Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions

Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly dif...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Verma, Amit [verfasserIn] Lewis, Mark [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2019

Schlagwörter:	Quadratic unconstrained binary optimization Quadratic reformulation Rosenberg quadratization Nonlinear optimization Pseudo-Boolean optimization Preprocessing

Übergeordnetes Werk:	Enthalten in: Optimization letters - Berlin : Springer, 2007, 14(2019), 6 vom: 03. Aug., Seite 1557-1569
Übergeordnetes Werk:	volume:14 ; year:2019 ; number:6 ; day:03 ; month:08 ; pages:1557-1569

Links:	Volltext

DOI / URN:	10.1007/s11590-019-01460-7

Katalog-ID:	SPR040543137

Internformat


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520			\|a Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude.
650		4	\|a Quadratic unconstrained binary optimization \|7 (dpeaa)DE-He213
650		4	\|a Quadratic reformulation \|7 (dpeaa)DE-He213
650		4	\|a Rosenberg quadratization \|7 (dpeaa)DE-He213
650		4	\|a Nonlinear optimization \|7 (dpeaa)DE-He213
650		4	\|a Pseudo-Boolean optimization \|7 (dpeaa)DE-He213
650		4	\|a Preprocessing \|7 (dpeaa)DE-He213
700	1		\|a Lewis, Mark \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Optimization letters \|d Berlin : Springer, 2007 \|g 14(2019), 6 vom: 03. Aug., Seite 1557-1569 \|w (DE-627)534676499 \|w (DE-600)2374345-1 \|x 1862-4480 \|7 nnns
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856	4	0	\|u https://dx.doi.org/10.1007/s11590-019-01460-7 \|z lizenzpflichtig \|3 Volltext
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912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
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912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2057
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912			\|a GBV_ILN_2061
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912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2070
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912			\|a GBV_ILN_2106
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912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2116
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2119
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912			\|a GBV_ILN_2129
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912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
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912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
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matchkey_str	article:18624480:2019----::piaqartceomltosforheresu
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publishDate	2019
allfields	10.1007/s11590-019-01460-7 doi (DE-627)SPR040543137 (SPR)s11590-019-01460-7-e DE-627 ger DE-627 rakwb eng 510 ASE Verma, Amit verfasserin aut Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude. Quadratic unconstrained binary optimization (dpeaa)DE-He213 Quadratic reformulation (dpeaa)DE-He213 Rosenberg quadratization (dpeaa)DE-He213 Nonlinear optimization (dpeaa)DE-He213 Pseudo-Boolean optimization (dpeaa)DE-He213 Preprocessing (dpeaa)DE-He213 Lewis, Mark verfasserin aut Enthalten in Optimization letters Berlin : Springer, 2007 14(2019), 6 vom: 03. Aug., Seite 1557-1569 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:14 year:2019 number:6 day:03 month:08 pages:1557-1569 https://dx.doi.org/10.1007/s11590-019-01460-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 14 2019 6 03 08 1557-1569
spelling	10.1007/s11590-019-01460-7 doi (DE-627)SPR040543137 (SPR)s11590-019-01460-7-e DE-627 ger DE-627 rakwb eng 510 ASE Verma, Amit verfasserin aut Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude. Quadratic unconstrained binary optimization (dpeaa)DE-He213 Quadratic reformulation (dpeaa)DE-He213 Rosenberg quadratization (dpeaa)DE-He213 Nonlinear optimization (dpeaa)DE-He213 Pseudo-Boolean optimization (dpeaa)DE-He213 Preprocessing (dpeaa)DE-He213 Lewis, Mark verfasserin aut Enthalten in Optimization letters Berlin : Springer, 2007 14(2019), 6 vom: 03. Aug., Seite 1557-1569 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:14 year:2019 number:6 day:03 month:08 pages:1557-1569 https://dx.doi.org/10.1007/s11590-019-01460-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 14 2019 6 03 08 1557-1569
allfields_unstemmed	10.1007/s11590-019-01460-7 doi (DE-627)SPR040543137 (SPR)s11590-019-01460-7-e DE-627 ger DE-627 rakwb eng 510 ASE Verma, Amit verfasserin aut Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude. Quadratic unconstrained binary optimization (dpeaa)DE-He213 Quadratic reformulation (dpeaa)DE-He213 Rosenberg quadratization (dpeaa)DE-He213 Nonlinear optimization (dpeaa)DE-He213 Pseudo-Boolean optimization (dpeaa)DE-He213 Preprocessing (dpeaa)DE-He213 Lewis, Mark verfasserin aut Enthalten in Optimization letters Berlin : Springer, 2007 14(2019), 6 vom: 03. Aug., Seite 1557-1569 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:14 year:2019 number:6 day:03 month:08 pages:1557-1569 https://dx.doi.org/10.1007/s11590-019-01460-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 14 2019 6 03 08 1557-1569
