Maximizing a Class of Utility Functions Over the Vertices of a Polytope
Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabili...
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| Autor*in: |
Atamtürk, Alper [verfasserIn] |
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Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017 |
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| Übergeordnetes Werk: |
Enthalten in: Operations research - Catonsville, MD : INFORMS, 1956, 65(2017), 2, Seite 433-445 |
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| Übergeordnetes Werk: |
volume:65 ; year:2017 ; number:2 ; pages:433-445 |
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| DOI / URN: |
10.1287/opre.2016.1570 |
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| 520 | |a Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. | ||
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10.1287/opre.2016.1570 doi PQ20170501 (DE-627)OLC1992122571 (DE-599)GBVOLC1992122571 (PRQ)c770-52d941406674ba1faa99fe5dbe57dd3c2e089e2c1c8f8148ee4ab6a5991aff0e0 (KEY)0032304720170000065000200433maximizingaclassofutilityfunctionsovertheverticeso DE-627 ger DE-627 rakwb eng 650 DE-600 SA 7410 AVZ rvk 85.03 bkl 30.00 bkl 85.00 bkl Atamtürk, Alper verfasserin aut Maximizing a Class of Utility Functions Over the Vertices of a Polytope 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. submodularity conic quadratic optimization value-at-risk assortment multinomial logit reliability PERT robust optimization combinatorial optimization Gómez, Andrés oth Enthalten in Operations research Catonsville, MD : INFORMS, 1956 65(2017), 2, Seite 433-445 (DE-627)129302287 (DE-600)123389-0 (DE-576)014495856 0030-364X nnns volume:65 year:2017 number:2 pages:433-445 http://dx.doi.org/10.1287/opre.2016.1570 Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW GBV_ILN_21 GBV_ILN_22 GBV_ILN_26 GBV_ILN_30 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_110 GBV_ILN_130 GBV_ILN_134 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4322 GBV_ILN_4700 SA 7410 85.03 AVZ 30.00 AVZ 85.00 AVZ AR 65 2017 2 433-445 |
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10.1287/opre.2016.1570 doi PQ20170501 (DE-627)OLC1992122571 (DE-599)GBVOLC1992122571 (PRQ)c770-52d941406674ba1faa99fe5dbe57dd3c2e089e2c1c8f8148ee4ab6a5991aff0e0 (KEY)0032304720170000065000200433maximizingaclassofutilityfunctionsovertheverticeso DE-627 ger DE-627 rakwb eng 650 DE-600 SA 7410 AVZ rvk 85.03 bkl 30.00 bkl 85.00 bkl Atamtürk, Alper verfasserin aut Maximizing a Class of Utility Functions Over the Vertices of a Polytope 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. submodularity conic quadratic optimization value-at-risk assortment multinomial logit reliability PERT robust optimization combinatorial optimization Gómez, Andrés oth Enthalten in Operations research Catonsville, MD : INFORMS, 1956 65(2017), 2, Seite 433-445 (DE-627)129302287 (DE-600)123389-0 (DE-576)014495856 0030-364X nnns volume:65 year:2017 number:2 pages:433-445 http://dx.doi.org/10.1287/opre.2016.1570 Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW GBV_ILN_21 GBV_ILN_22 GBV_ILN_26 GBV_ILN_30 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_110 GBV_ILN_130 GBV_ILN_134 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4322 GBV_ILN_4700 SA 7410 85.03 AVZ 30.00 AVZ 85.00 AVZ AR 65 2017 2 433-445 |
