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doi: 10.3934/jimo.2021171
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## Penalized NCP-functions for nonlinear complementarity problems and a scaling algorithm

 1 School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China 2 College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China

* Corresponding author: Chao Gu

Received  April 2021 Revised  August 2021 Early access October 2021

Fund Project: This work is supported by National Natural Science Foundation of China grant 11971302

In this paper, we systematically study the properties of penalized NCP-functions in derivative-free algorithms for nonlinear complementarity problems (NCPs), and give some regular conditions for stationary points of penalized NCP-functions to be solutions of NCPs. The main contribution is to unify and generalize previous results. Based on one of above penalized NCP-functions, we analyze a scaling algorithm for NCPs. The numerical results show that the scaling can greatly improve the effectiveness of the algorithm.

Citation: Jueyu Wang, Chao Gu, Guoqiang Wang. Penalized NCP-functions for nonlinear complementarity problems and a scaling algorithm. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021171
##### References:
 [1] B. Chen, X. Chen and C. Kanzow, A penalized Fischer-Burmeister NCP-function, Math. Programm., 88 (2000), 211-216.  doi: 10.1007/PL00011375. [2] J.-S. Chen, H.-T. Gao and S. Pan, An $R$-linearly convergent derivative-free algorithm for the NCPs based on the generalized Fischer-Burmeister merit function, J. Comput. Appl. Math., 232 (2009), 455-471.  doi: 10.1016/j.cam.2009.06.022. [3] J.-S. Chen, Z.-H. Huang and C.-Y. She, A new class of penalized NCP-functions and its properties, Comput. Optim. Appl., 50 (2001), 49-73.  doi: 10.1007/s10589-009-9315-9. [4] J.-S. Chen and S. Pan, A family of NCP functions and a descent method for the nonlinear complementarity problem, Comput. Optim. Appl., 40 (2008), 389-404.  doi: 10.1007/s10589-007-9086-0. [5] X. Chi, M. Gowda and J. Tao, The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra, J. Glob. Optim., 73 (2019), 153-169.  doi: 10.1007/s10898-018-0689-z. [6] X. Chi, Y. Wang, Z. Zhu and Z. Wan, Jacobian consistency of a one-parametric class of smoothing Fischer-Burmeister functions for SOCCP, Comput. Appl. Math., 37 (2018), 439-455.  doi: 10.1007/s40314-016-0352-6. [7] X. Chi, Z. Wan, Z. Zhu and L. Yuan, A nonmonotone smoothing Newton method for circular cone programming, Optim., 65 (2016), 2227-2250.  doi: 10.1080/02331934.2016.1217861. [8] T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programm., 75 (1996), 407-439.  doi: 10.1007/BF02592192. [9] S. P. Dirkse and M. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5 (1995), 319-345.  doi: 10.1080/10556789508805619. [10] M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669-713.  doi: 10.1137/S0036144595285963. [11] C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems, Comput. Optim. Appl., 5 (1996), 155-173.  doi: 10.1007/BF00249054. [12] L. Grippo, F. Lampariello and S. Ludidi, A nonmonotone line search technique for Newton's method, SIAM J. Numer. Anal., 23 (1986), 707-716.  doi: 10.1137/0723046. [13] C. Gu, D. Zhu and Y. Pei, A new inexact SQP algorithm for nonlinear systems of mixed equalities and inequalities, Numer. Algor., 78 (2018), 1233-1253.  doi: 10.1007/s11075-017-0421-y. [14] W.-Z. Gu and L.-Y. Lu, The linear convergence of a derivative-free descent method for nonlinear complementarity problems, J. Indust. Manag. Optim., 13 (2017), 531-548.  doi: 10.3934/jimo.2016030. [15] Z. Hao, Z. Wan and X. Chi, A power penalty method for second-order cone nonlinear complementarity problems, J. Comput. Appl. Math., 290 (2015), 136-149.  doi: 10.1016/j.cam.2015.05.007. [16] P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programm., 48 (1990), 161-220.  doi: 10.1007/BF01582255. [17] S.-L. Hu, Z.-H. Huang and J.