allfieldsGer	10.1007/s11590-019-01460-7 doi (DE-627)SPR040543137 (SPR)s11590-019-01460-7-e DE-627 ger DE-627 rakwb eng 510 ASE Verma, Amit verfasserin aut Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude. Quadratic unconstrained binary optimization (dpeaa)DE-He213 Quadratic reformulation (dpeaa)DE-He213 Rosenberg quadratization (dpeaa)DE-He213 Nonlinear optimization (dpeaa)DE-He213 Pseudo-Boolean optimization (dpeaa)DE-He213 Preprocessing (dpeaa)DE-He213 Lewis, Mark verfasserin aut Enthalten in Optimization letters Berlin : Springer, 2007 14(2019), 6 vom: 03. Aug., Seite 1557-1569 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:14 year:2019 number:6 day:03 month:08 pages:1557-1569 https://dx.doi.org/10.1007/s11590-019-01460-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 14 2019 6 03 08 1557-1569
allfieldsSound	10.1007/s11590-019-01460-7 doi (DE-627)SPR040543137 (SPR)s11590-019-01460-7-e DE-627 ger DE-627 rakwb eng 510 ASE Verma, Amit verfasserin aut Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude. Quadratic unconstrained binary optimization (dpeaa)DE-He213 Quadratic reformulation (dpeaa)DE-He213 Rosenberg quadratization (dpeaa)DE-He213 Nonlinear optimization (dpeaa)DE-He213 Pseudo-Boolean optimization (dpeaa)DE-He213 Preprocessing (dpeaa)DE-He213 Lewis, Mark verfasserin aut Enthalten in Optimization letters Berlin : Springer, 2007 14(2019), 6 vom: 03. Aug., Seite 1557-1569 (DE-627)534676499 (DE-600)2374345-1 1862-4480 nnns volume:14 year:2019 number:6 day:03 month:08 pages:1557-1569 https://dx.doi.org/10.1007/s11590-019-01460-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 14 2019 6 03 08 1557-1569
language	English
source	Enthalten in Optimization letters 14(2019), 6 vom: 03. Aug., Seite 1557-1569 volume:14 year:2019 number:6 day:03 month:08 pages:1557-1569
sourceStr	Enthalten in Optimization letters 14(2019), 6 vom: 03. Aug., Seite 1557-1569 volume:14 year:2019 number:6 day:03 month:08 pages:1557-1569
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Quadratic unconstrained binary optimization Quadratic reformulation Rosenberg quadratization Nonlinear optimization Pseudo-Boolean optimization Preprocessing
dewey-raw	510
isfreeaccess_bool	false
container_title	Optimization letters
authorswithroles_txt_mv	Verma, Amit @@aut@@ Lewis, Mark @@aut@@
publishDateDaySort_date	2019-08-03T00:00:00Z
hierarchy_top_id	534676499
dewey-sort	3510
id	SPR040543137
language_de	englisch
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Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. 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author	Verma, Amit
spellingShingle	Verma, Amit ddc 510 misc Quadratic unconstrained binary optimization misc Quadratic reformulation misc Rosenberg quadratization misc Nonlinear optimization misc Pseudo-Boolean optimization misc Preprocessing Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions
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topic_title	510 ASE Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions Quadratic unconstrained binary optimization (dpeaa)DE-He213 Quadratic reformulation (dpeaa)DE-He213 Rosenberg quadratization (dpeaa)DE-He213 Nonlinear optimization (dpeaa)DE-He213 Pseudo-Boolean optimization (dpeaa)DE-He213 Preprocessing (dpeaa)DE-He213
topic	ddc 510 misc Quadratic unconstrained binary optimization misc Quadratic reformulation misc Rosenberg quadratization misc Nonlinear optimization misc Pseudo-Boolean optimization misc Preprocessing
topic_unstemmed	ddc 510 misc Quadratic unconstrained binary optimization misc Quadratic reformulation misc Rosenberg quadratization misc Nonlinear optimization misc Pseudo-Boolean optimization misc Preprocessing
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title	Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions
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title_full	Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions
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title_sort	optimal quadratic reformulations of fourth degree pseudo-boolean functions
title_auth	Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions
abstract	Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude.
abstractGer	Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude.
abstract_unstemmed	Abstract Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude.
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container_issue	6
title_short	Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions
url	https://dx.doi.org/10.1007/s11590-019-01460-7
remote_bool	true
author2	Lewis, Mark
author2Str	Lewis, Mark
ppnlink	534676499
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hochschulschrift_bool	false
doi_str	10.1007/s11590-019-01460-7
up_date	2024-07-03T16:41:18.539Z
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Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions

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