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April  2019, 15(2): 757-774. doi: 10.3934/jimo.2018069

A proximal-projection partial bundle method for convex constrained minimax problems

Chunming Tang 1, , Jinbao Jian 2,3,, and  Guoyin Li 4,

1. 

College of Mathematics and Information Science, Guangxi University, Nanning 530004, China
2. 

College of Science, Guangxi University of Nationalities, Nanning 530007, China
3. 

School of Mathematics and Statistics, Yulin Normal University, Yulin 537000, China
4. 

Department of Applied Mathematics, University of New South Wales, Kensington, 2052, Sydney, Australia

* Corresponding author: Jinbao Jian

Received  September 2016 Revised  January 2018 Published  April 2019 Early access  June 2018

Fund Project: Project supported by the National Natural Science Foundation (11761013, 11771383) and Guangxi Natural Science Foundation (2013GXNSFAA019013, 2014GXNSFFA118001, 2016GXNSFDA380019) of China. This work was essentially done while the first author was visiting The University of New South Wales, Australia.

In this paper, we propose a partial bundle method for a convex constrained minimax problem where the objective function is expressed as maximum of finitely many convex (not necessarily differentiable) functions. To avoid complete evaluation of all component functions of the objective, a partial cutting-planes model is adopted instead of the traditional one. Based on the proximal-projection idea, at each iteration, an unconstrained proximal subproblem is solved first to generate an aggregate linear model of the objective function, and then another subproblem based on this model is solved to obtain a trial point. Moreover, a new descent test criterion is proposed, which can not only simplify the presentation of the algorithm, but also improve the numerical performance significantly. An explicit upper bound for the number of bundle resets is also derived. Global convergence of the algorithm is established, and some preliminary numerical results show that our method is very encouraging.

Keywords: Minimax problem, partial bundle method, proximal-projection, descent test criterion, global convergence.
Mathematics Subject Classification: Primary: 90C47, 49K35.
Citation: Chunming Tang, Jinbao Jian, Guoyin Li. A proximal-projection partial bundle method for convex constrained minimax problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 757-774. doi: 10.3934/jimo.2018069
References:
[1]

A. Baums, Minimax method in optimizing energy consumption in real-time embedded systems, Automatic Control and Computer Sciences, 43 (2009), 57-62.  doi: 10.3103/S0146411609020011.

[2]

J. F. Bonnans, J. C. Gilbert, C. Lemaréchal and C. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, Second ed. Springer-Verlag, Berlin Heidelberg New York, 2006.

[3]

J. V. Burke, A. S. Lewis and M. L. Overton, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM Journal on Optimization, 15 (2005), 751-779.  doi: 10.1137/030601296.

[4]

F. L. Chernousko, Minimax control for a class of linear systems subject to disturbances, Journal of Optimization Theory and Applications, 127 (2005), 535-548.  doi: 10.1007/s10957-005-7501-1.

[5]

J. Dattorro, Convex Optimization † Euclidean Distance Geometry, second edn, $ \mathcal{M}\varepsilonβ oo$, 2015.

[6]

G. Di Pillo, L. Grippo and S. Lucidi, A smooth method for the finite minimax problem, Mathematical Programming, 60 (1993), 187-214.  doi: 10.1007/BF01580609.

[7]

A. Fuduli, M. Gaudioso, G. Giallombardo and G. Miglionico, A partially inexact bundle method for convex semi-infinite minmax problems, Communications in Nonlinear Science and Numerical Simulation, 21 (2015), 172-180.  doi: 10.1016/j.cnsns.2014.07.033.

[8]

M. Gaudioso, G. Giallombardo and G. Miglionico, An incremental method for solving convex finite min-max problems, Mathematics of Operations Research, 31 (2006), 173-187.  doi: 10.1287/moor.1050.0175.

[9]

M. Gaudioso, G. Giallombardo and G. Miglionico, On solving the Lagrangian dual of integer programs via an incremental approach, Computational Optimization and Applications, 44 (2009), 117-138.  doi: 10.1007/s10589-007-9149-2.

[10]

W. Hare and J. Nutini, A derivative-free approximate gradient sampling algorithm for finite minimax problems, Computational Optimization and Applications, 56 (2013), 1-38.  doi: 10.1007/s10589-013-9547-6.

[11]

W. Hare and M. Macklem, Derivative-free optimization methods for finite minimax problems, Optimization Methods and Software, 28 (2013), 300-312.  doi: 10.1080/10556788.2011.638923.

