese Academy of Sciences

3. 

Institute of Mathematics, Hebei University of Technology, Tianjin 300401, China

4. 

Institute of Mathematics, Hebei University of Technology, Tianjin China

5. 

School of Science, Curtin University, Perth, Australia

6. 

School of Business, National University of Singapore, Singapore

*Corresponding author

Received  July 2018 Revised  August 2018 Published  December 2018

Fund Project: The first draft of this paper was completed on December 2, 2014. The first author is supported by the Chinese NSF grants (nos. 11631013, 11331012 and 81173633) and the National Key Basic Research Program of China (no. 2015CB856000). The second author is supported by the Chinese NSF grants (nos. 11671116 and 11271107) and the Major Research Plan of the NSFC (no. 91630202). The third author is supported by Grant DP-160101819 of Australia Research Council

With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closed-form solutions of slack variables of logarithmic-barrier problems of nonlinear programs. As a result, a two-parameter primal-dual nonlinear system is proposed, which corresponds to the Karush-Kuhn-Tucker point and the infeasible stationary point of nonlinear programs, respectively, as one of two parameters vanishes. Based on this distinctive system, we present a primal-dual interior-point method capable of rapidly detecting infeasibility of nonlinear programs. The method generates interior-point iterates without truncation of the step. It is proved that our method converges to a Karush-Kuhn-Tucker point of the original problem as the barrier parameter tends to zero. Otherwise, the scaling parameter tends to zero, and the method converges to either an infeasible stationary point or a singular stationary point of the original problem. Moreover, our method has the capability to rapidly detect the infeasibility of the problem. Under suitable conditions, the method can be superlinearly or quadratically convergent to the Karush-Kuhn-Tucker point if the original problem is feasible, and it can be superlinearly or quadratically convergent to the infeasible stationary point when the problem is infeasible. Preliminary numerical results show that the method is efficient in solving some simple but hard problems, where the superlinear convergence to an infeasible stationary point is demonstrated when we solve two infeasible problems in the literature.

Citation: Yu-Hong Dai, Xin-Wei Liu, Jie Sun. A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018190
References:
[1]

R. AndreaniE. G. BirginJ. M. Martinez and M. L. Schuverdt, Augmented Lagrangian methods under the constant positive linear dependence constraint qualification, Math. Program., 111 (2008), 5-32. doi: 10.1007/s10107-006-0077-1. Google Scholar

[2]

P. Armand and J. Benoist, A local convergence property of primal-dual methods for nonlinear programming, Math. Program., 115 (2008), 199-222. doi: 10.1007/s10107-007-0136-2. Google Scholar

[3]

P. ArmandJ. C. Gilbert and S. Jan-Jégou, A feasible BFGS interior point algorithm for solving convex minimization problems, SIAM J. Optim., 11 (2000), 199-222. doi: 10.1137/S1052623498344720. Google Scholar

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I. BongartzA. R. ConnN. I. M. Gould and P. L. Toint, CUTE: Constrained and Unconstrained Testing Environment, ACM Tran. Math. Software, 21 (1995), 123-160. Google Scholar

[5]

J. V. BurkeF. E. Curtis and H. Wang, A sequential quadratic optimization algorithm with rapid infeasibility detection, SIAM J. Optim., 24 (2014), 839-872. doi: 10.1137/120880045. Google Scholar

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J. V. Burke and S. P. Han, A robust sequential quadratic programming method, Math. Program., 43 (1989), 277-303. doi: 10.1007/BF01582294. Google Scholar

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R. H. Byrd, Robust Trust-Region Method for Constrained Optimization, Paper presented at the SIAM Conference on Optimization, Houston, TX, 1987.Google Scholar

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R. H. ByrdF. E. Curtis and J. Nocedal, Infeasibility detection and SQP methods for nonlinear optimization, SIAM J. Optim., 20 (2010), 2281-2299. doi: 10.1137/080738222. Google Scholar

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R. H. ByrdJ. C. Gilbert and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Math. Program., 89 (2000), 149-185. doi: 10.1007/PL00011391. Google Scholar

[10]

R. H. ByrdM. E. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 9 (1999), 877-900. doi: 10.1137/S1052623497325107. Google Scholar

[11]

