Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it
Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What...
Ausführliche Beschreibung
Autor*in: |
Izmailov, A. F. [verfasserIn] |
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E-Artikel |
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Sprache: |
Englisch |
Erschienen: |
2015 |
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Anmerkung: |
© Sociedad de Estadística e Investigación Operativa 2015 |
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Übergeordnetes Werk: |
Enthalten in: Top - Berlin : Springer, 1986, 23(2015), 1 vom: 27. Feb., Seite 1-26 |
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Übergeordnetes Werk: |
volume:23 ; year:2015 ; number:1 ; day:27 ; month:02 ; pages:1-26 |
Links: |
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DOI / URN: |
10.1007/s11750-015-0372-1 |
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Katalog-ID: |
SPR022747141 |
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520 | |a Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. | ||
650 | 4 | |a Critical Lagrange multipliers |7 (dpeaa)DE-He213 | |
650 | 4 | |a Second-order sufficiency |7 (dpeaa)DE-He213 | |
650 | 4 | |a Newton-type methods |7 (dpeaa)DE-He213 | |
650 | 4 | |a Sequential quadratic programming |7 (dpeaa)DE-He213 | |
650 | 4 | |a Newton–Lagrange method |7 (dpeaa)DE-He213 | |
650 | 4 | |a Superlinear convergence |7 (dpeaa)DE-He213 | |
700 | 1 | |a Solodov, M. V. |4 aut | |
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773 | 1 | 8 | |g volume:23 |g year:2015 |g number:1 |g day:27 |g month:02 |g pages:1-26 |
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10.1007/s11750-015-0372-1 doi (DE-627)SPR022747141 (SPR)s11750-015-0372-1-e DE-627 ger DE-627 rakwb eng Izmailov, A. F. verfasserin aut Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad de Estadística e Investigación Operativa 2015 Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. Critical Lagrange multipliers (dpeaa)DE-He213 Second-order sufficiency (dpeaa)DE-He213 Newton-type methods (dpeaa)DE-He213 Sequential quadratic programming (dpeaa)DE-He213 Newton–Lagrange method (dpeaa)DE-He213 Superlinear convergence (dpeaa)DE-He213 Solodov, M. V. aut Enthalten in Top Berlin : Springer, 1986 23(2015), 1 vom: 27. Feb., Seite 1-26 (DE-627)531199711 (DE-600)2322573-7 1863-8279 nnns volume:23 year:2015 number:1 day:27 month:02 pages:1-26 https://dx.doi.org/10.1007/s11750-015-0372-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2015 1 27 02 1-26 |
spelling |
10.1007/s11750-015-0372-1 doi (DE-627)SPR022747141 (SPR)s11750-015-0372-1-e DE-627 ger DE-627 rakwb eng Izmailov, A. F. verfasserin aut Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad de Estadística e Investigación Operativa 2015 Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. Critical Lagrange multipliers (dpeaa)DE-He213 Second-order sufficiency (dpeaa)DE-He213 Newton-type methods (dpeaa)DE-He213 Sequential quadratic programming (dpeaa)DE-He213 Newton–Lagrange method (dpeaa)DE-He213 Superlinear convergence (dpeaa)DE-He213 Solodov, M. V. aut Enthalten in Top Berlin : Springer, 1986 23(2015), 1 vom: 27. Feb., Seite 1-26 (DE-627)531199711 (DE-600)2322573-7 1863-8279 nnns volume:23 year:2015 number:1 day:27 month:02 pages:1-26 https://dx.doi.org/10.1007/s11750-015-0372-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2015 1 27 02 1-26 |
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10.1007/s11750-015-0372-1 doi (DE-627)SPR022747141 (SPR)s11750-015-0372-1-e DE-627 ger DE-627 rakwb eng Izmailov, A. F. verfasserin aut Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad de Estadística e Investigación Operativa 2015 Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. Critical Lagrange multipliers (dpeaa)DE-He213 Second-order sufficiency (dpeaa)DE-He213 Newton-type methods (dpeaa)DE-He213 Sequential quadratic programming (dpeaa)DE-He213 Newton–Lagrange method (dpeaa)DE-He213 Superlinear convergence (dpeaa)DE-He213 Solodov, M. V. aut Enthalten in Top Berlin : Springer, 1986 23(2015), 1 vom: 27. Feb., Seite 1-26 (DE-627)531199711 (DE-600)2322573-7 1863-8279 nnns volume:23 year:2015 number:1 day:27 month:02 pages:1-26 https://dx.doi.org/10.1007/s11750-015-0372-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2015 1 27 02 1-26 |