[12]

S. X. He and Y. Y. Nie, A class of nonlinear Lagrangian algorithms for minimax problems, Journal of Industrial and Management Optimization, 9 (2013), 75-97.  doi: 10.3934/jimo.2013.9.75.

[13]

M. Huang, X. J. Liang, Y. Lu and L. P. Pang, The bundle scheme for solving arbitrary eigenvalue optimizations, Journal of Industrial and Management Optimization, 13 (2017), 659-680.  doi: 10.3934/jimo.2016039.

[14]

J. B. Jian, X. L. Zhang, R. Quan and Q. Ma, Generalized monotone line search SQP algorithm for constrained minimax problems, Optimization, 58 (2009), 101-131.  doi: 10.1080/02331930801951140.

[15]

J. B. Jian, X. D. Mo, L. J. Qiu, S. M. Yang and F. S. Wang, Simple sequential quadratically constrained quadratic programming feasible algorithm with active identification sets for constrained minimax problems, Journal of Optimization Theory and Applications, 160 (2014), 158-188.  doi: 10.1007/s10957-013-0339-z.

[16]

J. B. Jian, C. M. Tang and F. Tang, A feasible descent bundle method for inequality constrained minimax problems (in Chinese), Science China: Mathematics, 45 (2015), 2001-2024. 

[17]

E. Karas, A. Ribeiro, C. Sagastizábal and M. Solodov, A bundle-filter method for nonsmooth convex constrained optimization, Mathematical Programming, 116 (2009), 297-320.  doi: 10.1007/s10107-007-0123-7.

[18]

N. Karmitsa, Test Problems for Large-Scale Nonsmooth Minimization, Tech. Rep. No. B. 4/2007, Department of Mathematical Information Technology, University of Jyväskylä, Finland, 2007.

[19]

K. C. Kiwiel, A projection-proximal bundle method for convex nondifferentiable minimization, In: M. Théra, R. Tichatschke (eds.) Ill-posed Variational Problems and Regularization Techniques, Lecture Notes in Econom. Math. Systems, Springer-Verlag, Berlin, 477 (1999), 137–150. doi: 10.1007/978-3-642-45780-7_9.

[20]

K. C. Kiwiel, A proximal-projection bundle method for Lagrangian relaxation, including semidefinite programming, SIAM Journal on Optimization, 17 (2006), 1015-1034.  doi: 10.1137/050639284.

[21]

K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, 1133. Springer-Verlag, 1985. doi: 10.1007/BFb0074500.

[22]

C. Lemaréchal, An extension of Davidon methods to nondifferentiable problems, Mathematical Programming Study, 3 (1975), 95-109. 

[23]

X. S. Li and S. C. Fang, On the entropic regularization method for solving min-max problems with applications, Mathematical Methods of Operations Research, 46 (1997), 119-130.  doi: 10.1007/BF01199466.

[24]

Y. P. Li and G. H. Huang, Inexact minimax regret integer programming for long-term planning of municipal solid waste management -- part a: Methodology development, Environmental Engineering Science, 26 (2009), 209-218.  doi: 10.1089/ees.2007.0241.ptA.

[25]

G. Liuzzi, S. Lucidi and M. Sciandrone, A derivative-free algorithm for linearly constrained finite minimax problems, SIAM Journal on Optimization, 16 (2006), 1054-1075.  doi: 10.1137/040615821.

[26]

L. Lukšan and J. Vlček, Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization, Tech. Rep. No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 2000.

[27]

K. Madsen and H. Schjær-Jacobsen, Linearly constrained minimax optimization, Mathematical Programming, 14 (1978), 208-223.  doi: 10.1007/BF01588966.

[28]

C. Michelot and F. Plastria, An extended multifacility minimax location problemrevisited, Annals of Operations Research, 111 (2002), 167-179.  doi: 10.1023/A:1020953703533.

[29]

A. Nedić and D. P. Bertsekas, Incremental subgradient methods for nondifferentiable optimization, SIAM Journal on Optimization, 12 (2001), 109-138.  doi: 10.1137/S1052623499362111.

[30]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N. J., 1970.

[31]

B. Rustem and Q. Nguyen, An algorithm for the inequality-constrained discrete min-max problem, SIAM Journal on Optimization, 8 (1998), 265-283.  doi: 10.1137/S1056263493260386.

[32]

C. Sagastizábal and M. Solodov, An infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter, SIAM Journal on Optimization, 16 (2005), 146-169.  doi: 10.1137/040603875.

