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Optimization for Control Design
Published in Derek A. Linkens, CAD for Control Systems, 2020
David Q. Mayne, Hannah Michalska
It is easily shown that Ψi increases monotonically and converges to Ψ*≙min {Ψ(α) | α ∈ A} as i→∞. If Ψ is strictly convex, αi converges to the unique solution of PD; else, any accumulation point à of an infinite sequence generated by the algorithm is a solution to PD. At iteration i, Ψi is a lower bound, and Ψ(αi) an upper bound, to Ψ*. The number of constraints in the linear program in Step (a) increases at every iteration, increasing its complexity. However, there exists schemes for dropping constraints, without prejuducing convergence; these limit the complexity of the subproblem (a). The cutting plane algorithm is a member of a larger family of closely related algorithms, the outer approximations algorithms for semi-infinite programming [11–14].
Big Data in Computational Health Informatics
Published in Ayman El-Baz, Jasjit S. Suri, Big Data in Multimodal Medical Imaging, 2019
Ruogu Fang, Yao Xiao, Jianqiao Tian, Samira Pouyanfar, Yimin Yang, Shu-Ching Chen, S. S. Iyengar
Several feature selection algorithms have been recently proposed, dealing with big data. Unlike conventional feature selection algorithms, online learning, specifically online feature selection, is suitable for large-scale real-world applications in such a way that each feature is processed upon its arrival and each time the best feature set is maintained from all seen features [61]. Yu et al. [62] proposes a scalable and accurate online approach (SAOLA) to select important features from large datasets with high dimensionality. Using a sequential scan, the SAOLA overcomes critical challenges in online feature selection, including the computational cost of online processing specifically for large datasets, which keeps growing on and on. Tan et al. [63] also presents a new feature selection method for extremely high dimensional datasets on big data. In this paper, the problem is transformed into a convex semi-infinite programming (SIP) issue, which is solved by a new feature generating machine (FGM). FGM repeatedly extracts the most relevant features using a reduced and primal form of the multiple kernel learning (MKL) subproblems. FGM is appropriate for feature selection on big data due to its subproblem optimization which involves a small subset of features with reduced memory overhead.
An extended conjugate duality for generalized semi-infinite programming problems via a convex decomposition
Published in Optimization, 2020
A. Aboussoror, S. Adly, S. Salim
The problem with the same convex data has been considered, e.g. in [2,7]. The case where the function f is assumed to be convex has been also considered in [7,8,15]. Let us summarize some works on optimality conditions for the class of generalized semi-infinite programming problems. In [2], for problem necessary and sufficient global optimality conditions are given by using tools from convex analysis combined with a notion of stability of optimization problems. In particular, by means of an optimality condition, the initial problem is reduced to a - problem. In [7,8], Kanzi provided for problem necessary optimality conditions of Fritz-John type. The study in these papers was based on the respective use of the subdifferential of marginal functions and some constraint qualifications. In [13], the authors first established optimality conditions for a class of semi-infinite programming problems by using penalty functions. Then they derived optimality conditions for a class of generalized semi-infinite programming problems via semi-infinite programming ones. In [14], Ye and Wu provided first-order optimality conditions for a class of generalized semi-infinite programming problems where all functions are assumed to be continuously differentiable. These optimality conditions are obtained under various extended well-known constraint qualifications.
Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty
Published in Optimization, 2020
Xiangkai Sun, Kok Lay Teo, Jing Zeng, Liying Liu
Semi-infinite programming deals with optimization problems in which finitely many variables appear in infinitely many inequality constraints. It is a field of active research, not only because of its surprising structural aspects, but also due to a wide range of applications in different fields of mathematics, economics and engineering. In the last decades, many successful treatments of deterministic semi-infinite programming have been investigated from several different perspectives. We refer the readers to the books [1,2] and the survey papers [3,4] for more details.