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10.1007/s11750-015-0372-1 doi (DE-627)SPR022747141 (SPR)s11750-015-0372-1-e DE-627 ger DE-627 rakwb eng Izmailov, A. F. verfasserin aut Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad de Estadística e Investigación Operativa 2015 Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. Critical Lagrange multipliers (dpeaa)DE-He213 Second-order sufficiency (dpeaa)DE-He213 Newton-type methods (dpeaa)DE-He213 Sequential quadratic programming (dpeaa)DE-He213 Newton–Lagrange method (dpeaa)DE-He213 Superlinear convergence (dpeaa)DE-He213 Solodov, M. V. aut Enthalten in Top Berlin : Springer, 1986 23(2015), 1 vom: 27. Feb., Seite 1-26 (DE-627)531199711 (DE-600)2322573-7 1863-8279 nnns volume:23 year:2015 number:1 day:27 month:02 pages:1-26 https://dx.doi.org/10.1007/s11750-015-0372-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2015 1 27 02 1-26 |
allfieldsSound |
10.1007/s11750-015-0372-1 doi (DE-627)SPR022747141 (SPR)s11750-015-0372-1-e DE-627 ger DE-627 rakwb eng Izmailov, A. F. verfasserin aut Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Sociedad de Estadística e Investigación Operativa 2015 Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. Critical Lagrange multipliers (dpeaa)DE-He213 Second-order sufficiency (dpeaa)DE-He213 Newton-type methods (dpeaa)DE-He213 Sequential quadratic programming (dpeaa)DE-He213 Newton–Lagrange method (dpeaa)DE-He213 Superlinear convergence (dpeaa)DE-He213 Solodov, M. V. aut Enthalten in Top Berlin : Springer, 1986 23(2015), 1 vom: 27. Feb., Seite 1-26 (DE-627)531199711 (DE-600)2322573-7 1863-8279 nnns volume:23 year:2015 number:1 day:27 month:02 pages:1-26 https://dx.doi.org/10.1007/s11750-015-0372-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2015 1 27 02 1-26 |
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F.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2015</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Sociedad de Estadística e Investigación Operativa 2015</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Critical Lagrange multipliers</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Second-order sufficiency</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Newton-type methods</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Sequential quadratic programming</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Newton–Lagrange method</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Superlinear convergence</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Solodov, M. 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Izmailov, A. F. |
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Izmailov, A. F. misc Critical Lagrange multipliers misc Second-order sufficiency misc Newton-type methods misc Sequential quadratic programming misc Newton–Lagrange method misc Superlinear convergence Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it |
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Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it Critical Lagrange multipliers (dpeaa)DE-He213 Second-order sufficiency (dpeaa)DE-He213 Newton-type methods (dpeaa)DE-He213 Sequential quadratic programming (dpeaa)DE-He213 Newton–Lagrange method (dpeaa)DE-He213 Superlinear convergence (dpeaa)DE-He213 |
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misc Critical Lagrange multipliers misc Second-order sufficiency misc Newton-type methods misc Sequential quadratic programming misc Newton–Lagrange method misc Superlinear convergence |
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critical lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it |
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Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it |
abstract |
Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. © Sociedad de Estadística e Investigación Operativa 2015 |
abstractGer |
Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. © Sociedad de Estadística e Investigación Operativa 2015 |
abstract_unstemmed |
Abstract We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient. © Sociedad de Estadística e Investigación Operativa 2015 |
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container_issue |
1 |
title_short |
Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it |
url |
https://dx.doi.org/10.1007/s11750-015-0372-1 |
remote_bool |
true |
author2 |
Solodov, M. V. |
author2Str |
Solodov, M. V. |
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hochschulschrift_bool |
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doi_str |
10.1007/s11750-015-0372-1 |
up_date |
2024-07-03T14:36:47.878Z |
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score |
7.4010057 |