[33]

N. Z. Shor, Minimization Methods for Non-differentiable Functions, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-82118-9.

[34]

C. M. Tang, H. Y. Chen and J. B. Jian, An improved partial bundle method for linearly constrained minimax problems, Statistics, Optimization and Information Computing, 4 (2016), 84-98.  doi: 10.19139/soic.v4i1.205.

[35]

C. M. Tang and J. B. Jian, Strongly sub-feasible direction method for constrained optimization problems with nonsmooth objective functions, European Journal of Operational Research, 218 (2012), 28-37.  doi: 10.1016/j.ejor.2011.10.055.

[36]

C. M. Tang, S. Liu, J. B. Jian and J. L. Li, A feasible SQP-GS algorithm for nonconvex, nonsmooth constrained optimization, Numerical Algorithms, 65 (2014), 1-22.  doi: 10.1007/s11075-012-9692-5.

[37]

A. Vardi, New minimax algorithm, Journal of Optimization Theory and Applications, 75 (1992), 613-634.  doi: 10.1007/BF00940496.

[38]

F. S. Wang and K. C. Zhang, A hybrid algorithm for nonlinear minimax problems, Annals of Operations Research, 164 (2008), 167-191.  doi: 10.1007/s10479-008-0401-7.

[39]

F. S. Wang and K. C. Zhang, A hybrid algorithm for linearly constrained minimax problems, Annals of Operations Research, 206 (2013), 501-525.  doi: 10.1007/s10479-012-1274-3.

[40]

S. Y. Wang, Y. Yamamoto and M. Yu, A minimax rule for portfolio selection in frictional markets, Mathematical Methods of Operations Research, 57 (2003), 141-155.  doi: 10.1007/s001860200241.

[41]

S. Xu, Smoothing method for minimax problems, Computational Optimization and Applications, 20 (2001), 267-279.  doi: 10.1023/A:1011211101714.

[42]

MOSEK: The MOSEK optimization toolbox for MATLAB manual, Version 7.1 (2016). MOSEK ApS, Denmark, http://www.mosek.com.

show all references

References:
[1]

A. Baums, Minimax method in optimizing energy consumption in real-time embedded systems, Automatic Control and Computer Sciences, 43 (2009), 57-62.  doi: 10.3103/S0146411609020011.

[2]

J. F. Bonnans, J. C. Gilbert, C. Lemaréchal and C. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, Second ed. Springer-Verlag, Berlin Heidelberg New York, 2006.

[3]

J. V. Burke, A. S. Lewis and M. L. Overton, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM Journal on Optimization, 15 (2005), 751-779.  doi: 10.1137/030601296.

[4]

F. L. Chernousko, Minimax control for a class of linear systems subject to disturbances, Journal of Optimization Theory and Applications, 127 (2005), 535-548.  doi: 10.1007/s10957-005-7501-1.

[5]

J. Dattorro, Convex Optimization † Euclidean Distance Geometry, second edn, $ \mathcal{M}\varepsilonβ oo$, 2015.

[6]

G. Di Pillo, L. Grippo and S. Lucidi, A smooth method for the finite minimax problem, Mathematical Programming, 60 (1993), 187-214.  doi: 10.1007/BF01580609.

[7]

A. Fuduli, M. Gaudioso, G. Giallombardo and G. Miglionico, A partially inexact bundle method for convex semi-infinite minmax problems, Communications in Nonlinear Science and Numerical Simulation, 21 (2015), 172-180.  doi: 10.1016/j.cnsns.2014.07.033.

[8]

M. Gaudioso, G. Giallombardo and G. Miglionico, An incremental method for solving convex finite min-max problems, Mathematics of Operations Research, 31 (2006), 173-187.  doi: 10.1287/moor.1050.0175.

[9]

M. Gaudioso, G. Giallombardo and G. Miglionico, On solving the Lagrangian dual of integer programs via an incremental approach, Computational Optimization and Applications, 44 (2009), 117-138.  doi: 10.1007/s10589-007-9149-2.

[10]

W. Hare and J. Nutini, A derivative-free approximate gradient sampling algorithm for finite minimax problems, Computational Optimization and Applications, 56 (2013), 1-38.  doi: 10.1007/s10589-013-9547-6.

[11]

W. Hare and M. Macklem, Derivative-free optimization methods for finite minimax problems, Optimization Methods and Software, 28 (2013), 300-312.  doi: 10.1080/10556788.2011.638923.

