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On the application of the SCD semismooth* newton method to variational inequalities of the second kind

- Gfrerer, Helmut, Outrata, Jiri, Valdman, Jan

**Authors:**Gfrerer, Helmut , Outrata, Jiri , Valdman, Jan**Date:**2022**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 30, no. 4 (2022), p. 1453-1484**Relation:**https://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.

**Authors:**Gfrerer, Helmut , Outrata, Jiri , Valdman, Jan**Date:**2022**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 30, no. 4 (2022), p. 1453-1484**Relation:**https://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.

Isolated calmness and sharp minima via Hölder Graphical Derivatives

- Kruger, Alexander, López, Marco, Yang, Xiaoqi, Zhu, Jiangxing

**Authors:**Kruger, Alexander , López, Marco , Yang, Xiaoqi , Zhu, Jiangxing**Date:**2022**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 30, no. 4 (2022), p. 1423-1441**Relation:**https://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper utilizes Hölder graphical derivatives for characterizing Hölder strong subregularity, isolated calmness and sharp minimum. As applications, we characterize Hölder isolated calmness in linear semi-infinite optimization and Hölder sharp minimizers of some penalty functions for constrained optimization. © 2022, The Author(s).

**Authors:**Kruger, Alexander , López, Marco , Yang, Xiaoqi , Zhu, Jiangxing**Date:**2022**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 30, no. 4 (2022), p. 1423-1441**Relation:**https://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper utilizes Hölder graphical derivatives for characterizing Hölder strong subregularity, isolated calmness and sharp minimum. As applications, we characterize Hölder isolated calmness in linear semi-infinite optimization and Hölder sharp minimizers of some penalty functions for constrained optimization. © 2022, The Author(s).

**Authors:**Théra, Michel**Date:**2022**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 5-20**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:**

New tour on the subdifferential of supremum via finite sums and suprema

- Hantoute, Abderrahim, López-Cerdá, Marco

**Authors:**Hantoute, Abderrahim , López-Cerdá, Marco**Date:**2022**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 81-106**Full Text:****Reviewed:****Description:**This paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family. © 2021, The Author(s).

**Authors:**Hantoute, Abderrahim , López-Cerdá, Marco**Date:**2022**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 81-106**Full Text:****Reviewed:****Description:**This paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family. © 2021, The Author(s).

Linkedness of cartesian products of complete graphs

- Jorgensen, Leif, Pineda-Villavicencio, Guillermo, Ugon, Julien

**Authors:**Jorgensen, Leif , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2022**Type:**Text , Journal article**Relation:**Ars Mathematica Contemporanea Vol. 22, no. 2 (2022), p.**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2k vertices is k-linked if, for every set of 2k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product Kd1+1 × Kd2+1 of complete graphs Kd1+1 and Kd2+1 is

**Authors:**Jorgensen, Leif , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2022**Type:**Text , Journal article**Relation:**Ars Mathematica Contemporanea Vol. 22, no. 2 (2022), p.**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2k vertices is k-linked if, for every set of 2k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product Kd1+1 × Kd2+1 of complete graphs Kd1+1 and Kd2+1 is

The linkedness of cubical polytopes : beyond the cube

- Bui, Hoa, Pineda-Villavicencio, Guillermo, Ugon, Julien

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2024**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 347, no. 3 (2024), p.**Relation:**https://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. In a previous paper [3] we proved that every cubical d-polytope is ⌊d/2⌋-linked. Here we strengthen this result by establishing the ⌊(d+1)/2⌋-linkedness of cubical d-polytopes, for every d≠3. A graph G is strongly k-linked if it has at least 2k+1 vertices and, for every vertex v of G, the subgraph G−v is k-linked. We say that a polytope is (strongly) k-linked if its graph is (strongly) k-linked. In this paper, we also prove that every cubical d-polytope is strongly ⌊d/2⌋-linked, for every d≠3. These results are best possible for this class of polytopes.**Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. In a previous paper [3] we proved that every cubical d-polytope is

