Metric regularity relative to a cone
- Authors: Van Ngai, Huynh , Tron, Nguyen , Théra, Michel
- Date: 2019
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 47, no. 3 (2019), p. 733-756
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The purpose of this paper is to discuss some of the highlights of the theory of metric regularity relative to a cone. For example, we establish a slope and some coderivative characterizations of this concept, as well as some stability results with respect to a Lipschitz perturbation.
Stability of error bounds for convex constraints systems in Banach spaces
- Authors: Thera, Michel , Van Ngai, Huynh , Kruger, Alexander
- Date: 2010
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 20, no. 6 (2010), p. 3280-3296
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- Description: This paper studies stability of error bounds for convex constraints in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems are considered.
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Directional metric regularity of multifunctions
- Authors: Ngai, Huynh Van , Thera, Michel
- Date: 2015
- Type: Text , Journal article
- Relation: Mathematics of Operations Research Vol. 40, no. 4 (2015), p. 969-991
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.
- Description: In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity. © 2015 INFORMS.
Directional metric pseudo subregularity of set-valued mappings: a general model
- Authors: Van Ngai, Huynh , Tron, Nguyen , Van Vu, Nguyen , Théra, Michel
- Date: 2020
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 61-87
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- Description: This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established. © 2019, Springer Nature B.V.
Variational analysis of paraconvex multifunctions
- Authors: Van Ngai, Huynh , Tron, Nguyen , Van Vu, Nguyen , Théra, Michel
- Date: 2022
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 180-218
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: Our aim in this article is to study the class of so-called ρ- paraconvex multifunctions from a Banach space X into the subsets of another Banach space Y. These multifunctions are defined in relation with a modulus function ρ: X→ [0 , + ∞) satisfying some suitable conditions. This class of multifunctions generalizes the class of γ- paraconvex multifunctions with γ> 1 introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. We establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Stability of error bounds for semi-infinite convex constraint systems
- Authors: Van Ngai, Huynh , Kruger, Alexander , Théra, Michel
- Date: 2010
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 20, no. 4 (2010), p. 2080-2096
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- Description: In this paper, we are concerned with the stability of the error bounds for semi-infinite convex constraint systems. Roughly speaking, the error bound of a system of inequalities is said to be stable if all its "small" perturbations admit a (local or global) error bound. We first establish subdifferential characterizations of the stability of error bounds for semi-infinite systems of convex inequalities. By applying these characterizations, we extend some results established by Azé and Corvellec [SIAM J. Optim., 12 (2002), pp. 913-927] on the sensitivity analysis of Hoffman constants to semi-infinite linear constraint systems. Copyright © 2010, Society for Industrial and Applied Mathematics.
Metric Regularity of the Sum of Multifunctions and Applications
- Authors: Van Ngai, Huynh , Tron, Nguyen Tron , Thera, Michel
- Date: 2014
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 160, no. 2 (2014), p. 355-390
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported. © 2013 Springer Science+Business Media New York.
Directional Holder metric regularity
- Authors: Ngai, Huynh Van , Tron, Nguyen Huu , Thera, Michel
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 785-819
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- Description: This paper sheds new light on regularity of multifunctions through various characterizations of directional Holder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Holder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Holder/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.
Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions
- Authors: Huynh, Van Ngai , Théra, Michel
- Date: 2018
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 457, no. 2 (2018), p. 1403-1421
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: In this paper, we present some inverse function theorems and implicit function theorems for set-valued mappings between Fréchet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence of solutions of differential equations in Fréchet spaces with non-smooth data is given.
Characterizations of stability of error bounds for convex inequality constraint systems
- Authors: Wei, Zhou , Théra, Michel , Yao, Jen-Chih
- Date: 2022
- Type: Text , Journal article
- Relation: Open Journal of Mathematical Optimization Vol. 3, no. (2022), p. 1-17
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- Description: In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimum of the directional derivative at a reference point over the unit sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the directional derivatives, at all points in the boundary of the solution set, over the unit sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations of stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constraint systems have the same linear perturbation. Our work demonstrates that verifying the stability of error bounds for convex inequality constraint systems is, to some degree, equivalent to solving convex minimization problems (defined by directional derivatives) over the unit sphere. © Zhou Wei & Michel Théra & Jen-Chih Yao.
Perturbation of error bounds
- Authors: Kruger, Alexander , López, Marco , Théra, Michel
- Date: 2018
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.
Slopes of multifunctions and extensions of metric regularity
- Authors: Ngai, Huynh Van , Kruger, Alexander , Thera, Michel
- Date: 2012
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics (Tạp chí toán học) Vol. 40, no. 2/3 (2012), p. 355-369
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article aims to demonstrate how the definitions of slopes can be extended to multi-valued mappings between metric spaces and applied for characterizing metric regularity. Several kinds of local and nonlocal slopes are defined and several metric regularity properties for set-valued mappings between metric spaces are investigated.
Error bounds revisited
- Authors: Cuong, Nguyen , Kruger, Alexander
- Date: 2022
- Type: Text , Journal article
- Relation: Optimization Vol. 71, no. 4 (2022), p. 1021-1053
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. Employing special collections of slope operators, we introduce a succinct form of sufficient error bound conditions, which allows one to combine in a single statement several different assertions: nonlocal and local primal space conditions in complete metric spaces, and subdifferential conditions in Banach and Asplund spaces. © 2022 Informa UK Limited, trading as Taylor & Francis Group.
Soil moisture, organic carbon, and nitrogen content prediction with hyperspectral data using regression models
- Authors: Datta, Dristi , Paul, Manoranjan , Murshed, Manzur , Teng, Shyh Wei , Schmidtke, Leigh
- Date: 2022
- Type: Text , Journal article
- Relation: Sensors (Basel, Switzerland) Vol. 22, no. 20 (2022), p.