[12]

S. X. He and Y. Y. Nie, A class of nonlinear Lagrangian algorithms for minimax problems, Journal of Industrial and Management Optimization, 9 (2013), 75-97.  doi: 10.3934/jimo.2013.9.75.

[13]

M. Huang, X. J. Liang, Y. Lu and L. P. Pang, The bundle scheme for solving arbitrary eigenvalue optimizations, Journal of Industrial and Management Optimization, 13 (2017), 659-680.  doi: 10.3934/jimo.2016039.

[14]

J. B. Jian, X. L. Zhang, R. Quan and Q. Ma, Generalized monotone line search SQP algorithm for constrained minimax problems, Optimization, 58 (2009), 101-131.  doi: 10.1080/02331930801951140.

[15]

J. B. Jian, X. D. Mo, L. J. Qiu, S. M. Yang and F. S. Wang, Simple sequential quadratically constrained quadratic programming feasible algorithm with active identification sets for constrained minimax problems, Journal of Optimization Theory and Applications, 160 (2014), 158-188.  doi: 10.1007/s10957-013-0339-z.

[16]

J. B. Jian, C. M. Tang and F. Tang, A feasible descent bundle method for inequality constrained minimax problems (in Chinese), Science China: Mathematics, 45 (2015), 2001-2024. 

[17]

E. Karas, A. Ribeiro, C. Sagastizábal and M. Solodov, A bundle-filter method for nonsmooth convex constrained optimization, Mathematical Programming, 116 (2009), 297-320.  doi: 10.1007/s10107-007-0123-7.

[18]

N. Karmitsa, Test Problems for Large-Scale Nonsmooth Minimization, Tech. Rep. No. B. 4/2007, Department of Mathematical Information Technology, University of Jyväskylä, Finland, 2007.

[19]

K. C. Kiwiel, A projection-proximal bundle method for convex nondifferentiable minimization, In: M. Théra, R. Tichatschke (eds.) Ill-posed Variational Problems and Regularization Techniques, Lecture Notes in Econom. Math. Systems, Springer-Verlag, Berlin, 477 (1999), 137–150. doi: 10.1007/978-3-642-45780-7_9.

[20]

K. C. Kiwiel, A proximal-projection bundle method for Lagrangian relaxation, including semidefinite programming, SIAM Journal on Optimization, 17 (2006), 1015-1034.  doi: 10.1137/050639284.

[21]

K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, 1133. Springer-Verlag, 1985. doi: 10.1007/BFb0074500.

[22]

C. Lemaréchal, An extension of Davidon methods to nondifferentiable problems, Mathematical Programming Study, 3 (1975), 95-109. 

[23]

X. S. Li and S. C. Fang, On the entropic regularization method for solving min-max problems with applications, Mathematical Methods of Operations Research, 46 (1997), 119-130.  doi: 10.1007/BF01199466.

[24]

Y. P. Li and G. H. Huang, Inexact minimax regret integer programming for long-term planning of municipal solid waste management -- part a: Methodology development, Environmental Engineering Science, 26 (2009), 209-218.  doi: 10.1089/ees.2007.0241.ptA.

[25]

G. Liuzzi, S. Lucidi and M. Sciandrone, A derivative-free algorithm for linearly constrained finite minimax problems, SIAM Journal on Optimization, 16 (2006), 1054-1075.  doi: 10.1137/040615821.

[26]

L. Lukšan and J. Vlček, Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization, Tech. Rep. No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 2000.

[27]

K. Madsen and H. Schjær-Jacobsen, Linearly constrained minimax optimization, Mathematical Programming, 14 (1978), 208-223.  doi: 10.1007/BF01588966.

[28]

C. Michelot and F. Plastria, An extended multifacility minimax location problemrevisited, Annals of Operations Research, 111 (2002), 167-179.  doi: 10.1023/A:1020953703533.

[29]

A. Nedić and D. P. Bertsekas, Incremental subgradient methods for nondifferentiable optimization, SIAM Journal on Optimization, 12 (2001), 109-138.  doi: 10.1137/S1052623499362111.

[30]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N. J., 1970.

[31]

B. Rustem and Q. Nguyen, An algorithm for the inequality-constrained discrete min-max problem, SIAM Journal on Optimization, 8 (1998), 265-283.  doi: 10.1137/S1056263493260386.

[32]

C. Sagastizábal and M. Solodov, An infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter, SIAM Journal on Optimization, 16 (2005), 146-169.  doi: 10.1137/040603875.