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2024**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 347, no. 3 (2024), p.**Relation:**https://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. In a previous paper [3] we proved that every cubical d-polytope is ⌊d/2⌋-linked. Here we strengthen this result by establishing the ⌊(d+1)/2⌋-linkedness of cubical d-polytopes, for every d≠3. A graph G is strongly k-linked if it has at least 2k+1 vertices and, for every vertex v of G, the subgraph G−v is k-linked. We say that a polytope is (strongly) k-linked if its graph is (strongly) k-linked. In this paper, we also prove that every cubical d-polytope is strongly ⌊d/2⌋-linked, for every d≠3. These results are best possible for this class of polytopes.**Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. In a previous paper [3] we proved that every cubical d-polytope is

Nonconvex bundle method with application to a delamination problem

- Dao, Minh, Gwinner, Joachim, Noll, Dominikus, Ovcharova, Nina

**Authors:**Dao, Minh , Gwinner, Joachim , Noll, Dominikus , Ovcharova, Nina**Date:**2016**Type:**Text , Journal article**Relation:**Computational optimization and applications Vol. 65, no. 1 (2016), p. 173-203**Full Text:**false**Reviewed:****Description:**Delamination is a typical failure mode of composite materials caused by weak bonding. It arises when a crack initiates and propagates under a destructive loading. Given the physical law characterizing the properties of the interlayer adhesive between the bonded bodies, we consider the problem of computing the propagation of the crack front and the stress field along the contact boundary. This leads to a hemivariational inequality, which after discretization by finite elements we solve by a nonconvex bundle method, where upper- C 1 criteria have to be minimized. As this is in contrast with other classes of mechanical problems with non-monotone friction laws and in other applied fields, where criteria are typically lower- C 1 , we propose a bundle method suited for both types of nonsmoothness. We prove its global convergence in the sense of subsequences and test it on a typical delamination problem of material sciences.

The lower bound theorem for d - polytopes with 2d + 1 vertices

- Pineda-Villavicencio, Guillermo, Yost, David

**Authors:**Pineda-Villavicencio, Guillermo , Yost, David**Date:**2022**Type:**Text , Journal article**Relation:**SIAM Journal on Discrete Mathematics Vol. 36, no. 4 (2022), p. 2920-2941**Full Text:****Reviewed:****Description:**The problem of calculating exact lower bounds for the number of k-faces of dpolytopes with n vertices, for each value of k, and characterizing the minimizers has recently been solved for n not exceeding 2d. We establish the corresponding result for n = 2d+ 1; the nature of the lower bounds and the minimizing polytopes are quite different in this case. As a byproduct, we also characterize all d-polytopes with d + 3 vertices and only one or two edges more than the minimum. © 2022 Society for Industrial and Applied Mathematics.

**Authors:**Pineda-Villavicencio, Guillermo , Yost, David**Date:**2022**Type:**Text , Journal article**Relation:**SIAM Journal on Discrete Mathematics Vol. 36, no. 4 (2022), p. 2920-2941**Full Text:****Reviewed:****Description:**The problem of calculating exact lower bounds for the number of k-faces of dpolytopes with n vertices, for each value of k, and characterizing the minimizers has recently been solved for n not exceeding 2d. We establish the corresponding result for n = 2d+ 1; the nature of the lower bounds and the minimizing polytopes are quite different in this case. As a byproduct, we also characterize all d-polytopes with d + 3 vertices and only one or two edges more than the minimum. © 2022 Society for Industrial and Applied Mathematics.

Robust and continuous metric subregularity for linear inequality systems

- Camacho, J., Cánovas, Maria, López, Marco, Parra, Juan

**Authors:**Camacho, J. , Cánovas, Maria , López, Marco , Parra, Juan**Date:**2023**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 86, no. 3 (2023), p. 967-988**Full Text:****Reviewed:****Description:**This paper introduces two new variational properties, robust and continuous metric subregularity, for finite linear inequality systems under data perturbations. The motivation of this study goes back to the seminal work by Dontchev, Lewis, and Rockafellar (2003) on the radius of metric regularity. In contrast to the metric regularity, the unstable continuity behavoir of the (always finite) metric subregularity modulus leads us to consider the aforementioned properties. After characterizing both of them, the radius of robust metric subregularity is computed and some insights on the radius of continuous metric subregularity are provided. © 2022, The Author(s).