R. H. Byrd, G. Liu and J. Nocedal, On the local behaviour of an interior point method for nonlinear programming, In, Numerical Analysis 1997 (eds. D. F. Griffiths and D. J. Higham), Addison-Wesley Longman, Reading, MA, 380 (1998), 37-56. Google Scholar

[12]

R. H. ByrdM. Marazzi and J. Nocedal, On the convergence of Newton iterations to non-stationary points, Math. Program., 99 (2004), 127-148. doi: 10.1007/s10107-003-0376-8. Google Scholar

[13]

L. F. Chen and D. Goldfarb, Interior-point $\ell_2$-penalty methods for nonlinear programming with strong global convergence properties, Math. Program., 108 (2006), 1-36. doi: 10.1007/s10107-005-0701-5. Google Scholar

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F. E. Curtis, A penalty-interior-point algorithm for nonlinear constrained optimization, Math. Program. Comput., 4 (2012), 181-209. doi: 10.1007/s12532-012-0041-4. Google Scholar

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A. S. El-BakryR. A. TapiaT. Tsuchiya and Y. Zhang, On the formulation and theory of the Newton interior-point method for nonlinear programming, J. Optim. Theory Appl., 89 (1996), 507-541. doi: 10.1007/BF02275347. Google Scholar

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A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, 1968; republished as Classics in Appl. Math. 4, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971316. Google Scholar

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R. Fletcher, Practical Methods for Optimization. Vol. 2: Constrained Optimization, John Wiley and Sons, Chichester, 1980. Google Scholar

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A. Forsgren and P. E. Gill, Primal-dual interior methods for nonconvex nonlinear programming, SIAM J. Optim., 8 (1998), 1132-1152. doi: 10.1137/S1052623496305560. Google Scholar

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A. ForsgrenPh. E. Gill and M. H. Wright, Interior methods for nonlinear optimization, SIAM Review, 44 (2002), 525-597. doi: 10.1137/S0036144502414942. Google Scholar

[20]

D. M. Gay, M. L. Overton and M. H. Wright, A primal-dual interior method for nonconvex nonlinear programming, in Advances in Nonlinear Programming, (ed. Y.-X. Yuan), Kluwer Academic Publishers, Dordrecht, 14 (1998), 31-56. doi: 10.1007/978-1-4613-3335-7_2. Google Scholar

[21]

E. M. Gertz and Ph. E. Gill, A primal-dual trust region algorithm for nonlinear optimization, Math. Program., 100 (2004), 49-94. doi: 10.1007/s10107-003-0486-3. Google Scholar

[22]

N. I. M. Gould, D. Orban and Ph. L. Toint, An interior-point $\ell_1$-penalty method for nonlinear optimization, in Recent Developments in Numerical Analysis and Optimization, Proceedings of NAOIII 2014, Springer, Verlag, 134 (2015), 117-150. doi: 10.1007/978-3-319-17689-5_6. Google Scholar

[23]

W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, Lecture Notes in Eco. and Math. Systems 187, Springer-Verlag, Berlin, New York, 1981. doi: 10.1007/BF00934594. Google Scholar

[24]

X.-W. LiuG. Perakis and J. Sun, A robust SQP method for mathematical programs with linear complementarity constraints, Comput. Optim. Appl., 34 (2006), 5-33. doi: 10.1007/s10589-005-3075-y. Google Scholar

[25]

X.-W. Liu and J. Sun, A robust primal-dual interior point algorithm for nonlinear programs, SIAM J. Optim., 14 (2004), 1163-1186. doi: 10.1137/S1052623402400641. Google Scholar

[26]

X.-W. Liu and Y.-X. Yuan, A robust algorithm for optimization with general equality and inequality constraints, SIAM J. Sci. Comput., 22 (2000), 517-534. doi: 10.1137/S1064827598334861. Google Scholar

[27]

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: 10.1007/s10107-009-0272-y. Google Scholar

[28]

J. NocedalF. Öztoprak and R. A. Waltz, An interior point method for nonlinear programming with infeasibility detection capabilities, Optim. Methods Softw., 29 (2014), 837-854. doi: 10.1080/10556788.2013.858156. Google Scholar

[29]

