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A modified Nelder-Mead barrier method for constrained optimization
Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand |
An interior point modified Nelder Mead method for nonlinearly constrained optimization is described. This method neither uses nor estimates objective function or constraint gradients. A modified logarithmic barrier function is used. The method generates a sequence of points which converges to KKT point(s) under mild conditions including existence of a Slater point. Numerical results are presented that show the algorithm performs well in practice.
References:
[1] |
C. Audet and J. E. Dennis Jr.,
Analysis of generalized pattern searches, SIAM J. Optim., 13 (2003), 889-903.
doi: 10.1137/S1052623400378742. |
[2] |
C. Audet and J. E. Dennis Jr.,
Mesh adaptive direct search algorithms for constrained optimization, SIAM J. Optim., 17 (2006), 188-217.
doi: 10.1137/040603371. |
[3] |
C. Audet and J. E. Dennis Jr.,
A progressive barrier for derivative free nonlinear programming, SIAM J. Optim., 20 (2009), 445-472.
doi: 10.1137/070692662. |
[4] |
M. A. Abramson, C. Audet, J. E. Dennis Jr. and S. Le Digabel, orthoMADS: a deterministic MADS instance with orthogonal directions, SIAM J. Optim., 20 (2009), 948-966.
doi: 10.1137/080716980. |
[5] |
F. H. Clarke, Optimization and Non-smooth Analysis, SIAM classics in applied mathematics, New York, 1990.
doi: 10.1137/1.9781611971309. |
[6] |
I. D. Coope and C. J. Price,
Frame based methods for unconstrained optimization, J. Optim. Theory and Appl., 107 (2000), 261-274.
doi: 10.1023/A:1026429319405. |
[7] |
I. D. Coope and C. J. Price,
On the convergence of grid based methods for unconstrained optimization, SIAM J. Optim., 11 (2001), 859-869.
doi: 10.1137/S1052623499354989. |
[8] |
I. D. Coope and C. J. Price,
Positive bases in numerical optimization, Comput. Optim. and Appl., 21 (2002), 169-175.
doi: 10.1023/A:1013760716801. |
[9] |
C. Davis, Theory of positive linear dependence, Amer. J. Math., 76 (1954), 733-746.
doi: 10.2307/2372648. |
[10] |
A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968. |
[11] |
F. Gao and L. Han, Implementing the Nelder Mead simplex algorithm with adaptive parameters, Comput. Optim. Appl., 51 (2012), 259-277. Google Scholar |
[12] |
W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, Springer, Berlin, 1981.
doi: 10.1007/BF00934594. |
[13] |
N. Karmitsa, Test Problems for Large-Scale Nonsmooth Minimization, Department of Mathematical Information Technology Report B. 4/2007, University of Jyväskylä, Finland, 2007. Google Scholar |
[14] |
C. T. Kelley,
Detection and remediation of stagnation in the Nelder-Mead algorithm using a sufficient decrease condition, SIAM J. Optim., 10 (1999), 43-55.
doi: 10.1137/S1052623497315203. |
[15] |
T. G. Kolda, R. M. Lewis and V. Torczon,
Optimization by direct search: New perspectives on some classical and modern methods, SIAM Rev., 45 (2003), 385-482.
doi: 10.1137/S003614450242889. |
[16] |
J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright,
Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim., 9 (1999), 112-147.
doi: 10.1137/S1052623496303470. |
[17] |
J. C. Lagarius, B. Poonen and M. H. Wright,
Convergence of the restricted Nelder Mead algorithm in two dimensions, SIAM J. Optim., 22 (2012), 501-532.
doi: 10.1137/110830150. |
[18] |
L. Lukšan and J. Vlček, Test problems for nonsmooth unconstrained and linearly constrained optimization, Tech. Report 798, Prague: Institute of Computer Science, Academy of Sciences of the Czech Republic, 2000. Google Scholar |
[19] |
K. I. M. McKinnon,
A simplex method for function minimization, SIAM J. Optim., 9 (1999), 148-158.
doi: 10.1137/S1052623496303482. |
[20] |
J. J. More, B. S. Garbow and K. E. Hillstrom,
Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41.
doi: 10.1145/355934.355936. |
[21] |
L. Nazareth and P. Tseng,
Gilding the lily: a variant of the Nelder Mead algorithm based on golden-section search, Comput. Optim. Appl., 22 (2002), 133-144.
doi: 10.1023/A:1014842520519. |
[22] |
J. A. Nelder and R. Mead,
A simplex method for function minimization, Computer J., 7 (1965), 308-313.
