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4López, Marco
3Correa, Rafael
3Hantoute, Abderrahim
1Baca, Jose
1Beer, Gerald
1Cánovas, Maria
1Lopez, Marco
1Morales-Silva, Daniel
1Nghiem, Truong
1Nguyen, Linh
1Nguyen, Thang
1Nguyen, Thanh
1Parra, Juan
1Rangel, Pablo
1Rubinov, Alex
1Sharikov, Evgenii
1Song, Hyoung-Kyu
1Vera De Serio, Virginia

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30101 Pure Mathematics
3Fenchel subdifferential
2Australian Digital Thesis
2Mathematical optimization
2Pointwise supremum function
2Subdifferential calculus rules
2Valadier-like formulas
10102 Applied Mathematics
1Collision avoidance
1Compact index set
1Convergence of numerical methods
1Convex
1Convex infinite-dimensional
1Functional analysis
1Functions
1Hausdorff excess
1Hölder calmness
1Increasing convex-along-rays functions
1Lagrangian functions

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Stability of the lower level sets of ICAR functions

- López, Marco, Rubinov, Alex, Vera De Serio, Virginia

**Authors:**López, Marco , Rubinov, Alex , Vera De Serio, Virginia**Date:**2005**Type:**Text , Journal article**Relation:**Numerical Functional Analysis and Optimization Vol. 26, no. 1 (2005), p. 113-127**Full Text:**false**Reviewed:****Description:**In this paper, we study the stability of the lower level set {x E R++n | f (x) ≤ 0} of a finite valued increasing convex-along-rays (ICAR) function f defined on R++n. In monotonic analysis, ICAR functions play the role of usual convex functions in classical convex analysis. We show that each ICAR function f is locally Lipschitz on int dom f and that the pointwise convergence of a sequence of ICAR functions implies its uniform convergence on each compact subset of R ++n. The latter allows us to establish stability results for ICAR functions in some sense similar to those for convex functions. Copyright © Taylor & Francis, Inc.**Description:**C1**Description:**2003001419

Conditions for global minimum through abstract convexity

**Authors:**Sharikov, Evgenii**Date:**2008**Type:**Text , Thesis , PhD**Full Text:****Description:**The theory of abstract convexity generalizes ideas of convex analysis by using the notion of global supports and the global definition of subdifferential. In order to apply this theory to optimization, we need to extend subdifferential calculus and separation properties into the area of abstract convexity.**Description:**Doctor of Philosophy

**Authors:**Sharikov, Evgenii**Date:**2008**Type:**Text , Thesis , PhD**Full Text:****Description:**The theory of abstract convexity generalizes ideas of convex analysis by using the notion of global supports and the global definition of subdifferential. In order to apply this theory to optimization, we need to extend subdifferential calculus and separation properties into the area of abstract convexity.**Description:**Doctor of Philosophy

G-coupling functions and properties of strongly star-shaped cones

**Authors:**Morales-Silva, Daniel**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**The main part of this thesis presents a new approach to the topic of conjugation, with applications to various optimization problems. It does so by introducing (what we call) G-coupling functions.**Description:**Doctor of Philosophy

**Authors:**Morales-Silva, Daniel**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**The main part of this thesis presents a new approach to the topic of conjugation, with applications to various optimization problems. It does so by introducing (what we call) G-coupling functions.**Description:**Doctor of Philosophy

Weaker conditions for subdifferential calculus of convex functions

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , López, Marco**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Functional Analysis Vol. 271, no. 5 (2016), p. 1177-1212**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15]), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau-Rockafellar formula (Rockafellar 1970, [23]; Moreau 1966, [20]), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization.**Description:**In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15]), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau-Rockafellar formula (Rockafellar 1970, [23]; Moreau 1966, [20]), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization. (C) 2016 Elsevier Inc. All rights reserved.