[33]

N. Z. Shor, Minimization Methods for Non-differentiable Functions, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-82118-9.

[34]

C. M. Tang, H. Y. Chen and J. B. Jian, An improved partial bundle method for linearly constrained minimax problems, Statistics, Optimization and Information Computing, 4 (2016), 84-98.  doi: 10.19139/soic.v4i1.205.

[35]

C. M. Tang and J. B. Jian, Strongly sub-feasible direction method for constrained optimization problems with nonsmooth objective functions, European Journal of Operational Research, 218 (2012), 28-37.  doi: 10.1016/j.ejor.2011.10.055.

[36]

C. M. Tang, S. Liu, J. B. Jian and J. L. Li, A feasible SQP-GS algorithm for nonconvex, nonsmooth constrained optimization, Numerical Algorithms, 65 (2014), 1-22.  doi: 10.1007/s11075-012-9692-5.

[37]

A. Vardi, New minimax algorithm, Journal of Optimization Theory and Applications, 75 (1992), 613-634.  doi: 10.1007/BF00940496.

[38]

F. S. Wang and K. C. Zhang, A hybrid algorithm for nonlinear minimax problems, Annals of Operations Research, 164 (2008), 167-191.  doi: 10.1007/s10479-008-0401-7.

[39]

F. S. Wang and K. C. Zhang, A hybrid algorithm for linearly constrained minimax problems, Annals of Operations Research, 206 (2013), 501-525.  doi: 10.1007/s10479-012-1274-3.

[40]

S. Y. Wang, Y. Yamamoto and M. Yu, A minimax rule for portfolio selection in frictional markets, Mathematical Methods of Operations Research, 57 (2003), 141-155.  doi: 10.1007/s001860200241.

[41]

S. Xu, Smoothing method for minimax problems, Computational Optimization and Applications, 20 (2001), 267-279.  doi: 10.1023/A:1011211101714.

[42]

MOSEK: The MOSEK optimization toolbox for MATLAB manual, Version 7.1 (2016). MOSEK ApS, Denmark, http://www.mosek.com.

Table 1.  Numerical results for $C = \{x:\ l\leq x\leq u\}$.
Problem $n, m$ Algorithm NI ND N$f_i$ $f^*$ N$_f^{eq}$
CB2$^\Box$ $2,3$ $ \verb"PPPBM"^+$ 3 2 6 20.000000 2
$ \verb"PPPBM"^-$ 11 3 17 20.000000 6
$ \verb"PPBM"$ 6 5 18 20.000000 6
CB3$^\Box$ $2,3$ $ \verb"PPPBM"^+$ 3 2 6 16.000000 2
$ \verb"PPPBM"^-$ 38 7 51 16.000000 17
$ \verb"PPBM"$ 9 7 27 16.000000 9
DEM$^\Box$ $2,3$ $ \verb"PPPBM"^+$ 2 2 6 -2.500000 2
$ \verb"PPPBM"^-$ 2 2 6 -2.500000 2
$ \verb"PPBM"$ 2 1 6 -2.500000 2
QL$^\Box$ $2,3$ $ \verb"PPPBM"^+$ 2 2 6 7.940000 2
$ \verb"PPPBM"^-$ 4 2 8 7.940000 3
$ \verb"PPBM"$ 2 1 6 7.940000 2
LQ$^\Box$ $2,2$ $ \verb"PPPBM"^+$ 2 2 4 -1.311371 2
$ \verb"PPPBM"^-$ 4 2 6 -1.311371 3
$ \verb"PPBM"$ 2 1 4 -1.311371 2
Mifflin1$^\Box$ $2,2$ $ \verb"PPPBM"^+$ 3 2 4 3.300000 2
$ \verb"PPPBM"^-$ 10 2 12 3.300000 6
$ \verb"PPBM"$ 3 2 6 3.300000 3
Rosen-Suzuki$^\Box$ $4,4$ $ \verb"PPPBM"^+$ 46 14 101 -43.853572 25
$ \verb"PPPBM"^-$ 177 10 247 -43.853353 62
$ \verb"PPBM"$ 87 14 348 -43.853158 87
Shor$^\Box$ $5,10$ $ \verb"PPPBM"^+$ 43 16 241 23.418920 24
$ \verb"PPPBM"^-$ 197 7 343 23.418952 34
$ \verb"PPBM"$ 46 16 460 23.418938 46
Maxquad$^\Box$ $10,5$ $ \verb"PPPBM"^+$ 38 15 145 -0.841407 29
$ \verb"PPPBM"^-$ 149 30 416 -0.841408 83
$ \verb"PPBM"$ 69 31 345 -0.841408 69
Maxq$^\Box$ $20,20$ $ \verb"PPPBM"^+$ 49 13 512 0.010000 26
$ \verb"PPPBM"^-$ 663 16 2967 0.010000 148
$ \verb"PPBM"$ 45 9 900 0.010000 45
Maxl$^\Box$ $20,20$ $ \verb"PPPBM"^+$ 19 11 269 0.100000 13
$ \verb"PPPBM"^-$ 61 24 623 0.100000 31
$ \verb"PPBM"$ 46 18 920 0.100000 46
Goffin$^\Box$ $50,50$ $ \verb"PPPBM"^+$ 27 3 1074 25.000000 21
$ \verb"PPPBM"^-$ 28 4 1124 25.000000 22
$ \verb"PPBM"$ 29 4 1450 25.000000 29
MXHILB$^\Box$ $50,50$ $ \verb"PPPBM"^+$ 41 16 830 0.000038 17
$ \verb"PPPBM"^-$ 148 8 858 0.000037 17
$ \verb"PPBM"$ 36 12 1800 0.000043 36
Table Options