J. Nocedal and S. Wright, Numerical Optimization, Springer-Verlag New York, Inc., 1999. doi: 10.1007/b98874. Google Scholar

[30] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York and London, 1970.
[31]

D. F. Shanno and R. J. Vanderbei, Interior-point methods for nonconvex nonlinear programming: Orderings and higher-order methods, Math. Program., 87 (2000), 303-316. doi: 10.1007/s101070050116. Google Scholar

[32]

P. Tseng, Convergent infeasible interior-point trust-region methods for constrained minimization, SIAM J. Optim., 13 (2002), 432-469. doi: 10.1137/S1052623499357945. Google Scholar

[33]

M. UlbrichS. Ulbrich and L. N. Vicente, A globally convergent primal-dual interior-point filter method for nonlinear programming, Math. Program., 100 (2004), 379-410. doi: 10.1007/s10107-003-0477-4. Google Scholar

[34]

A. Wächter and L. T. Biegler, Failure of global convergence for a class of interior point methods for nonlinear programming, Math. Program., 88 (2000), 565-574. doi: 10.1007/PL00011386. Google Scholar

[35]

A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Motivation and global convergence, SIAM J. Optim., 16 (2005), 1-31. doi: 10.1137/S1052623403426556. Google Scholar

[36]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar

[37]

M. H. Wright, Why a pure primal Newton barrier step may be infeasible?, SIAM J. Optim., 5 (1995), 1-12. doi: 10.1137/0805001. Google Scholar

[38]

S. J. Wright, On the convergence of the Newton/Log-barrier method, Math. Program., 90 (2001), 71-100. doi: 10.1007/PL00011421. Google Scholar

[39]

Y.-X. Yuan, On the convergence of a new trust region algorithm, Numer. Math., 70 (1995), 515-539. doi: 10.1007/s002110050133. Google Scholar

[40]

Y. Zhang, Solving large-scale linear programs by interior-point methods under the MATLAB environment, Optim. Methods Softw., 10 (1998), 1-31. doi: 10.1080/10556789808805699. Google Scholar

show all references

References:
[1]

R. AndreaniE. G. BirginJ. M. Martinez and M. L. Schuverdt, Augmented Lagrangian methods under the constant positive linear dependence constraint qualification, Math. Program., 111 (2008), 5-32. doi: 10.1007/s10107-006-0077-1. Google Scholar

[2]

P. Armand and J. Benoist, A local convergence property of primal-dual methods for nonlinear programming, Math. Program., 115 (2008), 199-222. doi: 10.1007/s10107-007-0136-2. Google Scholar

[3]

P. ArmandJ. C. Gilbert and S. Jan-Jégou, A feasible BFGS interior point algorithm for solving convex minimization problems, SIAM J. Optim., 11 (2000), 199-222. doi: 10.1137/S1052623498344720. Google Scholar

[4]

I. BongartzA. R. ConnN. I. M. Gould and P. L. Toint, CUTE: Constrained and Unconstrained Testing Environment, ACM Tran. Math. Software, 21 (1995), 123-160. Google Scholar

[5]

J. V. BurkeF. E. Curtis and H. Wang, A sequential quadratic optimization algorithm with rapid infeasibility detection, SIAM J. Optim., 24 (2014), 839-872. doi: 10.1137/120880045. Google Scholar

[6]

J. V. Burke and S. P. Han, A robust sequential quadratic programming method, Math. Program., 43 (1989), 277-303. doi: 10.1007/BF01582294. Google Scholar

[7]

R. H. Byrd, Robust Trust-Region Method for Constrained Optimization, Paper presented at the SIAM Conference on Optimization, Houston, TX, 1987.Google Scholar

[8]

R. H. ByrdF. E. Curtis and J. Nocedal, Infeasibility detection and SQP methods for nonlinear optimization, SIAM J. Optim., 20 (2010), 2281-2299. doi: 10.1137/080738222. Google Scholar

[9]

R. H. ByrdJ. C. Gilbert and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Math. Program., 89 (2000), 149-185. doi: 10.1007/PL00011391. Google Scholar

[10]

R. H. ByrdM. E. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 9 (1999), 877-900. doi: 10.1137/S1052623497325107. Google Scholar

[11]

