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December  2021, 11(4): 613-631. doi: 10.3934/naco.2020058

A modified Nelder-Mead barrier method for constrained optimization

C. J. Price

Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

Received  October 2020 Revised  November 2020 Published  December 2021 Early access  December 2020

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.

Keywords: Derivative free, positive basis, convergence analysis, frames, direct search, numerical results.
Mathematics Subject Classification: Primary:65K05.
Citation: C. J. Price. A modified Nelder-Mead barrier method for constrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 613-631. doi: 10.3934/naco.2020058
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

F. H. Clarke, Optimization and Non-smooth Analysis, SIAM classics in applied mathematics, New York, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Davis, Theory of positive linear dependence, Amer. J. Math., 76 (1954), 733-746. doi: 10.2307/2372648.  Google Scholar

[10]

A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. J. D. Powell, Direct search algorithms for optimization calculations, Acta Numerica, 7 (1998), 287-336.  doi: 10.1017/S0962492900002841.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

,, Google Scholar

[28]

V. Torczon, On the convergence of pattern search algorithms, SIAM J. Optim., 7 (1997), 1-25.  doi: 10.1137/S1052623493250780.  Google Scholar

[29]

P. Tseng, Fortified-descent simplicial search method: a general approach, SIAM J. Optim., 10 (1999), 269-288.  doi: 10.1137/S1052623495282857.  Google Scholar

[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.  Google Scholar

[31]

W.-C. Yu, Positive basis and a class of direct search techniques, Scientia Sinica (special issue), 1 (1979).  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

F. H. Clarke, Optimization and Non-smooth Analysis, SIAM classics in applied mathematics, New York, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Davis, Theory of positive linear dependence, Amer. J. Math., 76 (1954), 733-746. doi: 10.2307/2372648.  Google Scholar

[10]

A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. J. D. Powell, Direct search algorithms for optimization calculations, Acta Numerica, 7 (1998), 287-336.  doi: 10.1017/S0962492900002841.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

,, Google Scholar

[28]

V. Torczon, On the convergence of pattern search algorithms, SIAM J. Optim., 7 (1997), 1-25.  doi: 10.1137/S1052623493250780.  Google Scholar

[29]

P. Tseng, Fortified-descent simplicial search method: a general approach, SIAM J. Optim., 10 (1999), 269-288.  doi: 10.1137/S1052623495282857.  Google Scholar

[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.  Google Scholar

[31]

W.-C. Yu, Positive basis and a class of direct search techniques, Scientia Sinica (special issue), 1 (1979).  Google Scholar

[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.   Google Scholar

Figure 1.  The current polytope $ x_0 $, $ x_1 $, and $ x_2 $ is shown in solid lines. The modified Nelder Mead sub-algorithm first tries to replace the worst polytope point $ x_2 $ with one of $ x_{ic} $, $ x_{oc} $, $ x_r $ or $ x_e $. These four points lie on the dashed line through $ x_2 $ and the centroid $ \bar{x} $ of the remaining polytope points. If this fails to give sufficient descent, the fictitious polytope $ y_1 $, $ y_2 $, $ H $ is used, where the fictitious polytope's worst point $ H $ is reflected through the centroid $ \bar{y} $ of that polytope's remaining points, giving a fictitous new low point $ x_0 $. The expand step is then tried from this fictitious polytope, giving the pseudo-expand point $ x_{pe} $. The points $ y_1,\ldots,y_n $ ($ y_1 $ and $ y_2 $ here) and $ x_{pe} $ form a frame around $ x_0 $
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24], in 12, 18, and 24 dimensions with 11 constraints. Each curve gives the number of problems solved to an accuracy $ E_f < 10^{-4} $ within $ N $ simplex gradients, which is $ N(n+1) $ evaluations of $ f $">Figure 2.  Data profiles for PENMECO and V-ORTHOMADS on 100 randomly generated cones test functions from [24], in 12, 18, and 24 dimensions with 11 constraints. Each curve gives the number of problems solved to an accuracy $ E_f < 10^{-4} $ within $ N $ simplex gradients, which is $ N(n+1) $ evaluations of $ f $
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Table 1.  Numerical results for the 26 test problems. Problems 1-11 are smooth, the rest are nonsmooth.
PENMECO ORTHOMADS PATTERNSEARCH
function $ n $ $ q $ nf nc(all) nc(I) $ E_f $ $ \max(c) $ nf nc $ E_f $ nf $ E_f $
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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PENMECO ORTHOMADS PATTERNSEARCH
function $ n $ $ q $ nf nc(all) nc(I) $ E_f $ $ \max(c) $ nf nc $ E_f $ nf $ E_f $
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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