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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**Full Text:**false**Reviewed:****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).

Alternative representations of the normal cone to the domain of supremum functions and subdifferential calculus

- Correa, R., Hantoute, A., López, Marco

**Authors:**Correa, R. , Hantoute, A. , López, Marco**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 683-699**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.

**Authors:**Correa, R. , Hantoute, A. , López, Marco**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 683-699**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.

Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems

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

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel**Date:**2020**Type:**Text , Conference paper**Relation:**Jonathan Borwein Commemorative Conference, JBCC 2017 Vol. 313, p. 43-74**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail. © Springer Nature Switzerland AG 2020.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel**Date:**2020**Type:**Text , Conference paper**Relation:**Jonathan Borwein Commemorative Conference, JBCC 2017 Vol. 313, p. 43-74**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail. © Springer Nature Switzerland AG 2020.

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

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

The linkedness of cubical polytopes: the cube

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

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2021**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 28, no. 3 (2021), p.**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**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. We establish that the d-dimensional cube is [(d + 1)/2]-linked, for every d ≠ 3; this is the maximum possible linkedness of a d-polytope. This result implies that, for every d ≥ 1, a cubical d-polytope is [d/2]-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. 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 show that cubical 4-polytopes are strongly 2-linked and that, for each d ≥ 1, d-dimensional cubes are strongly

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2021**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 28, no. 3 (2021), p.**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**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. We establish that the d-dimensional cube is [(d + 1)/2]-linked, for every d ≠ 3; this is the maximum possible linkedness of a d-polytope. This result implies that, for every d ≥ 1, a cubical d-polytope is [d/2]-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. 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 show that cubical 4-polytopes are strongly 2-linked and that, for each d ≥ 1, d-dimensional cubes are strongly

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

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

Subdifferential of the supremum function : moving back and forth between continuous and non-continuous settings

- Correa, R., Hantoute, A., López, M. A.

**Authors:**Correa, R. , Hantoute, A. , López, M. A.**Date:**2021**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 189, no. 1-2 (2021), p. 217-247**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**In this paper we establish general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

**Authors:**Correa, R. , Hantoute, A. , López, M. A.**Date:**2021**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 189, no. 1-2 (2021), p. 217-247**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**In this paper we establish general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

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

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