-S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math., 230 (2009), 69-82.  doi: 10.1016/j.cam.2008.10.056. [18] C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Anal. Theory Methods Appl., 75 (2012), 588-597.  doi: 10.1016/j.na.2011.08.061. [19] C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem, Oper. Res. Lett., 38 (2010), 72-76.  doi: 10.1016/j.orl.2009.09.009. [20] C.-H. Huang, K.-J. Weng, J.-S. Chen, H.-W. Chu and M.-Y. Li, On four discrete-type families of NCP-functions, J. Nonlinear Convex Anal., 20 (2019), 283-306. [21] C. Kanzow and H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl., 11 (1998), 227-251.  doi: 10.1023/A:1026424918464. [22] P.-F. Ma, J.-S. Chen, C.-H. Huang and C.-H. Ko, Discovery of new complementarity functions for NCP and SOCCP, Comput. Appl. Math., 37 (2018), 5727-5749.  doi: 10.1007/s40314-018-0660-0. [23] J.-S. Pang, Complementarity problems, Handbook of Global Optimization, 271–338, Nonconvex Optim. Appl., 2, Kluwer Acad. Publ., Dordrecht, (1995). doi: 10.1007/978-1-4615-2025-2_6. [24] J. M. Peng, Derivative-free methods for monotone variational inequality and complementarity problems, J. Optim. Theory Appl., 99 (1998), 235-252.  doi: 10.1023/A:1021712513685. [25] K. Su and D. Yang, A smooth Newton method with 3-1 piecewise NCP function for generalized nonlinear complementarity problem, Int. J. Comput. Math., 95 (2018), 1703-1713.  doi: 10.1080/00207160.2017.1329531. [26] S. Wang and C.-S. Huang, A power penalty method for solving a nonlinear parabolic complementarity problem, Nonlinear Anal. Theory Methods Appl., 69 (2008), 1125-1137.  doi: 10.1016/j.na.2007.06.014. [27] S. Wang and X. Yang, A power penalty method for a bounded nonlinear complementarity problem, Optim., 64 (2015), 2377-2394.  doi: 10.1080/02331934.2014.967236. [28] S. Wang and X. Yang, A power penalty method for linear complementarity problems, Oper. Res. Lett., 36 (2008), 211-214.  doi: 10.1016/j.orl.2007.06.006. [29] S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, J. Optim. Theory Appl., 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3. [30] S. Wang and K. Zhang, An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering, Optim. Lett., 12 (2018), 1161-1178.  doi: 10.1007/s11590-016-1050-4. [31] K. Yamada, N. Yamashita and M. Fukushima, A new derivative-free descent method for the nonlinear complementarity problems, in: G.D. Pillo, F. Giannessi (Eds.), Nonlinear Optimization and Related Topics, Kluwer Academic Publishers, Netherlands, (2000), 463–487. doi: 10.1007/978-1-4757-3226-9_25. [32] K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, J. Indust. Manag. Optim., 7 (2011), 435-447.  doi: 10.3934/jimo.2011.7.435. [33] J. Zhu, H. Liu, C. Liu and W. Cong, A nonmonotone derivative-free algorithmfor nonlinear complementarity problems based on the new generalized penalized Fischer-Burmeister merit function, Numer. Algor., 58 (2011), 573-591.  doi: 10.1007/s11075-011-9471-8.

show all references

##### References:
 [1] B. Chen, X. Chen and C. Kanzow, A penalized Fischer-Burmeister NCP-function, Math. Programm., 88 (2000), 211-216.  doi: 10.1007/PL00011375. [2] J.-S. Chen, H.-T. Gao and S. Pan, An $R$-linearly convergent derivative-free algorithm for the NCPs based on the generalized Fischer-Burmeister merit function, J. Comput. Appl. Math., 232 (2009), 455-471.  doi: 10.1016/j.cam.2009.06.022. [3] J.-S. Chen, Z.-H. Huang and C.-Y. She, A new class of penalized NCP-functions and its properties, Comput. Optim. Appl., 50 (2001), 49-73.  doi: 10.1007/s10589-009-9315-9. [4] J.-S. Chen and S. Pan, A family of NCP functions and a descent method for the nonlinear complementarity problem, Comput. Optim. Appl., 40 (2008), 389-404.  doi: 10.1007/s10589-007-9086-0. [5] X. Chi, M. Gowda and J. Tao, The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra, J. Glob. Optim., 73 (2019), 153-169.  doi: 10.1007/s10898-018-0689-z. [6] X. Chi, Y. Wang, Z. Zhu and Z. Wan, Jacobian consistency of a one-parametric class of smoothing Fischer-Burmeister functions for SOCCP, Comput. Appl. Math., 37 (2018), 439-455.  