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
Autor*in: |
Verma, Amit [verfasserIn] Lewis, Mark [verfasserIn] |
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E-Artikel |
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Sprache: |
Englisch |
Erschienen: |
2019 |
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Schlagwörter: |
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Übergeordnetes Werk: |
Enthalten in: Optimization letters - Berlin : Springer, 2007, 14(2019), 6 vom: 03. Aug., Seite 1557-1569 |
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Übergeordnetes Werk: |
volume:14 ; year:2019 ; number:6 ; day:03 ; month:08 ; pages:1557-1569 |
Links: |
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DOI / URN: |
10.1007/s11590-019-01460-7 |
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Katalog-ID: |
SPR040543137 |
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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 | |
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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 |
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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 |
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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 |
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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 |
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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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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 |
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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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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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Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions |
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Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions |
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optimal quadratic reformulations of fourth degree pseudo-boolean functions |
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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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Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR040543137</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20220111080715.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201007s2019 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s11590-019-01460-7</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR040543137</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s11590-019-01460-7-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Verma, Amit</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2019</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="520" ind1=" " ind2=" "><subfield code="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.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Quadratic unconstrained binary optimization</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Quadratic reformulation</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Rosenberg quadratization</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Nonlinear optimization</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Pseudo-Boolean optimization</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Preprocessing</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Lewis, Mark</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Optimization letters</subfield><subfield code="d">Berlin : Springer, 2007</subfield><subfield code="g">14(2019), 6 vom: 03. 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