R. H. Byrd, G. Liu and J. Nocedal, On the local behaviour of an interior point method for nonlinear programming, In, Numerical Analysis 1997 (eds. D. F. Griffiths and D. J. Higham), Addison-Wesley Longman, Reading, MA, 380 (1998), 37-56. Google Scholar

[12]

R. H. ByrdM. Marazzi and J. Nocedal, On the convergence of Newton iterations to non-stationary points, Math. Program., 99 (2004), 127-148. doi: 10.1007/s10107-003-0376-8. Google Scholar

[13]

L. F. Chen and D. Goldfarb, Interior-point $\ell_2$-penalty methods for nonlinear programming with strong global convergence properties, Math. Program., 108 (2006), 1-36. doi: 10.1007/s10107-005-0701-5. Google Scholar

[14]

F. E. Curtis, A penalty-interior-point algorithm for nonlinear constrained optimization, Math. Program. Comput., 4 (2012), 181-209. doi: 10.1007/s12532-012-0041-4. Google Scholar

[15]

A. S. El-BakryR. A. TapiaT. Tsuchiya and Y. Zhang, On the formulation and theory of the Newton interior-point method for nonlinear programming, J. Optim. Theory Appl., 89 (1996), 507-541. doi: 10.1007/BF02275347. Google Scholar

[16]

A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, 1968; republished as Classics in Appl. Math. 4, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971316. Google Scholar

[17]

R. Fletcher, Practical Methods for Optimization. Vol. 2: Constrained Optimization, John Wiley and Sons, Chichester, 1980. Google Scholar

[18]

A. Forsgren and P. E. Gill, Primal-dual interior methods for nonconvex nonlinear programming, SIAM J. Optim., 8 (1998), 1132-1152. doi: 10.1137/S1052623496305560. Google Scholar

[19]

A. ForsgrenPh. E. Gill and M. H. Wright, Interior methods for nonlinear optimization, SIAM Review, 44 (2002), 525-597. doi: 10.1137/S0036144502414942. Google Scholar

[20]

D. M. Gay, M. L. Overton and M. H. Wright, A primal-dual interior method for nonconvex nonlinear programming, in Advances in Nonlinear Programming, (ed. Y.-X. Yuan), Kluwer Academic Publishers, Dordrecht, 14 (1998), 31-56. doi: 10.1007/978-1-4613-3335-7_2. Google Scholar

[21]

E. M. Gertz and Ph. E. Gill, A primal-dual trust region algorithm for nonlinear optimization, Math. Program., 100 (2004), 49-94. doi: 10.1007/s10107-003-0486-3. Google Scholar

[22]

N. I. M. Gould, D. Orban and Ph. L. Toint, An interior-point $\ell_1$-penalty method for nonlinear optimization, in Recent Developments in Numerical Analysis and Optimization, Proceedings of NAOIII 2014, Springer, Verlag, 134 (2015), 117-150. doi: 10.1007/978-3-319-17689-5_6. Google Scholar

[23]

W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, Lecture Notes in Eco. and Math. Systems 187, Springer-Verlag, Berlin, New York, 1981. doi: 10.1007/BF00934594. Google Scholar

[24]

X.-W. LiuG. Perakis and J. Sun, A robust SQP method for mathematical programs with linear complementarity constraints, Comput. Optim. Appl., 34 (2006), 5-33. doi: 10.1007/s10589-005-3075-y. Google Scholar

[25]

X.-W. Liu and J. Sun, A robust primal-dual interior point algorithm for nonlinear programs, SIAM J. Optim., 14 (2004), 1163-1186. doi: 10.1137/S1052623402400641. Google Scholar

[26]

X.-W. Liu and Y.-X. Yuan, A robust algorithm for optimization with general equality and inequality constraints, SIAM J. Sci. Comput., 22 (2000), 517-534. doi: 10.1137/S1064827598334861. Google Scholar

[27]

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: 10.1007/s10107-009-0272-y. Google Scholar

[28]

J. NocedalF. Öztoprak and R. A. Waltz, An interior point method for nonlinear programming with infeasibility detection capabilities, Optim. Methods Softw., 29 (2014), 837-854. doi: 10.1080/10556788.2013.858156. Google Scholar

[29]

J. Nocedal and S. Wright, Numerical Optimization, Springer-Verlag New York, Inc., 1999. doi: 10.1007/b98874. Google Scholar