doi: 10.1007/s40314-016-0352-6. [7] X. Chi, Z. Wan, Z. Zhu and L. Yuan, A nonmonotone smoothing Newton method for circular cone programming, Optim., 65 (2016), 2227-2250.  doi: 10.1080/02331934.2016.1217861. [8] T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programm., 75 (1996), 407-439.  doi: 10.1007/BF02592192. [9] S. P. Dirkse and M. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5 (1995), 319-345.  doi: 10.1080/10556789508805619. [10] M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669-713.  doi: 10.1137/S0036144595285963. [11] C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems, Comput. Optim. Appl., 5 (1996), 155-173.  doi: 10.1007/BF00249054. [12] L. Grippo, F. Lampariello and S. Ludidi, A nonmonotone line search technique for Newton's method, SIAM J. Numer. Anal., 23 (1986), 707-716.  doi: 10.1137/0723046. [13] C. Gu, D. Zhu and Y. Pei, A new inexact SQP algorithm for nonlinear systems of mixed equalities and inequalities, Numer. Algor., 78 (2018), 1233-1253.  doi: 10.1007/s11075-017-0421-y. [14] W.-Z. Gu and L.-Y. Lu, The linear convergence of a derivative-free descent method for nonlinear complementarity problems, J. Indust. Manag. Optim., 13 (2017), 531-548.  doi: 10.3934/jimo.2016030. [15] Z. Hao, Z. Wan and X. Chi, A power penalty method for second-order cone nonlinear complementarity problems, J. Comput. Appl. Math., 290 (2015), 136-149.  doi: 10.1016/j.cam.2015.05.007. [16] P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programm., 48 (1990), 161-220.  doi: 10.1007/BF01582255. [17] S.-L. Hu, Z.-H. Huang and J.-S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math., 230 (2009), 69-82.  doi: 10.1016/j.cam.2008.10.056. [18] C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Anal. Theory Methods Appl., 75 (2012), 588-597.  doi: 10.1016/j.na.2011.08.061. [19] C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem, Oper. Res. Lett., 38 (2010), 72-76.  doi: 10.1016/j.orl.2009.09.009. [20] C.-H. Huang, K.-J. Weng, J.-S. Chen, H.-W. Chu and M.-Y. Li, On four discrete-type families of NCP-functions, J. Nonlinear Convex Anal., 20 (2019), 283-306. [21] C. Kanzow and H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl., 11 (1998), 227-251.  doi: 10.1023/A:1026424918464. [22] P.-F. Ma, J.-S. Chen, C.-H. Huang and C.-H. Ko, Discovery of new complementarity functions for NCP and SOCCP, Comput. Appl. Math., 37 (2018), 5727-5749.  doi: 10.1007/s40314-018-0660-0. [23] J.-S. Pang, Complementarity problems, Handbook of Global Optimization, 271–338, Nonconvex Optim. Appl., 2, Kluwer Acad. Publ., Dordrecht, (1995). doi: 10.1007/978-1-4615-2025-2_6. [24] J. M. Peng, Derivative-free methods for monotone variational inequality and complementarity problems, J. Optim. Theory Appl., 99 (1998), 235-252.  doi: 10.1023/A:1021712513685. [25] K. Su and D. Yang, A smooth Newton method with 3-1 piecewise NCP function for generalized nonlinear complementarity problem, Int. J. Comput. Math., 95 (2018), 1703-1713.  doi: 10.1080/00207160.2017.1329531. [26] S. Wang and C.-S. Huang, A power penalty method for solving a nonlinear parabolic complementarity problem, Nonlinear Anal. Theory Methods Appl., 69 (2008), 1125-1137.  doi: 10.1016/j.na.2007.06.014. [27] S. Wang and X. Yang, A power penalty method for a bounded nonlinear complementarity problem, Optim., 64 (2015), 2377-2394.  doi: 10.1080/02331934.2014.967236. [28] S. Wang and X. Yang, A power penalty method for linear complementarity problems, Oper. Res. Lett., 36 (2008), 211-214.  doi: 10.1016/j.orl.2007.06.006. [29] S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, J. Optim. Theory Appl., 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3. [30] S. Wang and K. Zhang, An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering, Optim. Lett., 12 (2018), 1161-1178.  