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- Description: Soil moisture, soil organic carbon, and nitrogen content prediction are considered significant fields of study as they are directly related to plant health and food production. Direct estimation of these soil properties with traditional methods, for example, the oven-drying technique and chemical analysis, is a time and resource-consuming approach and can predict only smaller areas. With the significant development of remote sensing and hyperspectral (HS) imaging technologies, soil moisture, carbon, and nitrogen can be estimated over vast areas. This paper presents a generalized approach to predicting three different essential soil contents using a comprehensive study of various machine learning (ML) models by considering the dimensional reduction in feature spaces. In this study, we have used three popular benchmark HS datasets captured in Germany and Sweden. The efficacy of different ML algorithms is evaluated to predict soil content, and significant improvement is obtained when a specific range of bands is selected. The performance of ML models is further improved by applying principal component analysis (PCA), a dimensional reduction method that works with an unsupervised learning method. The effect of soil temperature on soil moisture prediction is evaluated in this study, and the results show that when the soil temperature is considered with the HS band, the soil moisture prediction accuracy does not improve. However, the combined effect of band selection and feature transformation using PCA significantly enhances the prediction accuracy for soil moisture, carbon, and nitrogen content. This study represents a comprehensive analysis of a wide range of established ML regression models using data preprocessing, effective band selection, and data dimension reduction and attempt to understand which feature combinations provide the best accuracy. The outcomes of several ML models are verified with validation techniques and the best- and worst-case scenarios in terms of soil content are noted. The proposed approach outperforms existing estimation techniques.
Recent contributions to linear semi-infinite optimization
- Authors: Goberna, Miguel , López, Marco
- Date: 2017
- Type: Text , Journal article
- Relation: 4OR: A Quarterly Journal of Operations Research Vol. 15, no. 3 (2017), p. 221-264
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of fields. Extensions to related optimization areas, as convex semi-infinite optimization, linear infinite optimization, and multi-objective linear semi-infinite optimization, are also commented. © 2017, Springer-Verlag GmbH Germany.
On semiregularity of mappings
- Authors: Cibulka, Radek , Fabian, Marian , Kruger, Alexander
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 473, no. 2 (2019), p. 811-836
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part. © 2019 Elsevier Inc.
Outer limits of subdifferentials for min–max type functions
- Authors: Eberhard, Andrew , Roshchina, Vera , Sang, Tian
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization Vol. 68, no. 7 (2019), p. 1391-1409
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- Description: We generalize the outer subdifferential construction suggested by Cánovas, Henrion, López and Parra for max type functions to pointwise minima of regular Lipschitz functions. We also answer an open question about the relation between the outer subdifferential of the support of a regular function and the end set of its subdifferential posed by Li, Meng and Yang.
Some new characterizations of intrinsic transversality in hilbert spaces
- Authors: Thao, Nguyen , Bui, Hoa , Cuong, Nguyen , Verhaegen, Michel
- Date: 2020
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 5-39
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- Description: Motivated by a number of questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. New results in terms of normal vectors clarify the picture of intrinsic transversality, its variants and sufficient conditions for subtransversality, and unify several of them. For the first time, intrinsic transversality is characterized by an equivalent condition which does not involve normal vectors. This characterization offers another perspective on intrinsic transversality. As a consequence, the obtained results allow us to answer a number of important questions about transversality-type properties. © 2020, The Author(s).
An adaptive hierarchical sliding mode controller for autonomous underwater vehicles
- Authors: Van Vu, Quang , Dinh, Tuan , Van Nguyen, Thien , Tran, Hoang , Nguyen, Linh
- Date: 2021
- Type: Text , Journal article
- Relation: Electronics (Switzerland) Vol. 10, no. 18 (2021), p.
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- Description: The paper addresses a problem of efficiently controlling an autonomous underwater vehicle (AUV), where its typical underactuated model is considered. Due to critical uncertainties and nonlinearities in the system caused by unavoidable external disturbances such as ocean currents when it operates, it is paramount to robustly maintain motions of the vehicle over time as expected. Therefore, it is proposed to employ the hierarchical sliding mode control technique to design the closed-loop control scheme for the device. However, exactly determining parameters of the AUV control system is impractical since its nonlinearities and external disturbances can vary those parameters over time. Thus, it is proposed to exploit neural networks to develop an adaptive learning mechanism that allows the system to learn its parameters adaptively. More importantly, stability of the AUV system controlled by the proposed approach is theoretically proved to be guaranteed by the use of the Lyapunov theory. Effectiveness of the proposed control scheme was verified by the experiments implemented in a synthetic environment, where the obtained results are highly promising. © 2021 by the authors. Licensee MDPI, Basel, Switzerland. **Please note that there are multiple authors for this article therefore only the name of the first 5 including Federation University Australia affiliate “Linh Nguyen" is provided in this record**
Primal Characterizations of error bounds for composite-convex inequalities
- Authors: Wei, Zhou , Théra, Michel , Yao, Jen-Chih
- Date: 2023
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 30, no. 4 (2023), p. 1329-1350
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- Description: This paper is devoted to primal conditions of error bounds for a general function. In terms of Bouligand tangent cones, lower Hadamard directional derivatives and the Hausdorff-Pompeiu excess of subsets, we provide several necessary and/or sufficient conditions for error bounds with mild assumptions. Then we use these primal results to characterize error bounds for composite-convex functions (i.e. the composition of a convex function with a continuously differentiable mapping). It is proved that the primal characterization of error bounds can be established via Bouligand tangent cones, directional derivatives and the Hausdorff-Pompeiu excess if the mapping is metrically regular at the given point. The accurate estimate on the error bound modulus is also obtained. © 2023 Heldermann Verlag. All rights reserved.