{"title":"A new dual-based cutting plane algorithm for nonlinear adjustable robust optimization","authors":"","doi":"10.1007/s10898-023-01360-2","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>This paper explores a class of nonlinear Adjustable Robust Optimization (ARO) problems, containing here-and-now and wait-and-see variables, with uncertainty in the objective function and constraints. By applying Fenchel’s duality on the wait-and-see variables, we obtain an equivalent dual reformulation, which is a nonlinear static robust optimization problem. Using the dual formulation, we provide conditions under which the ARO problem is convex on the here-and-now decision. Furthermore, since the dual formulation contains a non-concave maximization on the uncertain parameter, we use perspective relaxation and an alternating method to handle the non-concavity. By employing the perspective relaxation, we obtain an upper bound, which we show is the same as the static relaxation of the considered problem. Moreover, invoking the alternating method, we design a new dual-based cutting plane algorithm that is able to find a reasonable lower bound for the optimal objective value of the considered nonlinear ARO model. In addition to sketching and establishing the theoretical features of the algorithms, including convergence analysis, by numerical experiments we reveal the abilities of our cutting plane algorithm in producing locally robust solutions with an acceptable optimality gap.</p>","PeriodicalId":15961,"journal":{"name":"Journal of Global Optimization","volume":"36 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Global Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10898-023-01360-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
This paper explores a class of nonlinear Adjustable Robust Optimization (ARO) problems, containing here-and-now and wait-and-see variables, with uncertainty in the objective function and constraints. By applying Fenchel’s duality on the wait-and-see variables, we obtain an equivalent dual reformulation, which is a nonlinear static robust optimization problem. Using the dual formulation, we provide conditions under which the ARO problem is convex on the here-and-now decision. Furthermore, since the dual formulation contains a non-concave maximization on the uncertain parameter, we use perspective relaxation and an alternating method to handle the non-concavity. By employing the perspective relaxation, we obtain an upper bound, which we show is the same as the static relaxation of the considered problem. Moreover, invoking the alternating method, we design a new dual-based cutting plane algorithm that is able to find a reasonable lower bound for the optimal objective value of the considered nonlinear ARO model. In addition to sketching and establishing the theoretical features of the algorithms, including convergence analysis, by numerical experiments we reveal the abilities of our cutting plane algorithm in producing locally robust solutions with an acceptable optimality gap.
摘要 本文探讨了一类非线性可调稳健优化（ARO）问题，该问题包含此时此地和等待观察变量，目标函数和约束条件具有不确定性。通过对 "等待-观察 "变量应用 Fenchel 对偶，我们得到了一个等价的对偶重述，即一个非线性静态鲁棒优化问题。利用对偶表述，我们提供了 ARO 问题在此时此地的决策上具有凸性的条件。此外，由于对偶表述包含对不确定参数的非凹性最大化，我们使用透视松弛和交替法来处理非凹性。通过使用透视松弛法，我们得到了一个上界，并证明它与所考虑问题的静态松弛法相同。此外，利用交替法，我们设计了一种新的基于对偶的切割面算法，能够为所考虑的非线性 ARO 模型的最优目标值找到一个合理的下界。除了勾勒和建立算法的理论特征（包括收敛性分析）外，我们还通过数值实验揭示了我们的切割面算法在产生具有可接受最优性差距的局部稳健解方面的能力。
期刊介绍：
The Journal of Global Optimization publishes carefully refereed papers that encompass theoretical, computational, and applied aspects of global optimization. While the focus is on original research contributions dealing with the search for global optima of non-convex, multi-extremal problems, the journal’s scope covers optimization in the widest sense, including nonlinear, mixed integer, combinatorial, stochastic, robust, multi-objective optimization, computational geometry, and equilibrium problems. Relevant works on data-driven methods and optimization-based data mining are of special interest.
In addition to papers covering theory and algorithms of global optimization, the journal publishes significant papers on numerical experiments, new testbeds, and applications in engineering, management, and the sciences. Applications of particular interest include healthcare, computational biochemistry, energy systems, telecommunications, and finance. Apart from full-length articles, the journal features short communications on both open and solved global optimization problems. It also offers reviews of relevant books and publishes special issues.
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