doi: 10.1093/comjnl/7.4.308. |
[23] |
M. J. D. Powell,
Direct search algorithms for optimization calculations, Acta Numerica, 7 (1998), 287-336.
doi: 10.1017/S0962492900002841. |
[24] |
C. J. Price., Direct search nonsmooth constrained optimization via rounded $\ell_1$ penalty functions, Optim. Methods & Software, (2020), DOI: 10.1080/10556788.2020.1746961 Google Scholar |
[25] |
C. J. Price and I. D. Coope,
Frames and grids in unconstrained and linearly constrained optimization: a non-smooth approach, SIAM J. Optim., 14 (2003), 415-438.
doi: 10.1137/S1052623402407084. |
[26] |
C. J. Price, I. D. Coope and D. Byatt,
A convergent variant of the Nelder Mead algorithm, J. Optim. Theory and Appl., 113 (2002), 5-19.
doi: 10.1023/A:1014849028575. |
[27] | |
[28] |
V. Torczon,
On the convergence of pattern search algorithms, SIAM J. Optim., 7 (1997), 1-25.
doi: 10.1137/S1052623493250780. |
[29] |
P. Tseng,
Fortified-descent simplicial search method: a general approach, SIAM J. Optim., 10 (1999), 269-288.
doi: 10.1137/S1052623495282857. |
[30] |
M. H. Wright, Direct search methods: once scorned now respectable, in Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis Addison-Wesley, Reading, MA and Longman, Harlow, UK, 1996. |
[31] |
W.-C. Yu, Positive basis and a class of direct search techniques, Scientia Sinica (special issue), 1 (1979). |
[32] |
W. C. Yu and Y.-X. Li,
A direct search method by the local positive basis for the nonlinearly constrained optimization, Chinese Annals of Math., 2 (1981), 269-280.
|
show all references
References:
[1] |
C. Audet and J. E. Dennis Jr.,
Analysis of generalized pattern searches, SIAM J. Optim., 13 (2003), 889-903.
doi: 10.1137/S1052623400378742. |
[2] |
C. Audet and J. E. Dennis Jr.,
Mesh adaptive direct search algorithms for constrained optimization, SIAM J. Optim., 17 (2006), 188-217.
doi: 10.1137/040603371. |
[3] |
C. Audet and J. E. Dennis Jr.,
A progressive barrier for derivative free nonlinear programming, SIAM J. Optim., 20 (2009), 445-472.
doi: 10.1137/070692662. |
[4] |
M. A. Abramson, C. Audet, J. E. Dennis Jr. and S. Le Digabel, orthoMADS: a deterministic MADS instance with orthogonal directions, SIAM J. Optim., 20 (2009), 948-966.
doi: 10.1137/080716980. |
[5] |
F. H. Clarke, Optimization and Non-smooth Analysis, SIAM classics in applied mathematics, New York, 1990.
doi: 10.1137/1.9781611971309. |
[6] |
I. D. Coope and C. J. Price,
Frame based methods for unconstrained optimization, J. Optim. Theory and Appl., 107 (2000), 261-274.
doi: 10.1023/A:1026429319405. |
[7] |
I. D. Coope and C. J. Price,
On the convergence of grid based methods for unconstrained optimization, SIAM J. Optim., 11 (2001), 859-869.
doi: 10.1137/S1052623499354989. |
[8] |
I. D. Coope and C. J. Price,
Positive bases in numerical optimization, Comput. Optim. and Appl., 21 (2002), 169-175.
doi: 10.1023/A:1013760716801. |
[9] |
C. Davis, Theory of positive linear dependence, Amer. J. Math., 76 (1954), 733-746.
doi: 10.2307/2372648. |
[10] |
A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968. |
[11] |
F. Gao and L. Han, Implementing the Nelder Mead simplex algorithm with adaptive parameters, Comput. Optim. Appl., 51 (2012), 259-277. Google Scholar |
[12] |
W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, Springer, Berlin, 1981.
doi: 10.1007/BF00934594. |
[13] |
N. Karmitsa, Test Problems for Large-Scale Nonsmooth Minimization, Department of Mathematical Information Technology Report B. 4/2007, University of Jyväskylä, Finland, 2007. Google Scholar |
[14] |
C. T. Kelley,
Detection and remediation of stagnation in the Nelder-Mead algorithm using a sufficient decrease condition, SIAM J. Optim., 10 (1999), 43-55.
