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Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty
November
2020, 16(6): 2675-2701.
doi: 10.3934/jimo.2019075
An adaptively regularized sequential quadratic programming method for equality constrained optimization
| 1. | School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, China |
| 2. | School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, China |
In this paper, we devise an adaptively regularized SQP method for equality constrained optimization problem that is resilient to constraint degeneracy, with a relatively small departure from classical SQP method. The main feature of our method is an adaptively choice of regularization parameter, embedded in a trust-funnel-like algorithmic scheme. Unlike general regularized methods, which update regularization parameter after a regularized problem is approximately solved, our method updates the regularization parameter at each iteration according to the infeasibility measure and the promised improvements achieved by the trial step. The sequence of regularization parameters is not necessarily monotonically decreasing. The whole algorithm is globalized by a trust-funnel-like strategy, in which neither a penalty function nor a filter is needed. We present global and fast local convergence under weak assumptions. Preliminary numerical results on a collection of degenerate problems are reported, which are encouraging.
Keywords:
Equality constrained optimization,
adaptive regularization,
trust-funnel-like methods,
global convergence,
local convergence.
Mathematics Subject Classification:
Primary: 65K05, 90C30, 90C55.
Citation:
Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial and Management Optimization,
2020, 16
(6)
: 2675-2701.
doi: 10.3934/jimo.2019075
![]()
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R. H. Bielschowsky and F. A. M. Gomes,
Dynamic control of infeasibility in equality constrained optimization, SIAM J. Optim., 19 (2008), 1299-1325.
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Some stable methods for calculating inertia and solving symmetric linear systems, Math. Comput., 31 (1977), 163-179.
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An inexact restoration strategy for the globalization of the sSQP method, Comput. Optim. Appl., 54 (2013), 595-617.
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A primal–dual regularized interior-point method for convex quadratic programs, Math. Program. Comput., 4 (2012), 71-107.
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M. Fukushima,
A successive quadratic programming algorithm with global and superlinear convergence properties, Math. Program., 35 (1986), 253-264.
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P. E. Gill, W. Murray, D. Ponceleón and M. Saunders, Solving Reduced KKT Systems in Barrier Methods for Linear and Quadratic Programming, Technical Report Technical Report SOL 91-7, Systems Optimization Laboratory, Stanford University, Stanford, 1991.
doi: 10.21236/ADA239191. |
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A globally convergent stabilized SQP method, SIAM J. Optim., 23 (2013), 1983-2010.
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P. E. Gill and E. Wong, Sequential quadratic programming methods, in Mixed Integer Nonlinear Programming (eds. J. Lee and S. Leyffer), Springer, New York, 2012,147–224. |
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J. Gondzio,
Matrix-free interior point method, Comput. Optim. Appl., 51 (2012), 457-480.
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N. I. M. Gould and Ph. L. Toint,
Nonlinear programming without a penalty function or a filter, Math. Program., 122 (2010), 155-196.
doi: 10.1007/s10107-008-0244-7. |
| [28] |
N. I. M. Gould and Ph. L. Toint, Global convergence of a non-monotone trust-region filter algorithm for nonlinear programming, in Multiscale Optimization Methods and Applications (eds. W. W. Hager, S.-J. Huang, P. M. Pardalos and O. A. Prokopyev), Springer US, Boston, MA, 2006,125–150.
doi: 10.1007/0-387-29550-X_5. |
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W. W. Hager,
Stabilized sequential quadratic programming, Comput. Optim. Appl., 12 (1999), 253-273.
doi: 10.1023/A:1008640419184. |
| [30] |
A. F. Izmailov, M. V. Solodov and E. I. Uskov,
Combining stabilized SQP with the augmented Lagrangian algorithm, Comput. Optim. Appl., 62 (2015), 405-429.
doi: 10.1007/s10589-015-9744-6. |
| [31] |
A. F. Izmailov and M. V. Solodov,
Stabilized SQP revisited, Math. Program., 133 (2012), 93-120.
doi: 10.1007/s10107-010-0413-3. |
| [32] |
K. Levenberg,
A method for the solution of certain problems in least squares, Q. Appl. Math., 2 (1944), 164-168.