**Authors:**Camacho, J. , Cánovas, Maria , López, Marco , Parra, Juan**Date:**2023**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 86, no. 3 (2023), p. 967-988**Full Text:****Reviewed:****Description:**This paper introduces two new variational properties, robust and continuous metric subregularity, for finite linear inequality systems under data perturbations. The motivation of this study goes back to the seminal work by Dontchev, Lewis, and Rockafellar (2003) on the radius of metric regularity. In contrast to the metric regularity, the unstable continuity behavoir of the (always finite) metric subregularity modulus leads us to consider the aforementioned properties. After characterizing both of them, the radius of robust metric subregularity is computed and some insights on the radius of continuous metric subregularity are provided. © 2022, The Author(s).

A novel optimization approach towards improving separability of clusters

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

**Authors:**Bagirov, Adil , Hoseini-Monjezi, Najmeh , Taheri, Sona**Date:**2023**Type:**Text , Journal article**Relation:**Computers and Operations Research Vol. 152, no. (2023), p.**Relation:**http://purl.org/au-research/grants/arc/DP190100580**Full Text:**false**Reviewed:****Description:**The objective functions in optimization models of the sum-of-squares clustering problem reflect intra-cluster similarity and inter-cluster dissimilarities and in general, optimal values of these functions can be considered as appropriate measures for compactness of clusters. However, the use of the objective function alone may not lead to the finding of separable clusters. To address this shortcoming in existing models for clustering, we develop a new optimization model where the objective function is represented as a sum of two terms reflecting the compactness and separability of clusters. Based on this model we develop a two-phase incremental clustering algorithm. In the first phase, the clustering function is minimized to find compact clusters and in the second phase, a new model is applied to improve the separability of clusters. The Davies–Bouldin cluster validity index is applied as an additional measure to compare the compactness of clusters and silhouette coefficients are used to estimate the separability of clusters. The performance of the proposed algorithm is demonstrated and compared with that of four other algorithms using synthetic and real-world data sets. Numerical results clearly show that in comparison with other algorithms the new algorithm is able to find clusters with better separability and similar compactness. © 2022

Relaxed Lagrangian duality in convex infinite optimization: Reverse strong duality and optimality

- Dinh, Nguyen, Goberna, Miguel, Lopez, Marco, Volle, Michel

**Authors:**Dinh, Nguyen , Goberna, Miguel , Lopez, Marco , Volle, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Journal of Applied and Numerical Optimization Vol. , no. (2021), p.**Full Text:**false**Reviewed:****Description:**We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems.

Virtual care initiatives for older adults in Australia : scoping review

- Savira, Feby, Gupta, Adyya, Gilbert, Cecily, Huggins, Catherine, Browning, Colette, Chapman, Wendy, Haines, Terry, Peeters, Anna

**Authors:**Savira, Feby , Gupta, Adyya , Gilbert, Cecily , Huggins, Catherine , Browning, Colette , Chapman, Wendy , Haines, Terry , Peeters, Anna**Date:**2023**Type:**Text , Journal article , Review**Relation:**Journal of Medical Internet Research Vol. 25, no. (2023), p.**Full Text:****Reviewed:****Description:**Background: There has been a rapid shift toward the adoption of virtual health care services in Australia. It is unknown how widely virtual care has been implemented or evaluated for the care of older adults in Australia. Objective: We aimed to review the literature evaluating virtual care initiatives for older adults across a wide range of health conditions and modalities and identify key challenges and opportunities for wider adoption at both patient and system levels in Australia. Methods: A scoping review of the literature was conducted. We searched MEDLINE, Embase, PsycINFO, CINAHL, AgeLine, and gray literature (January 1, 2011, to March 8, 2021) to identify virtual care initiatives for older Australians (aged