Download as excel

Problem $n, m$ Algorithm NI ND N$f_i$ $f^*$ N$_f^{eq}$
CB2$^\Box$ $2,3$ $ \verb"PPPBM"^+$ 3 2 6 20.000000 2
$ \verb"PPPBM"^-$ 11 3 17 20.000000 6
$ \verb"PPBM"$ 6 5 18 20.000000 6
CB3$^\Box$ $2,3$ $ \verb"PPPBM"^+$ 3 2 6 16.000000 2
$ \verb"PPPBM"^-$ 38 7 51 16.000000 17
$ \verb"PPBM"$ 9 7 27 16.000000 9
DEM$^\Box$ $2,3$ $ \verb"PPPBM"^+$ 2 2 6 -2.500000 2
$ \verb"PPPBM"^-$ 2 2 6 -2.500000 2
$ \verb"PPBM"$ 2 1 6 -2.500000 2
QL$^\Box$ $2,3$ $ \verb"PPPBM"^+$ 2 2 6 7.940000 2
$ \verb"PPPBM"^-$ 4 2 8 7.940000 3
$ \verb"PPBM"$ 2 1 6 7.940000 2
LQ$^\Box$ $2,2$ $ \verb"PPPBM"^+$ 2 2 4 -1.311371 2
$ \verb"PPPBM"^-$ 4 2 6 -1.311371 3
$ \verb"PPBM"$ 2 1 4 -1.311371 2
Mifflin1$^\Box$ $2,2$ $ \verb"PPPBM"^+$ 3 2 4 3.300000 2
$ \verb"PPPBM"^-$ 10 2 12 3.300000 6
$ \verb"PPBM"$ 3 2 6 3.300000 3
Rosen-Suzuki$^\Box$ $4,4$ $ \verb"PPPBM"^+$ 46 14 101 -43.853572 25
$ \verb"PPPBM"^-$ 177 10 247 -43.853353 62
$ \verb"PPBM"$ 87 14 348 -43.853158 87
Shor$^\Box$ $5,10$ $ \verb"PPPBM"^+$ 43 16 241 23.418920 24
$ \verb"PPPBM"^-$ 197 7 343 23.418952 34
$ \verb"PPBM"$ 46 16 460 23.418938 46
Maxquad$^\Box$ $10,5$ $ \verb"PPPBM"^+$ 38 15 145 -0.841407 29
$ \verb"PPPBM"^-$ 149 30 416 -0.841408 83
$ \verb"PPBM"$ 69 31 345 -0.841408 69
Maxq$^\Box$ $20,20$ $ \verb"PPPBM"^+$ 49 13 512 0.010000 26
$ \verb"PPPBM"^-$ 663 16 2967 0.010000 148
$ \verb"PPBM"$ 45 9 900 0.010000 45
Maxl$^\Box$ $20,20$ $ \verb"PPPBM"^+$ 19 11 269 0.100000 13
$ \verb"PPPBM"^-$ 61 24 623 0.100000 31
$ \verb"PPBM"$ 46 18 920 0.100000 46
Goffin$^\Box$ $50,50$ $ \verb"PPPBM"^+$ 27 3 1074 25.000000 21
$ \verb"PPPBM"^-$ 28 4 1124 25.000000 22
$ \verb"PPBM"$ 29 4 1450 25.000000 29
MXHILB$^\Box$ $50,50$ $ \verb"PPPBM"^+$ 41 16 830 0.000038 17
$ \verb"PPPBM"^-$ 148 8 858 0.000037 17
$ \verb"PPBM"$ 36 12 1800 0.000043 36
Table 2.  Numerical results for $C = \{x:\ \|x-a\|\leq b\}$.
Problem $n, m$ Algorithm NI ND N$f_i$ $f^*$ N$_f^{eq}$
CB2° $2,3$ $ \verb"PPPBM"^+$ 2 2 6 3.343146 2
$ \verb"PPPBM"^-$ 4 2 8 3.343146 3
$ \verb"PPBM"$ 2 1 6 3.343146 2
CB3° $2,3$ $ \verb"PPPBM"^+$ 8 4 14 24.479797 5
$ \verb"PPPBM"^-$ 16 7 30 24.479796 10
