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A new proof of Balinski's theorem on the connectivity of polytopes

- Pineda-Villavicencio, Guillermo

  • Authors: Pineda-Villavicencio, Guillermo
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Discrete Mathematics Vol. 344, no. 7 (2021), p.
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  • Description: Balinski (1961) proved that the graph of a d-dimensional convex polytope is d-connected. We provide a new proof of this result. Our proof provides details on the nature of a separating set with exactly d vertices; some of which appear to be new. © 2021 Elsevier B.V.

Aggregate subgradient method for nonsmooth DC optimization

- Bagirov, Adil, Taheri, Sona, Joki, Kaisa, Karmitsa, Napsu, Mäkelä, Marko

  • Authors: Bagirov, Adil , Taheri, Sona , Joki, Kaisa , Karmitsa, Napsu , Mäkelä, Marko
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Optimization Letters Vol. 15, no. 1 (2021), p. 83-96
  • Relation: http://purl.org/au-research/grants/arc/DP190100580
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  • Description: The aggregate subgradient method is developed for solving unconstrained nonsmooth difference of convex (DC) optimization problems. The proposed method shares some similarities with both the subgradient and the bundle methods. Aggregate subgradients are defined as a convex combination of subgradients computed at null steps between two serious steps. At each iteration search directions are found using only two subgradients: the aggregate subgradient and a subgradient computed at the current null step. It is proved that the proposed method converges to a critical point of the DC optimization problem and also that the number of null steps between two serious steps is finite. The new method is tested using some academic test problems and compared with several other nonsmooth DC optimization solvers. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

- Dao, Minh, Phan, Hung

  • Authors: Dao, Minh , Phan, Hung
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (2021), p.
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  • Description: Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem. © 2021, The Author(s).

An augmented subgradient method for minimizing nonsmooth DC functions

- Bagirov, Adil, Hoseini Monjezi, Najmeh, Taheri, Sona

  • Authors: Bagirov, Adil , Hoseini Monjezi, Najmeh , Taheri, Sona
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Computational Optimization and Applications Vol. 80, no. 2 (2021), p. 411-438
  • Relation: http://purl.org/au-research/grants/arc/DP190100580
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  • Description: A method, called an augmented subgradient method, is developed to solve unconstrained nonsmooth difference of convex (DC) optimization problems. At each iteration of this method search directions are found by using several subgradients of the first DC component and one subgradient of the second DC component of the objective function. The developed method applies an Armijo-type line search procedure to find the next iteration point. It is proved that the sequence of points generated by the method converges to a critical point of the unconstrained DC optimization problem. The performance of the method is demonstrated using academic test problems with nonsmooth DC objective functions and its performance is compared with that of two general nonsmooth optimization solvers and five solvers specifically designed for unconstrained DC optimization. Computational results show that the developed method is efficient and robust for solving nonsmooth DC optimization problems. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Assessing nitrate contamination risks in groundwater : a machine learning approach

- Awais, Muhammad, Aslam, Bilal, Maqsoom, Ahsen, Khalil, Umer, Imran, Muhammad

  • Authors: Awais, Muhammad , Aslam, Bilal , Maqsoom, Ahsen , Khalil, Umer , Imran, Muhammad
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Applied Sciences (Switzerland) Vol. 11, no. 21 (2021), p.
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  • Description: Groundwater is one of the primary sources for the daily water requirements of the masses, but it is subjected to contamination due to the pollutants, such as nitrate, percolating through the soil with water. Especially in built-up areas, groundwater vulnerability and contamination are of major concern, and require appropriate consideration. The present study develops a novel framework for assessing groundwater nitrate contamination risk for the area along the Karakoram Highway, which is a part of the China Pakistan Economic Corridor (CPEC) route in northern Pakistan. A groundwater vulnerability map was prepared using the DRASTIC model. The nitrate concentration data from a previous study were used to formulate the nitrate contamination map. Three machine learning (ML) models, i.e., Support Vector Machine (SVM), Multivariate Discriminant Analysis (MDA), and Boosted Regression Trees (BRT), were used to analyze the probability of groundwater contamination incidence. Furthermore, groundwater contamination probability maps were obtained utilizing the ensemble modeling approach. The models were calibrated and validated through calibration trials, using the area under the receiver operating characteristic curve method (AUC), where a minimum AUC threshold value of 80% was achieved. Results indicated the accuracy of the models to be in the range of 0.82–0.87. The final groundwater contamination risk map highlights that 34% of the area is moderately vulnerable to groundwater contamination, and 13% of the area is exposed to high groundwater contamination risk. The findings of this study can facilitate decision-making regarding the location of future built-up areas properly in order to mitigate the nitrate contamination that can further reduce the associated health risks. © 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 “Muhammad Imran” is provided in this record**