doi: 10.1090/qam/10666. |
| [33] |
D.-H. Li and L. Qi, A Stabilized SQP Method via Linear Equations, Applied mathematics technical report AMR00/5, The University of New South Wales, 2000. |
| [34] |
X. W. Liu and Y. -X Yuan,
A sequential quadratic programming method without a penalty function or a filter for nonlinear equality constrained optimization, SIAM J. Optim., 21 (2011), 545-571.
doi: 10.1137/080739884. |
| [35] |
C. M. Maes, A Regularized Active-Set Method for Sparse Convex Quadratic Programming, PhD thesis, Institute for Computational and Mathematical Engineering, Stanford University, CA, 2010. |
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D. Q. Mayne and E. Polak, A surperlinearly convergent algorithm for constrained optimization problems, in Algorithms for Constrained Minimization of Smooth Nonlinear Functions (eds. A. G. Buckley and J.-L. Goffin), Springer Berlin Heidelberg, Berlin, Heidelberg, 1982, 45–61. |
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L. Minchenko and S. Stakhovski,
On relaxed constant rank regularity condition in mathematical programming, Optimization, 60 (2011), 429-440.
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J. J. Moré, The Levenberg-Marquardt algorithm: Implementation and theory, in Numerical Analysis, Springer, Berlin, 1978, 105–116. |
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B. A. Murtagh and M. A. Saunders, Minos 5.4 User's Guide (Revised), Technical report, Technical Report SOL 83-20R, Department of Operations Research, Stanford University, Stanford, CA 94305, USA, 1993; Revised, 1995. |
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M. J. D. Powell and Y.-X. Yuan,
A recursive quadratic programming algorithm that uses differentiable exact penalty functions, Math. Program., 35 (1986), 265-278.
doi: 10.1007/BF01580880. |
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M. J. D. Powell and Y.-X. Yuan,
A trust region algorithm for equality constrained optimization, Math. Program., 49 (1990), 189-211.
doi: 10.1007/BF01588787. |
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S. Q. Qiu and Z. W. Chen,
Global and local convergence of a class of penalty-free-type methods for nonlinear programming, Appl. Math. Model., 36 (2012), 3201-3216.
doi: 10.1016/j.apm.2011.10.009. |
| [43] |
S. Q. Qiu and Z. W. Chen,
A globally convergent penalty-free method for optimization with equality constraints and simple bounds, Acta Appl. Math., 142 (2016), 39-60.
doi: 10.1007/s10440-015-0013-6. |
| [44] |
T. Rusten and R. Winther,
A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl., 13 (1992), 887-904.
doi: 10.1137/0613054. |
| [45] |
M. Saunders and J. Tomlin, Solving Regularized Linear Programs Using Barrier Methods and KKT Systems, Technical Report SOL Report 96-4, Dept. of EESOR, Stanford University, 1996. |
| [46] |
D. Silvester and A. Wathen,
Fast iterative solution of stabilised Stokes systems part Ⅱ: Using general block preconditioners, SIAM J. Numer. Anal., 31 (1994), 1352-1367.
doi: 10.1137/0731070. |
| [47] |
W. Y. Sun and Y.-X. Yuan, Optimization Theory and Methods, Springer, 2005. |
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M. Ulbrich and S. Ulbrich,
Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function, Math. Program., 95 (2003), 103-135.
doi: 10.1007/s10107-002-0343-9. |
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M. Ulbrich, S. Ulbrich and L. N. Vicente,
A globally convergent primal-dual interior-point filter method for nonlinear programming, Math. Program., 100 (2004), 379-410.
doi: 10.1007/s10107-003-0477-4. |
| [50] |
S. Ulbrich,
On the superlinear local convergence of a filter-SQP method, Math. Program., 100 (2004), 217-245.
doi: 10.1007/s10107-003-0491-6. |
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R. J. Vanderbei,
Symmetric quasidefinite matrices, SIAM J. Optim., 5 (1995), 100-113.
doi: 10.1137/0805005. |
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R. J. Vanderbei,
Loqo: An interior point code for quadratic programming, Optim. Methods Softw., 11 (1999), 451-484.
doi: 10.1080/10556789908805759. |
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R. J. Vanderbei and D. F. Shanno,
An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl., 13 (1999), 231-252.
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Superlinear convergence of a stabilized SQP method to a degenerate solution, Comput. Optim. Appl., 11 (1998), 253-275.
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W. J. Xue, C. G. Shen and D. G. Pu,
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H. Yamashita, H. Yabe and T. Tanabe,
A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization, Math. Program., 102 (2005), 111-151.