[30] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York and London, 1970.
[31]

D. F. Shanno and R. J. Vanderbei, Interior-point methods for nonconvex nonlinear programming: Orderings and higher-order methods, Math. Program., 87 (2000), 303-316. doi: 10.1007/s101070050116. Google Scholar

[32]

P. Tseng, Convergent infeasible interior-point trust-region methods for constrained minimization, SIAM J. Optim., 13 (2002), 432-469. doi: 10.1137/S1052623499357945. Google Scholar

[33]

M. UlbrichS. Ulbrich and L. N. Vicente, A globally convergent primal-dual interior-point filter method for nonlinear programming, Math. Program., 100 (2004), 379-410. doi: 10.1007/s10107-003-0477-4. Google Scholar

[34]

A. Wächter and L. T. Biegler, Failure of global convergence for a class of interior point methods for nonlinear programming, Math. Program., 88 (2000), 565-574. doi: 10.1007/PL00011386. Google Scholar

[35]

A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Motivation and global convergence, SIAM J. Optim., 16 (2005), 1-31. doi: 10.1137/S1052623403426556. Google Scholar

[36]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar

[37]

M. H. Wright, Why a pure primal Newton barrier step may be infeasible?, SIAM J. Optim., 5 (1995), 1-12. doi: 10.1137/0805001. Google Scholar

[38]

S. J. Wright, On the convergence of the Newton/Log-barrier method, Math. Program., 90 (2001), 71-100. doi: 10.1007/PL00011421. Google Scholar

[39]

Y.-X. Yuan, On the convergence of a new trust region algorithm, Numer. Math., 70 (1995), 515-539. doi: 10.1007/s002110050133. Google Scholar

[40]

Y. Zhang, Solving large-scale linear programs by interior-point methods under the MATLAB environment, Optim. Methods Softw., 10 (1998), 1-31. doi: 10.1080/10556789808805699. Google Scholar