doi: 10.1007/s11590-016-1050-4. [31] K. Yamada, N. Yamashita and M. Fukushima, A new derivative-free descent method for the nonlinear complementarity problems, in: G.D. Pillo, F. Giannessi (Eds.), Nonlinear Optimization and Related Topics, Kluwer Academic Publishers, Netherlands, (2000), 463–487. doi: 10.1007/978-1-4757-3226-9_25. [32] K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, J. Indust. Manag. Optim., 7 (2011), 435-447.  doi: 10.3934/jimo.2011.7.435. [33] J. Zhu, H. Liu, C. Liu and W. Cong, A nonmonotone derivative-free algorithmfor nonlinear complementarity problems based on the new generalized penalized Fischer-Burmeister merit function, Numer. Algor., 58 (2011), 573-591.  doi: 10.1007/s11075-011-9471-8.
Performance profile for bertsekas(1)
Performance profile for colvdual(1)
Performance profile for colvnlp(2)
Performance profile for gafni(1)
Performance profile for kojshin(1)
Performance profile for sppe(1)
Performance profile on numbers of iterations
Performance profile on numbers of function evaluations
Performance profile on NIT (monotone vs. nonmonotone)
Performance profile on NF (monotone vs. nonmonotone)
Algorithm 1 with $p = 1.1$
 Algorithm 1 ($\delta = 1$) Algorithm 1 ($\delta = \delta^{*}$) Problem NIT NF $\Psi_{\alpha, \theta, p}$ NIT NF $\Psi_{\alpha, \theta, p}$ $\delta^{*}$ bertsekas(1) 8266 24367 2.3812e-009 12160 12170 4.9639e-007 27 bertsekas(2) 7891 20336 9.9677e-007 11695 11702 9.1309e-007 26 bertsekas(3) - - - - - - - colvdual(1) - - - 17306 55349 5.5376e-009 14 colvdual(2) - - - - - - - colvnlp(1) - - - - - - - colvnlp(2) - - - 4577 7154 5.6383e-009 30 cycle 10 11 3.0849e-009 10 11 3.0849e-009 1 explcp 75 90 1.4837e-007 75 90 1.4837e-007 1 gafni(1) 34569 121975 9.1296e-008 112 157 7.5602e-007 15 gafni(2) 47404 167978 1.0529e-007 73 102 4.0892e-007 8 gafni(3) 51991 184361 1.0342e-007 118 165 5.0671e-007 12 hanskoop(1) - - - - - - - hanskoop(2) - - - - - - - hanskoop(3) - - - 555 556 9.9791e-007 18 hanskoop(4) - - - - - - - josephy(1) 97 176 1.3794e-007 80 82 5.2841e-007 3 josephy(2) 266 443 8.8353e-008 59 62 3.7396e-008 2 josephy(3) 1541 3081 2.4551e-008 290 466 6.2118e-007 5 josephy(4) 789 1580 1.3789e-008 32 35 3.1777e-007 2 josephy(5) 253 426 8.8353e-008 29 31 3.4038e-007 3 josephy(6) 1241 2484 2.4521e-008 73 90 9.4225e-007 2 kojshin(1) 1125 2250 2.6963e-008 64 74 1.8273e-009 2 kojshin(2) - - - - - - - kojshin(3) - - - 363 584 3.0811e-007 7 kojshin(4) 193 331 2.5809e-008 51 54 1.8472e-009 2 kojshin(5) 1012 2026 2.6726e-008 101 103 4.1455e-007 4 kojshin(6) 215 431 7.5751e-010 49 51 2.7706e-008 7 mathinum(1) - - - - - - - mathinum(2) 122 130 2.1945e-008 122 130 2.1945e-008 1 mathinum(3) - - - 557 600 3.3811e-008 8 mathinum(4) - - - - - - - mathisum(1) 872 1716 1.1988e-007 379 404 6.0656e-007 5 mathisum(2) 266 512 4.1344e-007 339 400 9.4181e-007 4 mathisum(3) - - - 303 388 7.7901e-007 3 mathisum(4) - - - 340 386 7.6970e-007 4 nash(1) 130 346 5.8724e-007 52 62 7.3456e-007 16 nash(2) 500000 2426781 9.4354e-004 57 64 9.9217e-007 16 sppe(1) 26463 122854 9.7144e-007 2491 2591 8.6671e-007 22 sppe(2) 23832 105901 4.9202e-007 2431 2528 9.3475e-007 22 tobin(1) 1510 3393 6.8669e-007 484 491 2.8497e-009 10 tobin(2) 1836 4126 7.5978e-007 543 549 6.4011e-008 10
 Algorithm 1 ($\delta = 1$) Algorithm 1 ($\delta = \delta^{*}$) Problem NIT NF $\Psi_{\alpha, \theta, p}$ NIT NF $\Psi_{\alpha, \theta, p}$ $\delta^{*}$ bertsekas(1) 8266 24367 2.3812e-009 12160 12170 4.9639e-007 27 bertsekas(2) 7891 20336 9.9677e-007 11695 11702 9.1309e-007 26 bertsekas(3) - - - - - - - colvdual(1) - - - 17306 55349 5.5376e-009 14 colvdual(2) - - - - - - - colvnlp(1) - - - - - - - colvnlp(2) - - - 4577 7154 5.6383e-009 30 cycle 10 11 3.0849e-009 10 11 3.0849e-009 1 explcp 75 90 1.4837e-007 75 90 1.4837e-007 1 gafni(1) 34569 121975 9.1296e-008 112 157 7.5602e-007 15 gafni(2) 47404 167978 1.0529e-007 73 102 4.0892e-007 8 gafni(3) 51991 184361 1.0342e-007 118 165 5.0671e-007 12 hanskoop(1) - - - - - - - hanskoop(2) - - - - - - - hanskoop(3) - - - 555 556 9.9791e-007 18 hanskoop(4) - - - - - - - josephy(1) 