**Authors:**Savira, Feby , Gupta, Adyya , Gilbert, Cecily , Huggins, Catherine , Browning, Colette , Chapman, Wendy , Haines, Terry , Peeters, Anna**Date:**2023**Type:**Text , Journal article , Review**Relation:**Journal of Medical Internet Research Vol. 25, no. (2023), p.**Full Text:****Reviewed:****Description:**Background: There has been a rapid shift toward the adoption of virtual health care services in Australia. It is unknown how widely virtual care has been implemented or evaluated for the care of older adults in Australia. Objective: We aimed to review the literature evaluating virtual care initiatives for older adults across a wide range of health conditions and modalities and identify key challenges and opportunities for wider adoption at both patient and system levels in Australia. Methods: A scoping review of the literature was conducted. We searched MEDLINE, Embase, PsycINFO, CINAHL, AgeLine, and gray literature (January 1, 2011, to March 8, 2021) to identify virtual care initiatives for older Australians (aged

- Assaf, Rama, Scheunemann, Lisa, Birk, Carolin, Schröder, Jörg, Ooi, Ean Tat

**Authors:**Assaf, Rama , Scheunemann, Lisa , Birk, Carolin , Schröder, Jörg , Ooi, Ean Tat**Date:**2021**Type:**Text , Journal article**Relation:**Proceedings in applied mathematics and mechanics Vol. 20, no. 1 (2021), p. n/a**Full Text:**false**Reviewed:****Description:**The paper presents a comparative study of the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for the numerical evaluation of the volume‐averaged stress of composites. Two‐dimensional meso‐scale models of concrete represented by digital images and discretized using an automatic mesh generation algorithm are considered. The different computational approaches are discussed and compared with respect to accuracy and efficiency for both scenarios.

Optimality conditions, approximate stationarity, and applications 'a story beyond lipschitzness

- Kruger, Alexander, Mehlitz, Patrick

**Authors:**Kruger, Alexander , Mehlitz, Patrick**Date:**2022**Type:**Text , Journal article**Relation:**ESAIM - Control, Optimisation and Calculus of Variations Vol. 28, no. (2022), p.**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established. © The authors.

**Authors:**Kruger, Alexander , Mehlitz, Patrick**Date:**2022**Type:**Text , Journal article**Relation:**ESAIM - Control, Optimisation and Calculus of Variations Vol. 28, no. (2022), p.**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established. © The authors.

Error bounds revisited

- Cuong, Nguyen, Kruger, Alexander

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

Robust piecewise linear L 1-regression via nonsmooth DC optimization

- Bagirov, Adil, Taheri, Sona, Karmitsa, Napsu, Sultanova, Nargiz, Asadi, Soodabeh

**Authors:**Bagirov, Adil , Taheri, Sona , Karmitsa, Napsu , Sultanova, Nargiz , Asadi, Soodabeh**Date:**2022**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 37, no. 4 (2022), p. 1289-1309**Relation:**http://purl.org/au-research/grants/arc/DP190100580**Full Text:**false**Reviewed:****Description:**Piecewise linear (Formula presented.) -regression problem is formulated as an unconstrained difference of convex (DC) optimization problem and an algorithm for solving this problem is developed. Auxiliary problems are introduced to design an adaptive approach to generate a suitable piecewise linear regression model and starting points for solving the underlying DC optimization problems. The performance of the proposed algorithm as both approximation and prediction tool is evaluated using synthetic and real-world data sets containing outliers. It is also compared with mainstream machine learning regression algorithms using various performance measures. Results demonstrate that the new algorithm is robust to outliers and in general, provides better predictions than the other alternative regression algorithms for most data sets used in the numerical experiments. © 2020 Informa UK Limited, trading as Taylor & Francis Group.