Table 1.  Output for test problem (TP1)
$ l $ $ f_l $ $ v_l $ $ \|\phi_l\|_{\infty} $ $ \|\psi_l\|_{\infty} $ $ \beta_l $ $ \rho_l $ $ k $
0 5 16.6132 129.6234 129.6234 0.1000 3.3226 -
1 0.1606 2.0205 4.8082 0.7313 0.1000 0.0972 3
2 -0.0149 2.0002 0.0989 0.0445 0.1000 0.0020 4
3 -0.0036 2.0000 0.0029 0.0018 0.1000 3.1595e-06 3
4 -0.0029 2.0000 3.1674e-06 2.8185e-06 0.1000 1.0000e-09 1
5 0.0018 2.0000 1.0011e-09 6.7212e-10 - - -
$ l $ $ f_l $ $ v_l $ $ \|\phi_l\|_{\infty} $ $ \|\psi_l\|_{\infty} $ $ \beta_l $ $ \rho_l $ $ k $
0 5 16.6132 129.6234 129.6234 0.1000 3.3226 -
1 0.1606 2.0205 4.8082 0.7313 0.1000 0.0972 3
2 -0.0149 2.0002 0.0989 0.0445 0.1000 0.0020 4
3 -0.0036 2.0000 0.0029 0.0018 0.1000 3.1595e-06 3
4 -0.0029 2.0000 3.1674e-06 2.8185e-06 0.1000 1.0000e-09 1
5 0.0018 2.0000 1.0011e-09 6.7212e-10 - - -
Table 2.  Output for test problem (TP2)
$ l $ $ f_l $ $ v_l $ $ \|\phi_l\|_{\infty} $ $ \|\psi_l\|_{\infty} $ $ \beta_l $ $ \rho_l $ $ k $
0 -20 126.6501 2.8052e+03 2.8052e+03 0.1000 6.3325 -
1 -172.5829 172.7978 1.0948e+03 6.2866 0.1000 0.8719 6
2 0.2155 0.7149 1.4269 0.7894 0.1000 0.3895 1
3 -0.1364 0.5550 0.3865 0.3865 0.0100 0.3895 3
4 -0.1416 0.5223 0.2864 0.2648 0.0100 0.1512 1
5 -0.1472 0.5140 0.1446 0.1446 0.0100 0.0209 4
6 -0.1997 0.4472 0.0084 0.0016 0.0100 2.6880e-06 3
7 -0.1999 0.4472 2.4923e-06 2.4923e-06 0.0100 1.0000e-09 1
8 -0.1999 0.4472 9.2732e-10 9.2732e-10 - - -
$ l $ $ f_l $ $ v_l $ $ \|\phi_l\|_{\infty} $ $ \|\psi_l\|_{\infty} $ $ \beta_l $ $ \rho_l $ $ k $
0 -20 126.6501 2.8052e+03 2.8052e+03 0.1000 6.3325 -
1 -172.5829 172.7978 1.0948e+03 6.2866 0.1000 0.8719 6
2 0.2155 0.7149 1.4269 0.7894 0.1000 0.3895 1
3 -0.1364 0.5550 0.3865 0.3865 0.0100 0.3895 3
4 -0.1416 0.5223 0.2864 0.2648 0.0100 0.1512 1
5 -0.1472 0.5140 0.1446 0.1446 0.0100 0.0209 4
6 -0.1997 0.4472 0.0084 0.0016 0.0100 2.6880e-06 3
7 -0.1999 0.4472 2.4923e-06 2.4923e-06 0.0100 1.0000e-09 1
8 -0.1999 0.4472 9.2732e-10 9.2732e-10 - - -
Table 3.  Output for test problem (TP4)
$ l $ $ f_l $ $ v_l $ $ \|\phi_l\|_{\infty} $ $ \|\psi_l\|_{\infty} $ $ \beta_l $ $ \rho_l $ $ k $
0 20 2.8284 9.9557 9.9557 0.1000 1 -
1 0.2305 0.4167 0.8900 0.7008 0.0100 1 4
2 0.1652 0.1687 0.1631 0.0771 0.0100 0.3268 4
3 0.1690 0.1630 0.0503 0.0022 0.0100 4.7328e-06 1
4 0.8561 2.9531e-04 3.1379e-06 3.1379e-06 0.0100 1.0000e-09 14
5 0.9028 1.2372e-04 9.3463e-08 9.3463e-08 - - -
$ l $ $ f_l $ $ v_l $ $ \|\phi_l\|_{\infty} $ $ \|\psi_l\|_{\infty} $ $ \beta_l $ $ \rho_l $ $ k $
0 20 2.8284 9.9557 9.9557 0.1000 1 -
1 0.2305 0.4167 0.8900 0.7008 0.0100 1 4
2 0.1652 0.1687 0.1631 0.0771 0.0100 0.3268 4
3 0.1690 0.1630 0.0503 0.0022 0.0100 4.7328e-06 1
4 0.8561 2.9531e-04 3.1379e-06 3.1379e-06 0.0100 1.0000e-09 14
5 0.9028 1.2372e-04 9.3463e-08 9.3463e-08 - - -
Table 4.  The last $ 4 $ inner iterations corresponding to $ l = 4 $ for test problem (TP4)
$ k $ $ f_k $ $ v_k $ $ \|\phi_k\|_{\infty} $ $ \|\psi_k\|_{\infty} $ $ x_{k1} $ $ x_{k2} $
11 0.8500 5.7136e-04 5.6703e-04 5.6703e-04 1.0780 0.0001
12 0.8548 3.0434e-04 1.2222e-05 1.2222e-05 1.0754 -0.0002
13 0.8556 2.9845e-04 6.2125e-06 6.2125e-06 1.0750 -0.0002
14 0.8561 2.9531e-04 3.1379e-06 3.1379e-06 1.0747 -0.0002
$ k $ $ f_k $ $ v_k $ $ \|\phi_k\|_{\infty} $ $ \|\psi_k\|_{\infty} $ $ x_{k1} $ $ x_{k2} $
11 0.8500 5.7136e-04 5.6703e-04 5.6703e-04 1.0780 0.0001
12 0.8548 3.0434e-04 1.2222e-05 1.2222e-05 1.0754 -0.0002
13 0.8556 2.9845e-04 6.2125e-06 6.2125e-06 1.0750 -0.0002
14 0.8561 2.9531e-04 3.1379e-06 3.1379e-06 1.0747 -0.0002
[1]

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