97 176 1.3794e-007 80 82 5.2841e-007 3 josephy(2) 266 443 8.8353e-008 59 62 3.7396e-008 2 josephy(3) 1541 3081 2.4551e-008 290 466 6.2118e-007 5 josephy(4) 789 1580 1.3789e-008 32 35 3.1777e-007 2 josephy(5) 253 426 8.8353e-008 29 31 3.4038e-007 3 josephy(6) 1241 2484 2.4521e-008 73 90 9.4225e-007 2 kojshin(1) 1125 2250 2.6963e-008 64 74 1.8273e-009 2 kojshin(2) - - - - - - - kojshin(3) - - - 363 584 3.0811e-007 7 kojshin(4) 193 331 2.5809e-008 51 54 1.8472e-009 2 kojshin(5) 1012 2026 2.6726e-008 101 103 4.1455e-007 4 kojshin(6) 215 431 7.5751e-010 49 51 2.7706e-008 7 mathinum(1) - - - - - - - mathinum(2) 122 130 2.1945e-008 122 130 2.1945e-008 1 mathinum(3) - - - 557 600 3.3811e-008 8 mathinum(4) - - - - - - - mathisum(1) 872 1716 1.1988e-007 379 404 6.0656e-007 5 mathisum(2) 266 512 4.1344e-007 339 400 9.4181e-007 4 mathisum(3) - - - 303 388 7.7901e-007 3 mathisum(4) - - - 340 386 7.6970e-007 4 nash(1) 130 346 5.8724e-007 52 62 7.3456e-007 16 nash(2) 500000 2426781 9.4354e-004 57 64 9.9217e-007 16 sppe(1) 26463 122854 9.7144e-007 2491 2591 8.6671e-007 22 sppe(2) 23832 105901 4.9202e-007 2431 2528 9.3475e-007 22 tobin(1) 1510 3393 6.8669e-007 484 491 2.8497e-009 10 tobin(2) 1836 4126 7.5978e-007 543 549 6.4011e-008 10
Algorithm 1 with $p = 1.5$
 Algorithm 1 ($\delta = 1$) Algorithm 1 ($\delta = \delta^{*}$) Problem NIT NF $\Psi_{\alpha, \theta, p}$ NIT NF $\Psi_{\alpha, \theta, p}$ $\delta^{*}$ bertsekas(1) 26767 89347 3.5032e-007 20934 30729 6.1796e-011 16 bertsekas(2) 25115 83355 3.5032e-007 20864 30647 6.1708e-011 16 bertsekas(3) - - - - - - - colvdual(1) - - - 21590 67480 3.3568e-009 10 colvdual(2) - - - - - - - colvnlp(1) - - - - - - - colvnlp(2) - - - 3474 7716 3.3348e-009 9 cycle 8 9 1.1847e-009 4 5 3.1656e-010 2 explcp 26 46 2.5022e-007 12 14 3.0423e-007 3 gafni(1) 25223 92784 2.0520e-008 51 84 3.2553e-007 27 gafni(2) 29772 108923 3.7445e-008 59 98 3.0945e-007 27 gafni(3) 24233 87773 6.3577e-007 38 63 7.9123e-007 30 hanskoop(1) - - - - - - - hanskoop(2) 379 1022 6.4234e-007 139 253 5.7983e-007 2 hanskoop(3) 21 23 1.5259e-007 21 23 1.5259e-007 1 hanskoop(4) 23 32 1.8634e-007 24 25 6.3337e-008 2 josephy(1) 359 720 3.9772e-008 13 15 9.7292e-007 6 josephy(2) 420 841 3.4525e-008 23 24 8.8124e-007 6 josephy(3) 764 1468 2.2183e-007 328 590 9.5774e-007 30 josephy(4) 423 773 2.2183e-007 26 28 1.5330e-007 5 josephy(5) 408 748 2.2183e-007 14 16 9.7662e-007 6 josephy(6) 420 842 3.4825e-008 19 21 1.0946e-007 8 kojshin(1) 350 702 2.7028e-007 85 87 6.1427e-008 6 kojshin(2) - - - - - - - kojshin(3) 708 1412 2.6187e-007 370 644 9.5223e-007 26 kojshin(4) 278 470 5.2362e-008 44 49 6.1203e-008 5 kojshin(5) 322 646 2.6572e-007 112 113 5.9773e-008 11 kojshin(6) 42 81 2.2097e-007 12 15 2.5547e-007 5 mathinum(1) 173 292 9.5496e-007 62 78 4.9068e-007 3 mathinum(2) 92 129 8.1284e-007 50 65 1.1306e-007 3 mathinum(3) - - - 60 84 5.1237e-007 2 mathinum(4) - - - 99 107 3.3206e-007 4 mathisum(1) - - - 140 195 6.6254e-008 4 mathisum(2) - - - 178 267 7.5195e-008 3 mathisum(3) - - - 234 252 7.1548e-008 9 mathisum(4) - - - 175 220 6.6787e-008 4 nash(1) 324 1021 3.0684e-007 121 172 1.7734e-008 15 nash(2) 432702 1796231 4.9703e-011 139 192 7.6530e-007 19 sppe(1) 7514 28762 2.4311e-008 2155 2582 4.0753e-009 27 sppe(2) 7532 28254 7.6481e-009 2110 2551 3.2455e-009 28 tobin(1) 138 296 5.3730e-007 91 100 2.0161e-009 5 tobin(2) 380 786 9.9524e-007 78 88 4.8676e-008 5