Valadier-like Formulas for the Supremum Function II: The Compactly Indexed Case

- Correa, Rafael, Hantoute, Abderrahim, Lopez, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 26, no. 1 (2019), p. 299-324**Full Text:**false**Reviewed:****Description:**Continuing with the work on the subdifferential of the pointwise supremum of convex functions, started in part I of this paper [R. Correa, A. Hantoute, M. A. Lopez, Valadier-like formulas for the supremum function I, J. Convex Analysis 25 (2018) 1253-1278], we focus now on the compactly indexed case. We assume that the index set is compact and that the data functions are upper semicontinuous with respect to the index variable (actually, this assumption will only affect the set of epsilon-active indices at the reference point). As in the previous work, we do not require any continuity assumption with respect to the decision variable. The current compact setting gives rise to more explicit formulas, which only involve subdifferentials at the reference point of active data functions. Other formulas are derived under weak continuity assumptions. These formulas reduce to the characterization given by M. Valadier [Sous-differentiels d'une borne superieure et d'une somme continue de fonctions convexes, C. R. Acad. Sci. Paris Ser. A-B Math. 268 (1969) 39-42, Theorem 2], when the supremum function is continuous.

Valadier-like formulas for the supremum function I

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , López, Marco**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 25, no. 4 (2018), p. 1253-1278**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We generalize and improve the original characterization given by Valadier [19, Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdifferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to the Valadier formula. Our starting result is the characterization given in [11, Theorem 4], which uses the e-subdifferential at the reference point.

**Authors:**Correa, Rafael , Hantoute, Abderrahim , López, Marco**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 25, no. 4 (2018), p. 1253-1278**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We generalize and improve the original characterization given by Valadier [19, Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdifferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to the Valadier formula. Our starting result is the characterization given in [11, Theorem 4], which uses the e-subdifferential at the reference point.

A uniform approach to hölder calmness of subdifferentials

- Beer, Gerald, Cánovas, Maria, López, Marco, Parra, Juan

**Authors:**Beer, Gerald , Cánovas, Maria , López, Marco , Parra, Juan**Date:**2020**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 27, no. 1 (2020), p.**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**For finite-valued convex functions f defined on the n-dimensional Euclidean space, we are interested in the set-valued mapping assigning to each pair (f, x) the subdifferential of f at x. Our approach is uniform with respect to f in the sense that it involves pairs of functions close enough to each other, but not necessarily around a nominal function. More precisely, we provide lower and upper estimates, in terms of Hausdorff excesses, of the subdifferential of one of such functions at a nominal point in terms of the subdifferential of nearby functions in a ball centered in such a point. In particular, we obtain the (1/2) - Hölder calmness of our mapping at a nominal pair (f, x) under the assumption that the subdifferential mapping viewed as a set-valued mapping from Rn to Rn with f fixed is calm at each point of {x} × ∂f(x). © Heldermann Verlag**Description:**Funding details: Australian Research Council, ARC, DP160100854 Funding details: European Commission, EU Funding details: Ministerio de Economía y Competitividad, MINECO Funding details: Federación Española de Enfermedades Raras, FEDER Funding text 1:

Collision-free minimum-time trajectory planning for multiple vehicles based on ADMM

- Nguyen, Thanh, Nguyen, Thang, Nghiem, Truong, Nguyen, Linh, Baca, Jose, Rangel, Pablo, Song, Hyoung-Kyu

**Authors:**Nguyen, Thanh , Nguyen, Thang , Nghiem, Truong , Nguyen, Linh , Baca, Jose , Rangel, Pablo , Song, Hyoung-Kyu**Date:**2022**Type:**Text , Conference paper**Relation:**2022 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2022, Kyoto, Japan, 23-27 October 2022, 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Vol. 2022-October, p. 13785-13790**Full Text:**false**Reviewed:****Description:**The paper presents a practical approach for planning trajectories for multiple vehicles where both collision avoidance and minimum travelling time are simultaneously considered. It is first proposed to exploit the mixed-integer programming (MIP) approach to formulate the collision avoidance paradigm, where the linear dynamic models are utilized to derive the linear constraints. Moreover, travelling time of each vehicle is compromised among them and set to be minimized so that all the vehicles can practically reach the expected destinations at the shortest time. Unfortunately, the formulated optimization problem is NP-hard. In order to effectively address it, we propose to employ the alternating direction method of multipliers (ADMM), which can share the computational burdens to distributive optimization solvers. Thus, the proposed method can enable each vehicle to obtain an expected trajectory in a practical time. Convergence of the proposed algorithm is also discussed. To verify effectiveness of our approach, we implemented it in a numerical example, where the obtained results are highly promising. © 2022 IEEE.

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