Chebyshev multivariate polynomial approximation and point reduction procedure

- Sukhorukova, Nadezda, Ugon, Julien, Yost, David

  • Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Constructive Approximation Vol. 53, no. 3 (2021), p. 529-544
  • Relation: http://purl.org/au-research/grants/arc/DP180100602
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  • Description: We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Constraint reduction reformulations for projection algorithms with applications to wavelet construction

- Dao, Minh, Dizon, Neil, Hogan, Jeffrey, Tam, Matthew

  • Authors: Dao, Minh , Dizon, Neil , Hogan, Jeffrey , Tam, Matthew
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Journal of Optimization Theory and Applications Vol. 190, no. 1 (2021), p. 201-233
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  • Description: We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra’s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas–Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas–Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas–Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Efficient deterministic algorithm for huge-sized noisy sensor localization problems via canonical duality theory

- Latorre, Vittorio, Gao, David

  • Authors: Latorre, Vittorio , Gao, David
  • Date: 2021
  • Type: Text , Journal article
  • Relation: IEEE Transactions on Cybernetics Vol. 51, no. 10 (2021), p. 5069-5081
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  • Description: This paper presents a new deterministic method and a polynomial-time algorithm for solving general huge-sized sensor network localization problems. The problem is first formulated as a nonconvex minimization, which was considered as an NP-hard based on conventional theories. However, by the canonical duality theory, this challenging problem can be equivalently converted into a convex dual problem. By introducing a new optimality measure, a powerful canonical primal-dual interior (CPDI) point algorithm is developed which can solve efficiently huge-sized problems with hundreds of thousands of sensors. The new method is compared with the popular methods in the literature. Results show that the CPDI algorithm is not only faster than the benchmarks but also much more accurate on networks affected by noise on the distances. © 2013 IEEE.

From general language understanding to noisy text comprehension

- Kasthuriarachchy, Buddhika, Chetty, Madhu, Shatte, Adrian, Walls, Darren

  • Authors: Kasthuriarachchy, Buddhika , Chetty, Madhu , Shatte, Adrian , Walls, Darren
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Applied Sciences (Switzerland) Vol. 11, no. 17 (2021), p.
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  • Description: Obtaining meaning-rich representations of social media inputs, such as Tweets (unstructured and noisy text), from general-purpose pre-trained language models has become challenging, as these inputs typically deviate from mainstream English usage. The proposed research establishes effective methods for improving the comprehension of noisy texts. For this, we propose a new generic methodology to derive a diverse set of sentence vectors combining and extracting various linguistic characteristics from latent representations of multi-layer, pre-trained language models. Further, we clearly establish how BERT, a state-of-the-art pre-trained language model, comprehends the linguistic attributes of Tweets to identify appropriate sentence representations. Five new probing tasks are developed for Tweets, which can serve as benchmark probing tasks to study noisy text comprehension. Experiments are carried out for classification accuracy by deriving the sentence vectors from GloVe-based pre-trained models and Sentence-BERT, and by using different hidden layers from the BERT model. We show that the initial and middle layers of BERT have better capability for capturing the key linguistic characteristics of noisy texts than its latter layers. With complex predictive models, we further show that the sentence vector length has lesser importance to capture linguistic information, and the proposed sentence vectors for noisy texts perform better than the existing state-of-the-art sentence vectors. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