 Algorithm 1 ($\delta = 1$) Algorithm 1 ($\delta = \delta^{*}$) Problem NIT NF $\Psi_{\alpha, \theta, p}$ NIT NF $\Psi_{\alpha, \theta, p}$ $\delta^{*}$ bertsekas(1) 26767 89347 3.5032e-007 20934 30729 6.1796e-011 16 bertsekas(2) 25115 83355 3.5032e-007 20864 30647 6.1708e-011 16 bertsekas(3) - - - - - - - colvdual(1) - - - 21590 67480 3.3568e-009 10 colvdual(2) - - - - - - - colvnlp(1) - - - - - - - colvnlp(2) - - - 3474 7716 3.3348e-009 9 cycle 8 9 1.1847e-009 4 5 3.1656e-010 2 explcp 26 46 2.5022e-007 12 14 3.0423e-007 3 gafni(1) 25223 92784 2.0520e-008 51 84 3.2553e-007 27 gafni(2) 29772 108923 3.7445e-008 59 98 3.0945e-007 27 gafni(3) 24233 87773 6.3577e-007 38 63 7.9123e-007 30 hanskoop(1) - - - - - - - hanskoop(2) 379 1022 6.4234e-007 139 253 5.7983e-007 2 hanskoop(3) 21 23 1.5259e-007 21 23 1.5259e-007 1 hanskoop(4) 23 32 1.8634e-007 24 25 6.3337e-008 2 josephy(1) 359 720 3.9772e-008 13 15 9.7292e-007 6 josephy(2) 420 841 3.4525e-008 23 24 8.8124e-007 6 josephy(3) 764 1468 2.2183e-007 328 590 9.5774e-007 30 josephy(4) 423 773 2.2183e-007 26 28 1.5330e-007 5 josephy(5) 408 748 2.2183e-007 14 16 9.7662e-007 6 josephy(6) 420 842 3.4825e-008 19 21 1.0946e-007 8 kojshin(1) 350 702 2.7028e-007 85 87 6.1427e-008 6 kojshin(2) - - - - - - - kojshin(3) 708 1412 2.6187e-007 370 644 9.5223e-007 26 kojshin(4) 278 470 5.2362e-008 44 49 6.1203e-008 5 kojshin(5) 322 646 2.6572e-007 112 113 5.9773e-008 11 kojshin(6) 42 81 2.2097e-007 12 15 2.5547e-007 5 mathinum(1) 173 292 9.5496e-007 62 78 4.9068e-007 3 mathinum(2) 92 129 8.1284e-007 50 65 1.1306e-007 3 mathinum(3) - - - 60 84 5.1237e-007 2 mathinum(4) - - - 99 107 3.3206e-007 4 mathisum(1) - - - 140 195 6.6254e-008 4 mathisum(2) - - - 178 267 7.5195e-008 3 mathisum(3) - - - 234 252 7.1548e-008 9 mathisum(4) - - - 175 220 6.6787e-008 4 nash(1) 324 1021 3.0684e-007 121 172 1.7734e-008 15 nash(2) 432702 1796231 4.9703e-011 139 192 7.6530e-007 19 sppe(1) 7514 28762 2.4311e-008 2155 2582 4.0753e-009 27 sppe(2) 7532 28254 7.6481e-009 2110 2551 3.2455e-009 28 tobin(1) 138 296 5.3730e-007 91 100 2.0161e-009 5 tobin(2) 380 786 9.9524e-007 78 88 4.8676e-008 5
Algorithm 1 with p = 5
 Algorithm 1 ($\delta = 1$) Algorithm 1 ($\delta = \delta^{*}$) Problem NIT NF $\Psi_{\alpha, \theta, p}$ NIT NF $\Psi_{\alpha, \theta, p}$ $\delta^{*}$ bertsekas(1) - - - - - - - bertsekas(2) - - - - - - - bertsekas(3) - - - 1384 2099 1.4364e-009 22 colvdual(1) - - - 22642 70980 3.3306e-009 10 colvdual(2) - - - 33701 94313 3.3442e-009 30 colvnlp(1) - - - 2128 3941 7.9056e-009 29 colvnlp(2) - - - 3831 8685 3.7521e-009 10 cycle 4 5 5.2542e-019 4 5 5.2542e-019 1 explcp 38 74 2.0424e-008 11 14 2.0714e-007 2 gafni(1) 19223 67755 2.0788e-008 130 249 9.9662e-007 30 gafni(2) 21722 76890 3.5635e-008 99 224 6.5953e-007 11 gafni(3) 20366 72489 3.5055e-008 341 683 1.0469e-007 19 hanskoop(1) 26 43 2.3821e-010 27 33 3.7916e-007 4 hanskoop(2) 75 127 7.0665e-010 23 24 8.5594e-009 4 hanskoop(3) 5 7 5.3588e-007 5 7 5.3588e-007 1 hanskoop(4) 47 88 2.2551e-012 11 12 1.6245e-013 4 josephy(1) 30 61 2.8350e-007 8 10 2.8479e-007 11 josephy(2) 365 731 2.7223e-008 17 19 5.2938e-007 6 josephy(3) 32 66 2.8351e-007 10 14 2.8479e-007 11 josephy(4) 321 644 2.4569e-008 18 19 7.2910e-007 6 josephy(5) 24 51 2.3684e-007 16 18 9.2798e-007 6 josephy(6) 53 106 1.8162e-007 17 19 6.1620e-007 6 kojshin(1) 292 586 3.1499e-007 67 69 5.3961e-008 7 kojshin(2) 311 623 3.0988e-007 57 68 4.6426e-008 5 kojshin(3) 294 591 3.1499e-007 54 63 4.7443e-008 5 kojshin(4) - - - 46 50 5.1553e-008 5 kojshin(5) - - - 119 120 6.1836e-008 13 kojshin(6) 53 102 3.3305e-008 43 45 6.5165e-008 5 mathinum(1) 75 125 8.7344e-007 42 53 1.6740e-007 3 mathinum(2) 40 55 9.0882e-007 47 48 2.6675e-007 5 mathinum(3) 85 141 1.0574e-007 40 41 9.8855e-007 6 mathinum(4) 77 119 2.0889e-007 47 58 4.0341e-007 3 mathisum(1) - - - - - - mathisum(2) - - - 424 578 7.2195e-008 14 mathisum(3) - - - 580 736 7.1810e-008 21 mathisum(4) - - - - - - - nash(1) 149 470 5.1348e-007 139 218 4.0121e-009 13 nash(2) 24 48 1.6145e-007 17 32 1.1904e-016 2 sppe(1) 10560 43706 1.5941e-008 2252 2606 2.5723e-009 29 sppe(2) 7731 29014 1.4354e-008 2121 2459 3.1054e-009 29 tobin(1) 343 701 1.8828e-007 109 112 1.2528e-009 6 tobin(2) 343 695 9.9819e-007 96 98 1.0439e-007 8