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10.1287/opre.2016.1570 doi PQ20170501 (DE-627)OLC1992122571 (DE-599)GBVOLC1992122571 (PRQ)c770-52d941406674ba1faa99fe5dbe57dd3c2e089e2c1c8f8148ee4ab6a5991aff0e0 (KEY)0032304720170000065000200433maximizingaclassofutilityfunctionsovertheverticeso DE-627 ger DE-627 rakwb eng 650 DE-600 SA 7410 AVZ rvk 85.03 bkl 30.00 bkl 85.00 bkl Atamtürk, Alper verfasserin aut Maximizing a Class of Utility Functions Over the Vertices of a Polytope 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. submodularity conic quadratic optimization value-at-risk assortment multinomial logit reliability PERT robust optimization combinatorial optimization Gómez, Andrés oth Enthalten in Operations research Catonsville, MD : INFORMS, 1956 65(2017), 2, Seite 433-445 (DE-627)129302287 (DE-600)123389-0 (DE-576)014495856 0030-364X nnns volume:65 year:2017 number:2 pages:433-445 http://dx.doi.org/10.1287/opre.2016.1570 Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW GBV_ILN_21 GBV_ILN_22 GBV_ILN_26 GBV_ILN_30 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_110 GBV_ILN_130 GBV_ILN_134 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4322 GBV_ILN_4700 SA 7410 85.03 AVZ 30.00 AVZ 85.00 AVZ AR 65 2017 2 433-445 |
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10.1287/opre.2016.1570 doi PQ20170501 (DE-627)OLC1992122571 (DE-599)GBVOLC1992122571 (PRQ)c770-52d941406674ba1faa99fe5dbe57dd3c2e089e2c1c8f8148ee4ab6a5991aff0e0 (KEY)0032304720170000065000200433maximizingaclassofutilityfunctionsovertheverticeso DE-627 ger DE-627 rakwb eng 650 DE-600 SA 7410 AVZ rvk 85.03 bkl 30.00 bkl 85.00 bkl Atamtürk, Alper verfasserin aut Maximizing a Class of Utility Functions Over the Vertices of a Polytope 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. submodularity conic quadratic optimization value-at-risk assortment multinomial logit reliability PERT robust optimization combinatorial optimization Gómez, Andrés oth Enthalten in Operations research Catonsville, MD : INFORMS, 1956 65(2017), 2, Seite 433-445 (DE-627)129302287 (DE-600)123389-0 (DE-576)014495856 0030-364X nnns volume:65 year:2017 number:2 pages:433-445 http://dx.doi.org/10.1287/opre.2016.1570 Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW GBV_ILN_21 GBV_ILN_22 GBV_ILN_26 GBV_ILN_30 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_110 GBV_ILN_130 GBV_ILN_134 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4322 GBV_ILN_4700 SA 7410 85.03 AVZ 30.00 AVZ 85.00 AVZ AR 65 2017 2 433-445 |
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10.1287/opre.2016.1570 doi PQ20170501 (DE-627)OLC1992122571 (DE-599)GBVOLC1992122571 (PRQ)c770-52d941406674ba1faa99fe5dbe57dd3c2e089e2c1c8f8148ee4ab6a5991aff0e0 (KEY)0032304720170000065000200433maximizingaclassofutilityfunctionsovertheverticeso DE-627 ger DE-627 rakwb eng 650 DE-600 SA 7410 AVZ rvk 85.03 bkl 30.00 bkl 85.00 bkl Atamtürk, Alper verfasserin aut Maximizing a Class of Utility Functions Over the Vertices of a Polytope 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. submodularity conic quadratic optimization value-at-risk assortment multinomial logit reliability PERT robust optimization combinatorial optimization Gómez, Andrés oth Enthalten in Operations research Catonsville, MD : INFORMS, 1956 65(2017), 2, Seite 433-445 (DE-627)129302287 (DE-600)123389-0 (DE-576)014495856 0030-364X nnns volume:65 year:2017 number:2 pages:433-445 http://dx.doi.org/10.1287/opre.2016.1570 Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW GBV_ILN_21 GBV_ILN_22 GBV_ILN_26 GBV_ILN_30 GBV_ILN_40 GBV_ILN_60 GBV_ILN_65 GBV_ILN_110 GBV_ILN_130 GBV_ILN_134 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_4012 GBV_ILN_4029 GBV_ILN_4126 GBV_ILN_4193 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4322 GBV_ILN_4700 SA 7410 85.03 AVZ 30.00 AVZ 85.00 AVZ AR 65 2017 2 433-445 |
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maximizing a class of utility functions over the vertices of a polytope |
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Maximizing a Class of Utility Functions Over the Vertices of a Polytope |
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Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. |
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Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. |
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Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives. |
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