$ \verb"PPBM"$ 11 8 33 24.479795 11
DEM° $2,3$ $ \verb"PPPBM"^+$ 22 11 45 -1.499974 15
$ \verb"PPPBM"^-$ 36 2 41 -1.499956 14
$ \verb"PPBM"$ 19 13 57 -1.499950 19
QL° $2,3$ $ \verb"PPPBM"^+$ 5 5 15 25.820215 5
$ \verb"PPPBM"^-$ 12 5 24 25.820216 8
$ \verb"PPBM"$ 12 9 36 25.820215 12
LQ° $2,2$ $ \verb"PPPBM"^+$ 21 7 27 -0.999989 14
$ \verb"PPPBM"^-$ 41 4 46 -0.999999 23
$ \verb"PPBM"$ 18 17 36 -0.999996 18
Mifflin1° $2,2$ $ \verb"PPPBM"^+$ 3 2 4 48.153612 2
$ \verb"PPPBM"^-$ 4 2 6 48.153612 3
$ \verb"PPBM"$ 2 1 4 48.153612 2
Rosen-Suzuki° $4,4$ $ \verb"PPPBM"^+$ 10 6 25 39.715617 6
$ \verb"PPPBM"^-$ 12 4 24 39.715617 6
$ \verb"PPBM"$ 9 8 36 39.715617 9
Shor° $5,10$ $ \verb"PPPBM"^+$ 3 2 20 50.250278 2
$ \verb"PPPBM"^-$ 6 2 24 50.250278 2
$ \verb"PPBM"$ 4 1 40 50.250278 4
Maxquad° $10,5$ $ \verb"PPPBM"^+$ 35 16 146 -0.841408 29
$ \verb"PPPBM"^-$ 146 30 413 -0.841406 83
$ \verb"PPBM"$ 84 20 420 -0.841408 84
Maxq° $20,20$ $ \verb"PPPBM"^+$ 78 8 780 0.011183 39
$ \verb"PPPBM"^-$ 612 18 4777 0.011193 239
$ \verb"PPBM"$ 84 18 1680 0.011197 84
Maxl° $20,20$ $ \verb"PPPBM"^+$ 22 3 269 0.552786 13
$ \verb"PPPBM"^-$ 24 5 309 0.552786 15
$ \verb"PPBM"$ 22 2 440 0.552786 22
Goffin° $50,50$ $ \verb"PPPBM"^+$ 27 3 1074 28.786797 21
$ \verb"PPPBM"^-$ 28 4 1124 28.786797 22
$ \verb"PPBM"$ 28 3 1400 28.786797 28
MXHILB° $50,50$ $ \verb"PPPBM"^+$ 61 16 847 0.613446 17
$ \verb"PPPBM"^-$ 250 14 937 0.613479 19
$ \verb"PPBM"$ 41 21 2050 0.613444 41
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Problem $n, m$ Algorithm NI ND N$f_i$ $f^*$ N$_f^{eq}$
CB2° $2,3$ $ \verb"PPPBM"^+$ 2 2 6 3.343146 2
$ \verb"PPPBM"^-$ 4 2 8 3.343146 3
$ \verb"PPBM"$ 2 1 6 3.343146 2
CB3° $2,3$ $ \verb"PPPBM"^+$ 8 4 14 24.479797 5
$ \verb"PPPBM"^-$ 16 7 30 24.479796 10
$ \verb"PPBM"$ 11 8 33 24.479795 11
DEM° $2,3$ $ \verb"PPPBM"^+$ 22 11 45 -1.499974 15
$ \verb"PPPBM"^-$ 36 2 41 -1.499956 14
$ \verb"PPBM"$ 19 13 57 -1.499950 19
QL° $2,3$ $ \verb"PPPBM"^+$ 5 5 15 25.820215 5
$ \verb"PPPBM"^-$ 12 5 24 25.820216 8
$ \verb"PPBM"$ 12 9 36 25.820215 12
LQ° $2,2$ $ \verb"PPPBM"^+$ 21 7 27 -0.999989 14
$ \verb"PPPBM"^-$ 41 4 46 -0.999999 23