doi: 10.1137/S1052623497315203. |
[15] |
T. G. Kolda, R. M. Lewis and V. Torczon,
Optimization by direct search: New perspectives on some classical and modern methods, SIAM Rev., 45 (2003), 385-482.
doi: 10.1137/S003614450242889. |
[16] |
J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright,
Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim., 9 (1999), 112-147.
doi: 10.1137/S1052623496303470. |
[17] |
J. C. Lagarius, B. Poonen and M. H. Wright,
Convergence of the restricted Nelder Mead algorithm in two dimensions, SIAM J. Optim., 22 (2012), 501-532.
doi: 10.1137/110830150. |
[18] |
L. Lukšan and J. Vlček, Test problems for nonsmooth unconstrained and linearly constrained optimization, Tech. Report 798, Prague: Institute of Computer Science, Academy of Sciences of the Czech Republic, 2000. Google Scholar |
[19] |
K. I. M. McKinnon,
A simplex method for function minimization, SIAM J. Optim., 9 (1999), 148-158.
doi: 10.1137/S1052623496303482. |
[20] |
J. J. More, B. S. Garbow and K. E. Hillstrom,
Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41.
doi: 10.1145/355934.355936. |
[21] |
L. Nazareth and P. Tseng,
Gilding the lily: a variant of the Nelder Mead algorithm based on golden-section search, Comput. Optim. Appl., 22 (2002), 133-144.
doi: 10.1023/A:1014842520519. |
[22] |
J. A. Nelder and R. Mead,
A simplex method for function minimization, Computer J., 7 (1965), 308-313.
doi: 10.1093/comjnl/7.4.308. |
[23] |
M. J. D. Powell,
Direct search algorithms for optimization calculations, Acta Numerica, 7 (1998), 287-336.
doi: 10.1017/S0962492900002841. |
[24] |
C. J. Price., Direct search nonsmooth constrained optimization via rounded $\ell_1$ penalty functions, Optim. Methods & Software, (2020), DOI: 10.1080/10556788.2020.1746961 Google Scholar |
[25] |
C. J. Price and I. D. Coope,
Frames and grids in unconstrained and linearly constrained optimization: a non-smooth approach, SIAM J. Optim., 14 (2003), 415-438.
doi: 10.1137/S1052623402407084. |
[26] |
C. J. Price, I. D. Coope and D. Byatt,
A convergent variant of the Nelder Mead algorithm, J. Optim. Theory and Appl., 113 (2002), 5-19.
doi: 10.1023/A:1014849028575. |
[27] | |
[28] |
V. Torczon,
On the convergence of pattern search algorithms, SIAM J. Optim., 7 (1997), 1-25.
doi: 10.1137/S1052623493250780. |
[29] |
P. Tseng,
Fortified-descent simplicial search method: a general approach, SIAM J. Optim., 10 (1999), 269-288.
doi: 10.1137/S1052623495282857. |
[30] |
M. H. Wright, Direct search methods: once scorned now respectable, in Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis Addison-Wesley, Reading, MA and Longman, Harlow, UK, 1996. |
[31] |
W.-C. Yu, Positive basis and a class of direct search techniques, Scientia Sinica (special issue), 1 (1979). |
[32] |
W. C. Yu and Y.-X. Li,
A direct search method by the local positive basis for the nonlinearly constrained optimization, Chinese Annals of Math., 2 (1981), 269-280.
