{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:29:25Z","timestamp":1760146165099,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T00:00:00Z","timestamp":1728345600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12171219","61877032"],"award-info":[{"award-number":["12171219","61877032"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Distributionally robust optimization (DRO) is an advanced framework within the realm of optimization theory that addresses scenarios where the underlying probability distribution governing the data is uncertain or ambiguous. In this paper, we introduce a novel class of DRO challenges where the probability distribution of random variables is contingent upon the decision variables, and the ambiguity set is defined through parameterization involving the mean and a covariance matrix, which also depend on the decision variables. This dependency makes DRO difficult to solve directly; therefore, first, we demonstrate that under the condition of a full-space support set, the original problem can be reduced to a second-order cone programming (SOCP) problem. Subsequently, we solve this second-order cone programming problem using a projection differential equation approach. Compared with the traditional methods, the differential equation method offers advantages in providing continuous and smooth solutions, offering inherent stability analysis, and possessing a rich mathematical toolbox, which make the differential equation a powerful and versatile tool for addressing complex optimization challenges.<\/jats:p>","DOI":"10.3390\/axioms13100699","type":"journal-article","created":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T12:03:49Z","timestamp":1728389029000},"page":"699","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reformulation and Enhancement of Distributed Robust Optimization Framework Incorporating Decision-Adaptive Uncertainty Sets"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6579-5890","authenticated-orcid":false,"given":"Jie","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics, Liaoning Normal University, Dalian 116029, China"}]},{"given":"Shuang","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Basic Courses Teaching, Dalian Polytechnic University, Dalian 116034, China"}]},{"given":"Yifei","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Liaoning Normal University, Dalian 116029, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2291","DOI":"10.1287\/opre.2022.2326","article-title":"Finite-sample guarantees for Wasserstein distributionally robust optimization: Breaking the curse of dimensionality","volume":"71","author":"Gao","year":"2023","journal-title":"Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s10107-023-02014-7","article-title":"Residuals-based distributionally robust optimization with covariate information","volume":"207","author":"Kannan","year":"2024","journal-title":"Math. Program."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/j.ejor.2021.04.015","article-title":"Wasserstein distributionally robust chance-constrained optimization for energy and reserve dispatch: An exact and physically-bounded formulation","volume":"296","author":"Arrigo","year":"2022","journal-title":"Eur. J. Oper. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"159","DOI":"10.3934\/naco.2021057","article-title":"Distributionally robust optimization: A review on theory and applications","volume":"12","author":"Lin","year":"2022","journal-title":"Numer. Algebr. Control Optim."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1287\/moor.2022.1275","article-title":"Distributionally robust stochastic optimization with Wasserstein distance","volume":"48","author":"Gao","year":"2023","journal-title":"Math. Oper. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1287\/opre.2022.2383","article-title":"Wasserstein distributionally robust optimization and variation regularization","volume":"72","author":"Gao","year":"2024","journal-title":"Oper. Res."},{"key":"ref_7","first-page":"780","article-title":"Optimal inequalities in probability theory: A convex optimization approach","volume":"15","author":"Bertsimas","year":"2005","journal-title":"Oper. Res."},{"key":"ref_8","first-page":"1358","article-title":"Distributionally robust convex optimization","volume":"62","author":"Wiesemann","year":"2014","journal-title":"SIAM J. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1287\/opre.1090.0741","article-title":"Distributionally robust optimization under moment uncertainty with application to data-driven problems","volume":"58","author":"Delage","year":"2010","journal-title":"Oper. Res."},{"key":"ref_10","unstructured":"Liu, Q. (2018). Model and Stability Research on Distributionally Robust Optimization. [Doctoral Thesis, Dalian University of Technology]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1016\/j.ejor.2020.11.002","article-title":"Distributionally robust facility location problem under decision-dependent stochastic demand","volume":"292","author":"Basciftci","year":"2019","journal-title":"Eur. J. Oper. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.ejor.2021.07.013","article-title":"Distributionally robust optimization under endogenous uncertainty with an application in retrofitting planning","volume":"300","author":"Doan","year":"2022","journal-title":"Eur. J. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1855","DOI":"10.1137\/15M1038529","article-title":"Quantitative stability analysis for distributionally robust optimization with moment constraints","volume":"26","author":"Zhang","year":"2016","journal-title":"SIAM J. Optim."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Lin, S., Zhang, J., and Shi, N. (2022). An Alternating Iteration Algorithm for a Parameter-Dependent Distributionally Robust Optimization Model. Mathematics, 10.","DOI":"10.3390\/math10071175"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1287\/ijoc.2021.1096","article-title":"Distributionally robust optimization under a decision-dependent ambiguity set with applications to machine scheduling and humanitarian logistics","volume":"34","author":"Noyan","year":"2022","journal-title":"INFORMS J. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1080\/24725854.2023.2219281","article-title":"Decision-dependent distributionally robust Markov decision process method in dynamic epidemic control","volume":"56","author":"Song","year":"2024","journal-title":"IISE Trans."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Li, M., Tong, X., and Sun, H. (2024). Discretization and quantification for distributionally robust optimization with decision-dependent ambiguity sets. Optim. Methods Softw.","DOI":"10.1080\/10556788.2024.2401975"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1023\/B:COAP.0000033964.95511.23","article-title":"Interior point methods for second-order cone programming and OR applications","volume":"28","author":"Kuo","year":"2004","journal-title":"Comput. Optim. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2943","DOI":"10.1137\/22M1507681","article-title":"A quadratically convergent sequential programming method for second-order cone programs capable of warm starts","volume":"34","author":"Luo","year":"2024","journal-title":"SIAM J. Optim."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/s10957-022-02056-5","article-title":"Global convergence of algorithms under constant rank conditions for nonlinear second-order cone programming","volume":"195","author":"Andreani","year":"2022","journal-title":"J. Optim. Theory Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1748","DOI":"10.1137\/20M1374262","article-title":"An inexact augmented Lagrangian method for second-order cone programming with applications","volume":"31","author":"Liang","year":"2021","journal-title":"SIAM J. Optim."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2250030","DOI":"10.1142\/S0217595922500300","article-title":"An Implementable Augmented Lagrangian Method for Solving Second-Order Cone Constrained Variational Inequalities","volume":"40","author":"Sun","year":"2023","journal-title":"Pac. J. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2925","DOI":"10.3233\/JIFS-210164","article-title":"A new projection neural network for linear and convex quadratic second-order cone programming","volume":"42","author":"Zhang","year":"2022","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.3233\/JIFS-220972","article-title":"A new neural network based on smooth function for SOCCVI problems","volume":"44","author":"Liu","year":"2023","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.neucom.2022.12.008","article-title":"Neural network models for time-varying tensor complementarity problems","volume":"523","author":"Wei","year":"2023","journal-title":"Neurocomputing"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1016\/j.jfranklin.2023.11.041","article-title":"Finite time convergent recurrent neural network for variational inequality problems subject to equality constraints","volume":"361","author":"Conchas","year":"2024","journal-title":"J. Frankl. Inst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5580","DOI":"10.1109\/TSMC.2023.3274222","article-title":"A novel projection neural network for solving a class of monotone variational inequalities","volume":"53","author":"Wen","year":"2023","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.neucom.2021.04.059","article-title":"Exponential convergence of a proximal projection neural network for mixed variational inequalities and applications","volume":"454","author":"Ju","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1080\/02331934.2019.1705822","article-title":"Notes on a neural network approach to inverse variational inequalities","volume":"70","author":"Xu","year":"2021","journal-title":"Optimization"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1007\/s10957-021-01915-x","article-title":"Global exponential stability of a neural network for inverse variational inequalities","volume":"190","author":"Vuong","year":"2021","journal-title":"J. Optim. Theory Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"108068","DOI":"10.1016\/j.engappai.2024.108068","article-title":"Enhanced fault tolerant kinematic control of redundant robots with linear-variational-inequality based zeroing neural network","volume":"133","author":"Yang","year":"2024","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_32","first-page":"5511978","article-title":"A Neural Network Based on a Nonsmooth Equation for a Box Constrained Variational Inequality Problem","volume":"1","author":"Wang","year":"2024","journal-title":"J. Math."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Goberna, M.A., and L\u00f3pez, M.A. (2001). On Duality Theory of Conic Linear Problems. Semi-Infinite Programming. Nonconvex Optimization and Its Applications, Springer.","DOI":"10.1007\/978-1-4757-3403-4"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Faraut, U., and Kor\u00e1nyi, A. (1994). Analysis on Symmetric Cones. Oxford Mathematical Monographs, Oxford University Press.","DOI":"10.1093\/oso\/9780198534778.001.0001"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Bonnans, J., and Shapiro, A. (2000). Perturbation Analysis of Optimization Problems, Springer.","DOI":"10.1007\/978-1-4612-1394-9"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Facchinei, F., and Pang, J.S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer.","DOI":"10.1007\/b97544"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1109\/TNN.2004.824252","article-title":"A general projection neural network for solving monotone variational inequalities and related optimization problems","volume":"15","author":"Xia","year":"2004","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1109\/81.995659","article-title":"A projection neural network and its application to constrained optimization problems","volume":"49","author":"Xia","year":"2002","journal-title":"IEEE Trans. Circuits Syst. I Fundam. Theory Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/699\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:09:10Z","timestamp":1760112550000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/699"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,8]]},"references-count":38,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["axioms13100699"],"URL":"https:\/\/doi.org\/10.3390\/axioms13100699","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,10,8]]}}}