Convex projection and convex multi-objective optimization
Abstract In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projecti...
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| Autor*in: |
Kováčová, Gabriela [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s) 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of global optimization - Springer US, 1991, 83(2021), 2 vom: 24. Nov., Seite 301-327 |
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| Übergeordnetes Werk: |
volume:83 ; year:2021 ; number:2 ; day:24 ; month:11 ; pages:301-327 |
| Links: |
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| DOI / URN: |
10.1007/s10898-021-01111-1 |
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OLC2078763098 |
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| 520 | |a Abstract In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations. | ||
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10.1007/s10898-021-01111-1 doi (DE-627)OLC2078763098 (DE-He213)s10898-021-01111-1-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn 83.00 bkl Kováčová, Gabriela verfasserin (orcid)0000-0003-2088-0597 aut Convex projection and convex multi-objective optimization 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations. Convex projection Convex vector optimization Convex multi-objective optimization Rudloff, Birgit (orcid)0000-0003-1675-5451 aut Enthalten in Journal of global optimization Springer US, 1991 83(2021), 2 vom: 24. Nov., Seite 301-327 (DE-627)130979074 (DE-600)1074566-X (DE-576)034188533 0925-5001 nnns volume:83 year:2021 number:2 day:24 month:11 pages:301-327 https://doi.org/10.1007/s10898-021-01111-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_2030 83.00 VZ AR 83 2021 2 24 11 301-327 |
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10.1007/s10898-021-01111-1 doi (DE-627)OLC2078763098 (DE-He213)s10898-021-01111-1-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn 83.00 bkl Kováčová, Gabriela verfasserin (orcid)0000-0003-2088-0597 aut Convex projection and convex multi-objective optimization 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations. Convex projection Convex vector optimization Convex multi-objective optimization Rudloff, Birgit (orcid)0000-0003-1675-5451 aut Enthalten in Journal of global optimization Springer US, 1991 83(2021), 2 vom: 24. Nov., Seite 301-327 (DE-627)130979074 (DE-600)1074566-X (DE-576)034188533 0925-5001 nnns volume:83 year:2021 number:2 day:24 month:11 pages:301-327 https://doi.org/10.1007/s10898-021-01111-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_2030 83.00 VZ AR 83 2021 2 24 11 301-327 |
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10.1007/s10898-021-01111-1 doi (DE-627)OLC2078763098 (DE-He213)s10898-021-01111-1-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn 83.00 bkl Kováčová, Gabriela verfasserin (orcid)0000-0003-2088-0597 aut Convex projection and convex multi-objective optimization 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations. Convex projection Convex vector optimization Convex multi-objective optimization Rudloff, Birgit (orcid)0000-0003-1675-5451 aut Enthalten in Journal of global optimization Springer US, 1991 83(2021), 2 vom: 24. Nov., Seite 301-327 (DE-627)130979074 (DE-600)1074566-X (DE-576)034188533 0925-5001 nnns volume:83 year:2021 number:2 day:24 month:11 pages:301-327 https://doi.org/10.1007/s10898-021-01111-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_2030 83.00 VZ AR 83 2021 2 24 11 301-327 |
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510 VZ 17,1 ssgn 83.00 bkl Convex projection and convex multi-objective optimization Convex projection Convex vector optimization Convex multi-objective optimization |
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Convex projection and convex multi-objective optimization |
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Convex projection and convex multi-objective optimization |
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Kováčová, Gabriela |
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Kováčová, Gabriela Rudloff, Birgit |
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Kováčová, Gabriela |
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convex projection and convex multi-objective optimization |
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Convex projection and convex multi-objective optimization |
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Abstract In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations. © The Author(s) 2021 |
| abstractGer |
Abstract In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations. © The Author(s) 2021 |
| abstract_unstemmed |
Abstract In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations. © The Author(s) 2021 |
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