|


PENMECO | ORTHOMADS | PATTERNSEARCH | |||||||||||
function | nf | nc(all) | nc(I) | nf | nc | nf | |||||||
1 | HS43 Rosen-Suzuki | 4 | 3 | 8186 | 8209 | - | 4e-9 | -3e-8 | 1328 | 2116 | 4e-4 | 2725 | 0.5 |
2 | HS44 | 4 | 10 | 6556 | 6587 | 17 | 4e-9 | -1e-9 | 26 | 365 | 0 | 4577 | 2e-7 |
3 | HS45 | 5 | 10 | 7896 | 7949 | 41 | 2e-9 | -4e-10 | 56 | 494 | 0 | 5359 | 0.43 |
4 | HS72 | 4 | 10 | 3802 | 4401 | - | 2e-8 | -2e-15 | 5074 | 6411 | 3e-4 | 15760 | 5e-4 |
5 | HS76 | 4 | 7 | 3194 | 3458 | 140 | 1e-10 | -2e-11 | 411 | 1156 | 0.01 | 76909 | 0.014 |
6 | HS93 | 6 | 1 | 14709 | 22001 | - | 6e-3 | -1e-13 | 4643 | 6827 | 3e-3 | 1105 | 2e-3 |
7 | HS100 | 7 | 4 | 45073 | 45163 | - | 4e-6 | -1e-3 | 2694 | 4792 | 5e-4 | 5205 | 5e-3 |
8 | HS108 | 9 | 14 | 24982 | 26198 | 1058 | 2e-8 | -2e-8 | 5226 | 14443 | 0.4 | 13307 | 0.26 |
9 | Audet-Dennis cresc. | 10 | 2 | 52532 | 52921 | - | 3e-7 | -4e-9 | 22267 | 29561 | 1e-4 | 26004 | 0.84 |
10 | mod. Jennrich-Sam. | 12 | 10 | 71702 | 80014 | - | 3.4 | -0.10 | 6336 | 33694 | 28.5 | 17145 | 29.4 |
11 | mod. vardim | 21 | 1 | 79667 | 80468 | 468 | 1e-7 | -1e-7 | 17997 | 80002 | 5e-7 | - | - |
12 | LV4.1 mad1 | 2 | 1 | 2421 | 2427 | - | 4e-9 | -3e-11 | 369 | 587 | 4e-9 | 635 | 3e-4 |
13 | LV4.2 mad2 | 2 | 1 | 1924 | 1932 | - | 2e-9 | -2e-9 | 323 | 504 | 8e-6 | 1827 | 1e-7 |
14 | LV4.3 mad4 | 2 | 1 | 3070 | 3076 | - | 4e-9 | -3e-11 | 4034 | 4639 | 4e-9 | 1995 | 1e-6 |
15 | LV4.4 mad5 | 2 | 1 | 3363 | 3410 | - | 5e-9 | -9e-11 | 147 | 265 | 5e-5 | 695 | 8e-5 |
16 | mod. Beale | 5 | 3 | 27525 | 27591 | 15 | 1e-9 | -8e-10 | 5022 | 9087 | 0.03 | 37462 | 0.24 |
17 | LV4.5 pentagon | 6 | 15 | 29246 | 29373 | 108 | 5e-6 | -1e-9 | 6948 | 12452 | 0.04 | 13472 | 0.16 |
18 | LV4.7 mod. equil | 7 | 8 | 23739 | 23821 | - | 4e-5 | -0.03 | 53633 | 53701 | 1.9 | 12322 | 3e-3 |
19 | mod. Zakharov | 8 | 2 | 32469 | 32921 | 86 | 1e-10 | -5e-11 | 10982 | 22115 | 1e-4 | 47017 | 8e-10 |
20 | LV4.8 Wong2 | 10 | 3 | 79589 | 80000 | - | 8e-5 | -3e-9 | 18436 | 24656 | 0.17 | 9420 | 0.2 |
21 | maxQ10 | 10 | 8 | 79777 | 80001 | - | 2e-4 | -3e-5 | 11916 | 22834 | 1.8 | 14195 | 3.4 |
22 | chainLQ | 10 | 8 | 79861 | 80995 | 995 | 2e-7 | -5e-7 | 0 | 821 | infeas. | 59740 | 15 |
23 | crescentI | 20 | 1 | 79720 | 80000 | - | 8e-5 | -9e-9 | 23788 | 33759 | 0.78 | 52699 | 0.013 |
24 | crescentII | 20 | 1 | 79603 | 80029 | 29 | 8e-10 | -8e-10 | 41328 | 50365 | 2e-9 | 29710 | 6e-9 |
25 | chainCB3I | 20 | 1 | 79712 | 80000 | - | 9e-11 | -3e-10 | 44512 | 80007 | 1e-3 | 55562 | 0.032 |
26 | chainCB3II | 20 | 1 | 66735 | 66752 | - | 9e-6 | -2e-3 | 69767 | 70523 | 4e-4 | 71610 | 0.14 |