Cryptography-based secure data storage and sharing using HEVC and public clouds

- Usman, Muhammad, Ahmad Jan, Mian, He, Xiangjian

**Authors:**Usman, Muhammad , Ahmad Jan, Mian , He, Xiangjian**Date:**2017**Type:**Text , Journal article**Relation:**Information sciences Vol. 387, no. (2017), p. 90-102**Full Text:**false**Reviewed:****Description:**Mobile devices are widely used for uploading/downloading media files such as audio, video and images to/from the remote servers. These devices have limited resources and are required to offload resource-consuming media processing tasks to the clouds for further processing. Migration of these tasks means that the media services provided by the clouds need to be authentic and trusted by the mobile users. The existing schemes for secure exchange of media files between the mobile devices and the clouds have limitations in terms of memory support, processing load, battery power, and data size. These schemes lack the support for large-sized video files and are not suitable for resource-constrained mobile devices. This paper proposes a secure, lightweight, robust, and efficient scheme for data exchange between the mobile users and the media clouds. The proposed scheme considers High Efficiency Video Coding (HEVC) Intra-encoded video streams in unsliced mode as a source for data hiding. Our proposed scheme aims to support real-time processing with power-saving constraint in mind. Advanced Encryption Standard (AES) is used as a base encryption technique by our proposed scheme. The simulation results clearly show that the proposed scheme outperforms AES-256 by decreasing the processing time up to 4.76% and increasing the data size up to 0.72% approximately. The proposed scheme can readily be applied to real-time cloud media streaming.

Multivariate approximation by polynomial and generalized rational functions

- Díaz Millán, Reinier, Peiris, Vinesha, Sukhorukova, Nadezda, Ugon, Julien

**Authors:**Díaz Millán, Reinier , Peiris, Vinesha , Sukhorukova, Nadezda , Ugon, Julien**Date:**2022**Type:**Text , Journal article**Relation:**Optimization Vol. 71, no. 4 (2022), p. 1171-1187**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

**Authors:**Díaz Millán, Reinier , Peiris, Vinesha , Sukhorukova, Nadezda , Ugon, Julien**Date:**2022**Type:**Text , Journal article**Relation:**Optimization Vol. 71, no. 4 (2022), p. 1171-1187**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

Radius theorems for subregularity in infinite dimensions

- Gfrerer, Helmut, Kruger, Alexander

**Authors:**Gfrerer, Helmut , Kruger, Alexander**Date:**2023**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 86, no. 3 (2023), p. 1117-1158**Relation:**https://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper continues our previous work (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) on the radius of subregularity that was initiated by Asen Dontchev. We extend the results of (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) to general Banach/Asplund spaces and to other classes of perturbations, and sharpen the coderivative tools used in the analysis of the robustness of well-posedness of mathematical problems and related regularity properties of mappings involved in the statements. We also expand the selection of classes of perturbations, for which the formula for the radius of strong subregularity is valid. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

**Authors:**Gfrerer, Helmut , Kruger, Alexander**Date:**2023**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 86, no. 3 (2023), p. 1117-1158**Relation:**https://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper continues our previous work (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) on the radius of subregularity that was initiated by Asen Dontchev. We extend the results of (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) to general Banach/Asplund spaces and to other classes of perturbations, and sharpen the coderivative tools used in the analysis of the robustness of well-posedness of mathematical problems and related regularity properties of mappings involved in the statements. We also expand the selection of classes of perturbations, for which the formula for the radius of strong subregularity is valid. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Optimality conditions in DC-constrained mathematical programming problems

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

**Authors:**Correa, Rafael , Lopez, Marco , Pérez-Aros, Pedro**Date:**2023**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 198, no. 3 (2023), p. 1191-1225**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**This paper provides necessary and sufficient optimality conditions for abstract-constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of certain mappings, in particular their structure as difference of convex functions, and uses techniques of generalized differentiation (subdifferential and coderivative). It turns out that these tools can be used fruitfully out of the scope of Asplund spaces. Applications to infinite, stochastic and semi-definite programming are developed in separate sections. © 2023, The Author(s).

**Authors:**Correa, Rafael , Lopez, Marco , Pérez-Aros, Pedro**Date:**2023**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 198, no. 3 (2023), p. 1191-1225**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**This paper provides necessary and sufficient optimality conditions for abstract-constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of certain mappings, in particular their structure as difference of convex functions, and uses techniques of generalized differentiation (subdifferential and coderivative). It turns out that these tools can be used fruitfully out of the scope of Asplund spaces. Applications to infinite, stochastic and semi-definite programming are developed in separate sections. © 2023, The Author(s).

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