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N. Yamashita and M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, in Topics in Numerical Analysis, Springer, 2001,239–249.
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show all references
References:
| [1] |
P. Armand, J. Benoist and D. Orban,
From global to local convergence of interior methods for nonlinear optimization, Optim. Method. Softw., 28 (2013), 1051-1080.
doi: 10.1080/10556788.2012.668905. |
| [2] |
S. Arreckx and D. Orban, A Regularized Factorization-Free Method for Equality-Constrained Optimization, Technical Report G-2016-65, GERAD HEC Montréal, 2016.
doi: 10.1137/16M1088570. |
| [3] |
R. H. Bielschowsky and F. A. M. Gomes,
Dynamic control of infeasibility in equality constrained optimization, SIAM J. Optim., 19 (2008), 1299-1325.
doi: 10.1137/070679557. |
| [4] |
P. T. Boggs and J. W. Tolle,
Sequential quadratic programming, Acta Numer., 4 (1995), 1-51.
doi: 10.1017/s0962492900002518. |
| [5] |
J. R. Bunch and L. Kaufman,
Some stable methods for calculating inertia and solving symmetric linear systems, Math. Comput., 31 (1977), 163-179.
doi: 10.1090/S0025-5718-1977-0428694-0. |
| [6] |
R. M. Chamberlain, M. J. D. Powell, C. Lemarechal and H. C. Pedersen, The watchdog technique for forcing convergence in algorithms for constrained optimization, in Algorithms for Constrained Minimization of Smooth Nonlinear Functions (eds. A. G. Buckley and J. L. Goffin), Springer, Berlin, Heidelberg, 1982, 1–17.
doi: 10.1007/bfb0120945. |
| [7] |
A. R. Conn, N. I. M. Gould, D. Orban and P. L. Toint,
A primal-dual trust-region algorithm for non-convex nonlinear programming, Math. Program., 87 (2000), 215-249.
doi: 10.1007/s101070050112. |
| [8] |
A. Conn, N. Gould and P. Toint, Trust-Region Methods, SIAM, Philadelphia, PA, USA, 2000.
doi: 10.1137/1.9780898719857. |
| [9] |
H. Dan, N. Yamashita and M. Fukushima,
Convergence properties of the inexact Levenberg-Marquardt method under local error bound conditions, Optim. Method. Softw., 17 (2002), 605-626.
doi: 10.1080/1055678021000049345. |
| [10] | |
| [11] |
E. D. Dolan and J. J. Moré,
Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.
doi: 10.1007/s101070100263. |
| [12] |
I. S. Duff,
Ma57–-a code for the solution of sparse symmetric definite and indefinite systems, ACM Trans. Math. Softw., 30 (2004), 118-144.
doi: 10.1145/992200.992202. |
| [13] |
J.-Y. Fan and Y.-X. Yuan,
On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74 (2005), 23-39.
doi: 10.1007/s00607-004-0083-1. |
| [14] |
D. Fernández, E. A. Pilotta and G. A. Torres,
An inexact restoration strategy for the globalization of the sSQP method, Comput. Optim. Appl., 54 (2013), 595-617.
doi: 10.1007/s10589-012-9502-y. |
| [15] |
R. Fletcher, Second order corrections for non-differentiable optimization, in Numerical Analysis: Proceedings of the 9th Biennial Conference Held at Dundee, Scotland, June 23–26, 1981 (ed. G. A. Watson), Springer, Berlin, Heidelberg, 1982, 85–114. |
| [16] |
R. Fletcher, S. Leyffer and Ph. L. Toint,
On the global convergence of a filter–SQP algorithm, SIAM J. Optim, 13 (2002), 44-59.
doi: 10.1137/S105262340038081X. |
| [17] |
R. Fletcher, N. I. M. Gould, S. Leyffer, Ph. L. Toint and A. Wächter,
Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming, SIAM J. Optim., 13 (2002), 635-659.
doi: 10.1137/S1052623499357258. |
| [18] |
R. Fletcher and S. Leyffer,
Nonlinear programming without a penalty function, Math. Program., 91 (2002), 239-269.
doi: 10.1007/s101070100244. |
| [19] |
R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, The Scientific Press, 1993.