$ \verb"PPBM"$ 18 17 36 -0.999996 18
Mifflin1° $2,2$ $ \verb"PPPBM"^+$ 3 2 4 48.153612 2
$ \verb"PPPBM"^-$ 4 2 6 48.153612 3
$ \verb"PPBM"$ 2 1 4 48.153612 2
Rosen-Suzuki° $4,4$ $ \verb"PPPBM"^+$ 10 6 25 39.715617 6
$ \verb"PPPBM"^-$ 12 4 24 39.715617 6
$ \verb"PPBM"$ 9 8 36 39.715617 9
Shor° $5,10$ $ \verb"PPPBM"^+$ 3 2 20 50.250278 2
$ \verb"PPPBM"^-$ 6 2 24 50.250278 2
$ \verb"PPBM"$ 4 1 40 50.250278 4
Maxquad° $10,5$ $ \verb"PPPBM"^+$ 35 16 146 -0.841408 29
$ \verb"PPPBM"^-$ 146 30 413 -0.841406 83
$ \verb"PPBM"$ 84 20 420 -0.841408 84
Maxq° $20,20$ $ \verb"PPPBM"^+$ 78 8 780 0.011183 39
$ \verb"PPPBM"^-$ 612 18 4777 0.011193 239
$ \verb"PPBM"$ 84 18 1680 0.011197 84
Maxl° $20,20$ $ \verb"PPPBM"^+$ 22 3 269 0.552786 13
$ \verb"PPPBM"^-$ 24 5 309 0.552786 15
$ \verb"PPBM"$ 22 2 440 0.552786 22
Goffin° $50,50$ $ \verb"PPPBM"^+$ 27 3 1074 28.786797 21
$ \verb"PPPBM"^-$ 28 4 1124 28.786797 22
$ \verb"PPBM"$ 28 3 1400 28.786797 28
MXHILB° $50,50$ $ \verb"PPPBM"^+$ 61 16 847 0.613446 17
$ \verb"PPPBM"^-$ 250 14 937 0.613479 19
$ \verb"PPBM"$ 41 21 2050 0.613444 41
Table 3.  Numerical results for Madsen-Schjær-Jacobsen's problem [27].
$n, m$ Algorithm NI ND N$f_i$ $f^*$ N$_f^{eq}$
$30, 58$ $ \verb"PPPBM"^+$ 99 15 3842 -19.336156 66
$ \verb"PPBM"$ 99 18 5742 -19.336155 99
$50, 98$ $ \verb"PPPBM"^+$ 138 22 10502 -62.867219 107
$ \verb"PPBM"$ 161 25 15778 -62.867219 161
$100,198$ $ \verb"PPPBM"^+$ 346 27 42247 -281.077609 213
$ \verb"PPBM"$ 358 25 70884 -281.077569 358
$150,298$ $ \verb"PPPBM"^+$ 388 31 81950 -655.540257 275
$ \verb"PPBM"$ 510 31 151980 -655.540202 510
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$n, m$ Algorithm NI ND N$f_i$ $f^*$ N$_f^{eq}$
$30, 58$ $ \verb"PPPBM"^+$ 99 15 3842 -19.336156 66
$ \verb"PPBM"$ 99 18 5742 -19.336155 99
$50, 98$ $ \verb"PPPBM"^+$ 138 22 10502 -62.867219 107
$ \verb"PPBM"$ 161 25 15778 -62.867219 161
$100,198$ $ \verb"PPPBM"^+$ 346 27 42247 -281.077609 213
$ \verb"PPBM"$ 358 25 70884 -281.077569 358
$150,298$ $ \verb"PPPBM"^+$ 388 31 81950 -655.540257 275
$ \verb"PPBM"$ 510 31 151980 -655.540202 510
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