Gateaux differentiability revisited

- Abbasi, Malek, Kruger, Alexander, Théra, Michel

Generalised rational approximation and its application to improve deep learning classifiers

- Peiris, V, Sharon, Nir, Sukhorukova, Nadezda, Ugon, Julien

  • Authors: Peiris, V , Sharon, Nir , Sukhorukova, Nadezda , Ugon, Julien
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Applied Mathematics and Computation Vol. 389, no. (2021), p.
  • Relation: http://purl.org/au-research/grants/arc/DP180100602
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  • Description: A rational approximation (that is, approximation by a ratio of two polynomials) is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non-Lipschitz functions, where polynomial approximations are not efficient. We prove that the optimisation problems appearing in the best uniform rational approximation and its generalisation to a ratio of linear combinations of basis functions are quasiconvex even when the basis functions are not restricted to monomials. Then we show how this fact can be used in the development of computational methods. This paper presents a theoretical study of the arising optimisation problems and provides results of several numerical experiments. We apply our approximation as a preprocessing step to deep learning classifiers and demonstrate that the classification accuracy is significantly improved compared to the classification of the raw signals. © 2020
  • Description: This research was supported by the Australian Research Council (ARC), Solving hard Chebyshev approximation problems through nonsmooth analysis (Discovery Project DP180100602 ).  This research was partially sponsored by Tel Aviv-Swinburne Research Collaboration Grant (2019).

Generalized bregman envelopes and proximity operators

- Burachik, Regina, Dao, Minh, Lindstrom, Scott

  • Authors: Burachik, Regina , Dao, Minh , Lindstrom, Scott
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Journal of Optimization Theory and Applications Vol. 190, no. 3 (2021), p. 744-778
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  • Description: Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martínez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural “extreme” cases that highlight the importance of which generalized Bregman distance is chosen. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Incremental DC optimization algorithm for large-scale clusterwise linear regression

- Bagirov, Adil, Taheri, Sona, Cimen, Emre

  • Authors: Bagirov, Adil , Taheri, Sona , Cimen, Emre
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Journal of Computational and Applied Mathematics Vol. 389, no. (2021), p. 1-17
  • Relation: https://purl.org/au-research/grants/arc/DP190100580
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  • Description: The objective function in the nonsmooth optimization model of the clusterwise linear regression (CLR) problem with the squared regression error is represented as a difference of two convex functions. Then using the difference of convex algorithm (DCA) approach the CLR problem is replaced by the sequence of smooth unconstrained optimization subproblems. A new algorithm based on the DCA and the incremental approach is designed to solve the CLR problem. We apply the Quasi-Newton method to solve the subproblems. The proposed algorithm is evaluated using several synthetic and real-world data sets for regression and compared with other algorithms for CLR. Results demonstrate that the DCA based algorithm is efficient for solving CLR problems with the large number of data points and in particular, outperforms other algorithms when the number of input variables is small. © 2020 Elsevier B.V.

Intelligent energy prediction techniques for fog computing networks

- Farooq, Umar, Shabir, Muhammad, Javed, Muhammad, Imran, Muhammad

  • Authors: Farooq, Umar , Shabir, Muhammad , Javed, Muhammad , Imran, Muhammad
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Applied Soft Computing Vol. 111, no. (2021), p.
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  • Description: Energy Efficiency is a key concern for future fog-enabled Internet of Things (IoT). Since Fog Nodes (FNs) are energy-constrained devices, task offloading techniques must consider the energy consumption of the FNs to maximize the performance of IoT applications. In this context, accurate energy prediction can enable the development of intelligent energy-aware task offloading techniques. In this paper, we present two energy prediction techniques, the first one is based on the Recursive Least Square (RLS) filter and the second one uses the Artificial Neural Network (ANN). Both techniques use inputs such as the number of tasks and size of the tasks to predict the energy consumption at different fog nodes. Simulation results show that both techniques have a root mean square error of less than 3%. However, the ANN-based technique shows up to 20% less root mean square error as compared to the RLS-based technique. © 2021 Elsevier B.V.

Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

- Beer, G., Cánovas, M. J., López, Marco, Parra, Juan

  • Authors: Beer, G. , Cánovas, M. J. , López, Marco , Parra, Juan
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Mathematical Programming Vol. 189, no. 1-2 (2021), p. 75-98. https://purl.org/au-research/grants/arc/DP180100602
  • Relation: http://purl.org/au-research/grants/arc/DP180100602
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  • Description: This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in Rn. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of Rn+1. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society. Correction: The article “Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric”, written by Beer,G., Cánovas, M.J., López, M.A., Parra, J.was originally published Online First without Open Access. After publication in volume 189, issue 1–2, page 75–98 the author decided to opt for Open Choice and to make the article an Open Access publication. Therefore, the copyright of the article has been changed to © The Author(s) 2020 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License. https://doi.org/10.1007/s10107-021-01751-x
  • Description: This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in Rn. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of Rn+1. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

Logarithmically improved regularity criterion for the 3D Hall-MHD equations

- Gala, Sadek, Théra, Michel

  • Authors: Gala, Sadek , Théra, Michel
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Computational and Applied Mathematics Vol. 40, no. 7 (2021), p.
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  • Description: In this work, we study the blow-up criterion of the smooth solutions of three-dimensional incompressible Hall-magnetohydrohynamics equations (in short, Hall-MHD). We obtain a logarithmically improved regularity criterion of smooth solutions in terms of the B˙∞,∞0 norm. We improve the blow-up criterion for smooth solutions established in Ye (Appl Anal 96:2669–2683, 2016). © 2021, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.

Malware detection in edge devices with fuzzy oversampling and dynamic class weighting

- Khoda, Mahbub, Kamruzzaman, Joarder, Gondal, Iqbal, Imam, Tasadduq, Rahman, Ashfaqur

  • Authors: Khoda, Mahbub , Kamruzzaman, Joarder , Gondal, Iqbal , Imam, Tasadduq , Rahman, Ashfaqur
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Applied Soft Computing Vol. 112, no. (2021), p.
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  • Description: In Internet-of-things (IoT) domain, edge devices are used increasingly for data accumulation, preprocessing, and analytics. Intelligent integration of edge devices with Artificial Intelligence (AI) facilitates real-time analysis and decision making. However, these devices simultaneously provide additional attack opportunities for malware developers, potentially leading to information and financial loss. Machine learning approaches can detect such attacks but their performance degrades when benign samples substantially outnumber malware samples in training data. Existing approaches for such imbalanced data assume samples represented as continuous features and thus can generate invalid samples when malware applications are represented by binary features. We propose a novel malware oversampling technique that addresses this issue. Further, we propose two approaches for malware detection. Our first approach uses fuzzy set theory, while the second approach dynamically assigns higher priority to malware samples using a novel loss function. Combining our oversampling technique with these approaches, the proposed approach attains over 9% improvement over competing methods in terms of F1_score. Our approaches can, therefore, result in enhanced privacy and security in edge computing services. © 2021 Elsevier B.V.

Necessary and sufficient optimality conditions in DC semi-infinite programming

- Correa, Rafael, López, Marco, Pérez-Aros, Pedro

  • Authors: Correa, Rafael , López, Marco , Pérez-Aros, Pedro
  • Date: 2021
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 837-865
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  • Description: This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and local) optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section. © 2021 Society for Industrial and Applied Mathematics

On a semismooth* Newton method for solving generalized equations

- Gfrerer, Helmut, Outrata, Jiri

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2021
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 489-517
  • Relation: https://doi.org/10.1137/19M1257408
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  • Description: In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multivalued part of the considered GE. The method is based on the new notion of semismoothness\ast, which, together with a suitable regularity condition, ensures the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models. © 2021 Society for Industrial and Applied Mathematics

Primal necessary characterizations of transversality properties

- Cuong, Nguyen, Kruger, Alexander

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Positivity Vol. 25, no. 2 (2021), p. 531-558
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings as well as two nonlinear transversality properties of a convex set-valued mapping to a convex set in the range space are discussed. © 2020, Springer Nature Switzerland AG.