|
| [20] |
M. P. Friedlander and D. Orban,
A primal–dual regularized interior-point method for convex quadratic programs, Math. Program. Comput., 4 (2012), 71-107.
doi: 10.1007/s12532-012-0035-2. |
| [21] |
M. Fukushima,
A successive quadratic programming algorithm with global and superlinear convergence properties, Math. Program., 35 (1986), 253-264.
doi: 10.1007/BF01580879. |
| [22] |
P. E. Gill, W. Murray, D. Ponceleón and M. Saunders, Solving Reduced KKT Systems in Barrier Methods for Linear and Quadratic Programming, Technical Report Technical Report SOL 91-7, Systems Optimization Laboratory, Stanford University, Stanford, 1991.
doi: 10.21236/ADA239191. |
| [23] |
P. E. Gill, V. Kungurtsev and D. P. Robinson, A stabilized SQP method: Superlinear convergence, Math. Program., 163 (2017), 369-?10.
doi: 10.1007/s10107-016-1066-7. |
| [24] |
P. E. Gill and D. P. Robinson,
A globally convergent stabilized SQP method, SIAM J. Optim., 23 (2013), 1983-2010.
doi: 10.1137/120882913. |
| [25] |
P. E. Gill and E. Wong, Sequential quadratic programming methods, in Mixed Integer Nonlinear Programming (eds. J. Lee and S. Leyffer), Springer, New York, 2012,147–224. |
| [26] |
J. Gondzio,
Matrix-free interior point method, Comput. Optim. Appl., 51 (2012), 457-480.
doi: 10.1007/s10589-010-9361-3. |
| [27] |
N. I. M. Gould and Ph. L. Toint,
Nonlinear programming without a penalty function or a filter, Math. Program., 122 (2010), 155-196.
doi: 10.1007/s10107-008-0244-7. |
| [28] |
N. I. M. Gould and Ph. L. Toint, Global convergence of a non-monotone trust-region filter algorithm for nonlinear programming, in Multiscale Optimization Methods and Applications (eds. W. W. Hager, S.-J. Huang, P. M. Pardalos and O. A. Prokopyev), Springer US, Boston, MA, 2006,125–150.
doi: 10.1007/0-387-29550-X_5. |
| [29] |
W. W. Hager,
Stabilized sequential quadratic programming, Comput. Optim. Appl., 12 (1999), 253-273.
doi: 10.1023/A:1008640419184. |
| [30] |
A. F. Izmailov, M. V. Solodov and E. I. Uskov,
Combining stabilized SQP with the augmented Lagrangian algorithm, Comput. Optim. Appl., 62 (2015), 405-429.
doi: 10.1007/s10589-015-9744-6. |
| [31] |
A. F. Izmailov and M. V. Solodov,
Stabilized SQP revisited, Math. Program., 133 (2012), 93-120.
doi: 10.1007/s10107-010-0413-3. |
| [32] |
K. Levenberg,
A method for the solution of certain problems in least squares, Q. Appl. Math., 2 (1944), 164-168.
doi: 10.1090/qam/10666. |
| [33] |
D.-H. Li and L. Qi, A Stabilized SQP Method via Linear Equations, Applied mathematics technical report AMR00/5, The University of New South Wales, 2000. |
| [34] |
X. W. Liu and Y. -X Yuan,
A sequential quadratic programming method without a penalty function or a filter for nonlinear equality constrained optimization, SIAM J. Optim., 21 (2011), 545-571.
doi: 10.1137/080739884. |
| [35] |
C. M. Maes, A Regularized Active-Set Method for Sparse Convex Quadratic Programming, PhD thesis, Institute for Computational and Mathematical Engineering, Stanford University, CA, 2010. |
| [36] |
D. Q. Mayne and E. Polak, A surperlinearly convergent algorithm for constrained optimization problems, in Algorithms for Constrained Minimization of Smooth Nonlinear Functions (eds. A. G. Buckley and J.-L. Goffin), Springer Berlin Heidelberg, Berlin, Heidelberg, 1982, 45–61. |
| [37] |
L. Minchenko and S. Stakhovski,
On relaxed constant rank regularity condition in mathematical programming, Optimization, 60 (2011), 429-440.
