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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.19185 |
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| _version_ | 1866911412469628928 |
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| author | Yu, Xian Basciftci, Beste |
| author_facet | Yu, Xian Basciftci, Beste |
| contents | We consider a two-stage distributionally robust optimization (DRO) model with multimodal uncertainty, where both the mode probabilities and uncertainty distributions could be affected by the first-stage decisions. To address this setting, we propose a generic framework by introducing a $ϕ$-divergence based ambiguity set to characterize the decision-dependent mode probabilities and further consider both moment-based and Wasserstein distance-based ambiguity sets to characterize the uncertainty distribution under each mode. We identify two special $ϕ$-divergence examples (variation distance and $χ^2$-distance) and provide specific forms of decision dependence relationships under which we can derive tractable reformulations. Furthermore, we investigate the benefits of considering multimodality in a DRO model compared to a single-modal counterpart through an analytical analysis. Additionally, we develop a separation-based decomposition algorithm to solve the resulting multimodal decision-dependent DRO models with finite convergence and optimality guarantee under certain settings. We provide a detailed computational study over two example problem settings, the facility location problem and shipment planning problem with pricing, to illustrate our results, which demonstrate that omission of multimodality or decision-dependent uncertainties within DRO frameworks result in inadequately performing solutions with worse in-sample and out-of-sample performances under various settings. We further demonstrate the speed-ups obtained by the solution algorithm against the off-the-shelf solver over various instances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19185 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distributionally Robust Optimization with Multimodal Decision-Dependent Ambiguity Sets Yu, Xian Basciftci, Beste Optimization and Control We consider a two-stage distributionally robust optimization (DRO) model with multimodal uncertainty, where both the mode probabilities and uncertainty distributions could be affected by the first-stage decisions. To address this setting, we propose a generic framework by introducing a $ϕ$-divergence based ambiguity set to characterize the decision-dependent mode probabilities and further consider both moment-based and Wasserstein distance-based ambiguity sets to characterize the uncertainty distribution under each mode. We identify two special $ϕ$-divergence examples (variation distance and $χ^2$-distance) and provide specific forms of decision dependence relationships under which we can derive tractable reformulations. Furthermore, we investigate the benefits of considering multimodality in a DRO model compared to a single-modal counterpart through an analytical analysis. Additionally, we develop a separation-based decomposition algorithm to solve the resulting multimodal decision-dependent DRO models with finite convergence and optimality guarantee under certain settings. We provide a detailed computational study over two example problem settings, the facility location problem and shipment planning problem with pricing, to illustrate our results, which demonstrate that omission of multimodality or decision-dependent uncertainties within DRO frameworks result in inadequately performing solutions with worse in-sample and out-of-sample performances under various settings. We further demonstrate the speed-ups obtained by the solution algorithm against the off-the-shelf solver over various instances. |
| title | Distributionally Robust Optimization with Multimodal Decision-Dependent Ambiguity Sets |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2404.19185 |