PENMECO | ORTHOMADS | PATTERNSEARCH | |||||||||||
function | nf | nc(all) | nc(I) | nf | nc | nf | |||||||
1 | HS43 Rosen-Suzuki | 4 | 3 | 8186 | 8209 | - | 4e-9 | -3e-8 | 1328 | 2116 | 4e-4 | 2725 | 0.5 |
2 | HS44 | 4 | 10 | 6556 | 6587 | 17 | 4e-9 | -1e-9 | 26 | 365 | 0 | 4577 | 2e-7 |
3 | HS45 | 5 | 10 | 7896 | 7949 | 41 | 2e-9 | -4e-10 | 56 | 494 | 0 | 5359 | 0.43 |
4 | HS72 | 4 | 10 | 3802 | 4401 | - | 2e-8 | -2e-15 | 5074 | 6411 | 3e-4 | 15760 | 5e-4 |
5 | HS76 | 4 | 7 | 3194 | 3458 | 140 | 1e-10 | -2e-11 | 411 | 1156 | 0.01 | 76909 | 0.014 |
6 | HS93 | 6 | 1 | 14709 | 22001 | - | 6e-3 | -1e-13 | 4643 | 6827 | 3e-3 | 1105 | 2e-3 |
7 | HS100 | 7 | 4 | 45073 | 45163 | - | 4e-6 | -1e-3 | 2694 | 4792 | 5e-4 | 5205 | 5e-3 |
8 | HS108 | 9 | 14 | 24982 | 26198 | 1058 | 2e-8 | -2e-8 | 5226 | 14443 | 0.4 | 13307 | 0.26 |
9 | Audet-Dennis cresc. | 10 | 2 | 52532 | 52921 | - | 3e-7 | -4e-9 | 22267 | 29561 | 1e-4 | 26004 | 0.84 |
10 | mod. Jennrich-Sam. | 12 | 10 | 71702 | 80014 | - | 3.4 | -0.10 | 6336 | 33694 | 28.5 | 17145 | 29.4 |
11 | mod. vardim | 21 | 1 | 79667 | 80468 | 468 | 1e-7 | -1e-7 | 17997 | 80002 | 5e-7 | - | - |
12 | LV4.1 mad1 | 2 | 1 | 2421 | 2427 | - | 4e-9 | -3e-11 | 369 | 587 | 4e-9 | 635 | 3e-4 |
13 | LV4.2 mad2 | 2 | 1 | 1924 | 1932 | - | 2e-9 | -2e-9 | 323 | 504 | 8e-6 | 1827 | 1e-7 |
14 | LV4.3 mad4 | 2 | 1 | 3070 | 3076 | - | 4e-9 | -3e-11 | 4034 | 4639 | 4e-9 | 1995 | 1e-6 |
15 | LV4.4 mad5 | 2 | 1 | 3363 | 3410 | - | 5e-9 | -9e-11 | 147 | 265 | 5e-5 | 695 | 8e-5 |
16 | mod. Beale | 5 | 3 | 27525 | 27591 | 15 | 1e-9 | -8e-10 | 5022 | 9087 | 0.03 | 37462 | 0.24 |
17 | LV4.5 pentagon | 6 | 15 | 29246 | 29373 | 108 | 5e-6 | -1e-9 | 6948 | 12452 | 0.04 | 13472 | 0.16 |
18 | LV4.7 mod. equil | 7 | 8 | 23739 | 23821 | - | 4e-5 | -0.03 | 53633 | 53701 | 1.9 | 12322 | 3e-3 |
19 | mod. Zakharov | 8 | 2 | 32469 | 32921 | 86 | 1e-10 | -5e-11 | 10982 | 22115 | 1e-4 | 47017 | 8e-10 |
20 | LV4.8 Wong2 | 10 | 3 | 79589 | 80000 | - | 8e-5 | -3e-9 | 18436 | 24656 | 0.17 | 9420 | 0.2 |
21 | maxQ10 | 10 | 8 | 79777 | 80001 | - | 2e-4 | -3e-5 | 11916 | 22834 | 1.8 | 14195 | 3.4 |
22 | chainLQ | 10 | 8 | 79861 | 80995 | 995 | 2e-7 | -5e-7 | 0 | 821 | infeas. | 59740 | 15 |
23 | crescentI | 20 | 1 | 79720 | 80000 | - | 8e-5 | -9e-9 | 23788 | 33759 | 0.78 | 52699 | 0.013 |
24 | crescentII | 20 | 1 | 79603 | 80029 | 29 | 8e-10 | -8e-10 | 41328 | 50365 | 2e-9 | 29710 | 6e-9 |
25 | chainCB3I | 20 | 1 | 79712 | 80000 | - | 9e-11 | -3e-10 | 44512 | 80007 | 1e-3 | 55562 | 0.032 |
26 | chainCB3II | 20 | 1 | 66735 | 66752 | - | 9e-6 | -2e-3 | 69767 | 70523 | 4e-4 | 71610 | 0.14 |
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