 Algorithm 1 ($\delta = 1$) Algorithm 1 ($\delta = \delta^{*}$) Problem NIT NF $\Psi_{\alpha, \theta, p}$ NIT NF $\Psi_{\alpha, \theta, p}$ $\delta^{*}$ bertsekas(1) - - - - - - - bertsekas(2) - - - - - - - bertsekas(3) - - - 1384 2099 1.4364e-009 22 colvdual(1) - - - 22642 70980 3.3306e-009 10 colvdual(2) - - - 33701 94313 3.3442e-009 30 colvnlp(1) - - - 2128 3941 7.9056e-009 29 colvnlp(2) - - - 3831 8685 3.7521e-009 10 cycle 4 5 5.2542e-019 4 5 5.2542e-019 1 explcp 38 74 2.0424e-008 11 14 2.0714e-007 2 gafni(1) 19223 67755 2.0788e-008 130 249 9.9662e-007 30 gafni(2) 21722 76890 3.5635e-008 99 224 6.5953e-007 11 gafni(3) 20366 72489 3.5055e-008 341 683 1.0469e-007 19 hanskoop(1) 26 43 2.3821e-010 27 33 3.7916e-007 4 hanskoop(2) 75 127 7.0665e-010 23 24 8.5594e-009 4 hanskoop(3) 5 7 5.3588e-007 5 7 5.3588e-007 1 hanskoop(4) 47 88 2.2551e-012 11 12 1.6245e-013 4 josephy(1) 30 61 2.8350e-007 8 10 2.8479e-007 11 josephy(2) 365 731 2.7223e-008 17 19 5.2938e-007 6 josephy(3) 32 66 2.8351e-007 10 14 2.8479e-007 11 josephy(4) 321 644 2.4569e-008 18 19 7.2910e-007 6 josephy(5) 24 51 2.3684e-007 16 18 9.2798e-007 6 josephy(6) 53 106 1.8162e-007 17 19 6.1620e-007 6 kojshin(1) 292 586 3.1499e-007 67 69 5.3961e-008 7 kojshin(2) 311 623 3.0988e-007 57 68 4.6426e-008 5 kojshin(3) 294 591 3.1499e-007 54 63 4.7443e-008 5 kojshin(4) - - - 46 50 5.1553e-008 5 kojshin(5) - - - 119 120 6.1836e-008 13 kojshin(6) 53 102 3.3305e-008 43 45 6.5165e-008 5 mathinum(1) 75 125 8.7344e-007 42 53 1.6740e-007 3 mathinum(2) 40 55 9.0882e-007 47 48 2.6675e-007 5 mathinum(3) 85 141 1.0574e-007 40 41 9.8855e-007 6 mathinum(4) 77 119 2.0889e-007 47 58 4.0341e-007 3 mathisum(1) - - - - - - mathisum(2) - - - 424 578 7.2195e-008 14 mathisum(3) - - - 580 736 7.1810e-008 21 mathisum(4) - - - - - - - nash(1) 149 470 5.1348e-007 139 218 4.0121e-009 13 nash(2) 24 48 1.6145e-007 17 32 1.1904e-016 2 sppe(1) 10560 43706 1.5941e-008 2252 2606 2.5723e-009 29 sppe(2) 7731 29014 1.4354e-008 2121 2459 3.1054e-009 29 tobin(1) 343 701 1.8828e-007 109 112 1.2528e-009 6 tobin(2) 343 695 9.9819e-007 96 98 1.0439e-007 8
Algorithm 1 with monotone line search
 $p=1.1$ $p=1.5$ $p=5$ Problem NIT NF NIT NF NIT NF bertsekas(1) 12125 41027 21659 74329 - - bertsekas(2) - - 22409 78381 - - bertsekas(3) - - - - - - colvdual(1) - - - - - - colvdual(2) - - - - - - colvnlp(1) - - - - - - colvnlp(2) - - - - - - cycle 10 11 8 9 4 5 explcp 80 100 27 48 39 76 gafni(1) 304691 1207169 24788 89141 25304 91005 gafni(2) 144756 557579 25562 91844 25020 89860 gafni(3) 269424 1058494 26240 94133 25002 89336 hanskoop(1) - - - - - - hanskoop(2) - - - - - - hanskoop(3) - - - - 5 7 hanskoop(4) - - - - 16 27 josephy(1) 111 197 376 751 - - josephy(2) 1082 2160 420 841 365 731 josephy(3) 1602 3204 446 891 - - josephy(4) 29 49 98 194 374 749 josephy(5) 20 33 30 59 26 54 josephy(6) 1241 2484 420 842 51 102 kojshin(1) - - 476 953 24 46 kojshin(2) - - - - - - kojshin(3) - - - - - - kojshin(4) 22 31 35 68 18 35 kojshin(5) 1010 2020 288 577 274 550 kojshin(6) - - 46 91 48 96 mathinum(1) - - 158 358 81 176 mathinum(2) - - 116 236 88 230 mathinum(3) - - - - 122 323 mathinum(4) - - - - 101 237 mathisum(1) - - - - - - mathisum(2) - - - - - - mathisum(3) 1372 2701 - - - - mathisum(4) - - - - - - nash(1) 500000 2560023 250 502 73 149 nash(2) 500000 2678901 432702 1796231 24 48 sppe(1) 7828 26193 6076 22833 5156 18401 sppe(2) 7887 26310 7178 27719 5081 18251 tobin(1) 2166 4875 395 801 461 930 tobin(2) 2524 5702 477 972 493 993
 $p=1.1$ $p=1.5$ $p=5$ Problem NIT NF NIT NF NIT NF bertsekas(1) 12125 41027 21659 74329 - - bertsekas(2) - - 22409 78381 - - bertsekas(3) - - - - - - colvdual(1) - - - - - - colvdual(2) - - - - - - colvnlp(1) - - - - - - colvnlp(2) - - - - - - cycle 10 11 8 9 4 5 explcp 80 100 27 48 39 76 gafni(1) 304691 1207169 24788 89141 25304 91005 gafni(2) 144756 557579 25562 91844 25020 89860 gafni(3) 269424 1058494 26240 94133 25002 89336 hanskoop(1) - - - - - - hanskoop(2) - - - - - - hanskoop(3) - - - - 5 7 hanskoop(4) - - - - 16 27 josephy(1) 111 197 376 751 - - josephy(2) 1082 2160 420 841 365 731 josephy(3) 1602 3204 446 891 - - josephy(4) 29 49 98 194 374 749 josephy(5) 20 33 30 59 26 54 josephy(6) 1241 2484 420 842 51 102 kojshin(1) - - 476 953 24 46 kojshin(2) - - - - - - kojshin(3) - - - - - - kojshin(4) 22 31 35 68 18 35 kojshin(5) 1010 2020 288 577 274 550 kojshin(6) - - 46 91 48 96 mathinum(1) - - 158 358 81 176 mathinum(2) - - 116 236 88 230 mathinum(3) - - - - 122 323 mathinum(4) - - - - 101 237 mathisum(1) - - - - - - mathisum(2) - - - - - - mathisum(3) 1372 2701 - - - - mathisum(4) - - - - - - nash(1) 500000 2560023 250 502 73 149 nash(2) 500000 2678901 432702 1796231 24 48 sppe(1) 7828 26193 6076 22833 5156 18401 sppe(2) 7887 26310 7178 27719 5081 18251 tobin(1) 2166 4875 395 801 461 930 tobin(2) 2524 5702 477 972 493 993