doi: 10.1080/02331930902971377. |
| [38] |
J. J. Moré, The Levenberg-Marquardt algorithm: Implementation and theory, in Numerical Analysis, Springer, Berlin, 1978, 105–116. |
| [39] |
B. A. Murtagh and M. A. Saunders, Minos 5.4 User's Guide (Revised), Technical report, Technical Report SOL 83-20R, Department of Operations Research, Stanford University, Stanford, CA 94305, USA, 1993; Revised, 1995. |
| [40] |
M. J. D. Powell and Y.-X. Yuan,
A recursive quadratic programming algorithm that uses differentiable exact penalty functions, Math. Program., 35 (1986), 265-278.
doi: 10.1007/BF01580880. |
| [41] |
M. J. D. Powell and Y.-X. Yuan,
A trust region algorithm for equality constrained optimization, Math. Program., 49 (1990), 189-211.
doi: 10.1007/BF01588787. |
| [42] |
S. Q. Qiu and Z. W. Chen,
Global and local convergence of a class of penalty-free-type methods for nonlinear programming, Appl. Math. Model., 36 (2012), 3201-3216.
doi: 10.1016/j.apm.2011.10.009. |
| [43] |
S. Q. Qiu and Z. W. Chen,
A globally convergent penalty-free method for optimization with equality constraints and simple bounds, Acta Appl. Math., 142 (2016), 39-60.
doi: 10.1007/s10440-015-0013-6. |
| [44] |
T. Rusten and R. Winther,
A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl., 13 (1992), 887-904.
doi: 10.1137/0613054. |
| [45] |
M. Saunders and J. Tomlin, Solving Regularized Linear Programs Using Barrier Methods and KKT Systems, Technical Report SOL Report 96-4, Dept. of EESOR, Stanford University, 1996. |
| [46] |
D. Silvester and A. Wathen,
Fast iterative solution of stabilised Stokes systems part Ⅱ: Using general block preconditioners, SIAM J. Numer. Anal., 31 (1994), 1352-1367.
doi: 10.1137/0731070. |
| [47] |
W. Y. Sun and Y.-X. Yuan, Optimization Theory and Methods, Springer, 2005. |
| [48] |
M. Ulbrich and S. Ulbrich,
Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function, Math. Program., 95 (2003), 103-135.
doi: 10.1007/s10107-002-0343-9. |
| [49] |
M. Ulbrich, S. Ulbrich and L. N. Vicente,
A globally convergent primal-dual interior-point filter method for nonlinear programming, Math. Program., 100 (2004), 379-410.
doi: 10.1007/s10107-003-0477-4. |
| [50] |
S. Ulbrich,
On the superlinear local convergence of a filter-SQP method, Math. Program., 100 (2004), 217-245.
doi: 10.1007/s10107-003-0491-6. |
| [51] |
R. J. Vanderbei,
Symmetric quasidefinite matrices, SIAM J. Optim., 5 (1995), 100-113.
doi: 10.1137/0805005. |
| [52] |
R. J. Vanderbei,
Loqo: An interior point code for quadratic programming, Optim. Methods Softw., 11 (1999), 451-484.
doi: 10.1080/10556789908805759. |
| [53] |
R. J. Vanderbei and D. F. Shanno,
An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl., 13 (1999), 231-252.
doi: 10.1023/A:1008677427361. |
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Figure 2.
Performance compared to LOQO
Table 1.
Compared to MINOS
| Use MINOS'Stopping rule. | ||
| MINOS | New Alg. | |
| Problem solved |
100 | 103 |
| Robustness ( |
95.24% | 98.10% |
| Average evals ( |
84.04 | 67.5 |
| Median evals ( |
27 | 21 |
| Use MINOS'Stopping rule. | ||
| MINOS | New Alg. | |
| Problem solved |
100 | 103 |
| Robustness ( |
95.24% | 98.10% |
| Average evals ( |
84.04 | 67.5 |
| Median evals ( |
27 | 21 |
Table 2.
Compared to LOQO.
| Use LOQO's Stopping rule. | ||
| LOQO | New Alg. | |
| Problem solved ( | 78 | 96 |
| Robustness ( | 74.30% | 91.43% |
| Average evals ( | 26.6 | 62.1 |
| Median evals ( | 20 | 21 |
| Use LOQO's Stopping rule. | ||
| LOQO | New Alg. | |
| Problem solved ( | 78 | 96 |
| Robustness ( | 74.30% | 91.43% |
| Average evals ( | 26.6 | 62.1 |
| Median evals ( | 20 | 21 |
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