Algorithm 1 with nonmonotone line search (p = 5)
 $M=5, s=1$ $M=5, s=5$ $M=10, s=1$ $M=10, s=5$ Problem NIT NF NIT NF NIT NF NIT NF bertsekas(1) - - - - - - - - bertsekas(2) - - - - 26236 90444 20236 90444 bertsekas(3) - - - - - - - - colvdual(1) - - - - - - - - colvdual(2) - - - - - - - - colvnlp(1) - - - - - - - - colvnlp(2) - - - - - - - - cycle 4 5 4 5 4 5 4 5 explcp 38 74 38 74 38 74 38 74 gafni(1) - - 19223 67755 7250 25114 4346 15115 gafni(2) 21722 76890 21722 76890 28597 104363 28597 104363 gafni(3) 20366 72489 20366 72489 27454 100831 27454 100831 hanskoop(1) - - 26 43 161 287 171 434 hanskoop(2) - - 75 127 175 341 - - hanskoop(3) 13 16 5 7 11 12 5 7 hanskoop(4) 17 24 47 88 - - 46 86 josephy(1) 67 124 30 61 - - 435 819 josephy(2) 365 731 365 731 365 731 365 731 josephy(3) 167 295 32 66 990 1820 523 985 josephy(4) 36 71 321 644 872 1629 321 644 josephy(5) 45 87 24 51 1058 2009 651 1225 josephy(6) 53 106 53 106 513 967 513 967 kojshin(1) 67 126 292 586 959 1647 292 586 kojshin(2) 311 623 311 623 311 623 311 623 kojshin(3) 138 237 294 591 1227 2091 294 591 kojshin(4) 69 122 - - 913 1552 - - kojshin(5) 75 135 - - 936 1594 - - kojshin(6) 106 202 53 102 932 1604 913 1557 mathinum(1) 69 111 75 125 76 119 123 192 mathinum(2) 77 110 40 55 106 154 40 55 mathinum(3) 116 199 85 141 259 457 128 203 mathinum(4) 86 134 77 119 203 340 94 142 mathisum(1) - - - - - - - - mathisum(2) - - - - - - - - mathisum(3) - - - - - - - - mathisum(4) - - - - - - - - nash(1) - - 149 470 - - 1163 3868 nash(2) 24 48 24 48 71 150 24 48 sppe(1) 9429 38110 10560 43706 28627 140093 27303 130659 sppe(2) 7822 28858 7731 29014 29515 144976 26991 129993 tobin(1) 377 771 343 701 355 728 330 667 tobin(2) 147 297 343 695 417 871 350 704
 $M=5, s=1$ $M=5, s=5$ $M=10, s=1$ $M=10, s=5$ Problem NIT NF NIT NF NIT NF NIT NF bertsekas(1) - - - - - - - - bertsekas(2) - - - - 26236 90444 20236 90444 bertsekas(3) - - - - - - - - colvdual(1) - - - - - - - - colvdual(2) - - - - - - - - colvnlp(1) - - - - - - - - colvnlp(2) - - - - - - - - cycle 4 5 4 5 4 5 4 5 explcp 38 74 38 74 38 74 38 74 gafni(1) - - 19223 67755 7250 25114 4346 15115 gafni(2) 21722 76890 21722 76890 28597 104363 28597 104363 gafni(3) 20366 72489 20366 72489 27454 100831 27454 100831 hanskoop(1) - - 26 43 161 287 171 434 hanskoop(2) - - 75 127 175 341 - - hanskoop(3) 13 16 5 7 11 12 5 7 hanskoop(4) 17 24 47 88 - - 46 86 josephy(1) 67 124 30 61 - - 435 819 josephy(2) 365 731 365 731 365 731 365 731 josephy(3) 167 295 32 66 990 1820 523 985 josephy(4) 36 71 321 644 872 1629 321 644 josephy(5) 45 87 24 51 1058 2009 651 1225 josephy(6) 53 106 53 106 513 967 513 967 kojshin(1) 67 126 292 586 959 1647 292 586 kojshin(2) 311 623 311 623 311 623 311 623 kojshin(3) 138 237 294 591 1227 2091 294 591 kojshin(4) 69 122 - - 913 1552 - - kojshin(5) 75 135 - - 936 1594 - - kojshin(6) 106 202 53 102 932 1604 913 1557 mathinum(1) 69 111 75 125 76 119 123 192 mathinum(2) 77 110 40 55 106 154 40 55 mathinum(3) 116 199 85 141 259 457 128 203 mathinum(4) 86 134 77 119 203 340 94 142 mathisum(1) - - - - - - - - mathisum(2) - - - - - - - - mathisum(3) - - - - - - - - mathisum(4) - - - - - - - - nash(1) - - 149 470 - - 1163 3868 nash(2) 24 48 24 48 71 150 24 48 sppe(1) 9429 38110 10560 43706 28627 140093 27303 130659 sppe(2) 7822 28858 7731 29014 29515 144976 26991 129993 tobin(1) 377 771 343 701 355 728 330 667 tobin(2) 147 297 343 695 417 871 350 704
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