Please note that this is a historical record related to WoMBaT 2018 workshop.
Title: DP Programming: optimality conditions and numerical methods
Speaker: Adil Bagirov, Federation University Australia
Abstract: In this talk, we discuss optimization problems where the objective function is represented as a difference of two convex polyhedral (convex piecewise linear) (DP) functions. First we consider unconstrained minimization of such functions, formulate necessary and sufficient conditions for local and global optimality. Then we extend these results to problems on minimization of DP functions subject to linear equalities and inequalities. We present numerical algorithms for finding both local and global solutions to the DP programming problems.
Joint work with J. Ugon.
Title: About Hölder error bounds
Speaker: Alex Kruger, Federation University Australia
Abstract: We continue the study of necessary and sufficient subdifferential conditions for Hölder error bounds, particularly merging the conventional approach with the new advancements proposed recently in Yao, Zheng (2016). We formulate a general lemma collecting the main arguments used in the proofs of the subdifferential sufficient error bound conditions and demonstrate that both linear and Hölder type conditions, conventional and the new advancements, can be obtained as direct consequences of this lemma. Some new estimates for the modulus of Hölder error bounds will be presented.
References:
A. Y. Kruger, M. A. López, X. Q. Yang and J. Zhu, Hölder error bounds and Hölder calmness with applications to convex semi-infinite optimization, arXiv:1806.06442 (2018)
Title: Extremal Principle and its extension
Speaker: Bui Thi Hoa, Federation University Australia
Title: On the FPV property, Monotone operator structure and the Monotone Polar of Representable Monotone Sets
Speaker: Andrew Eberhard, RMIT University
Abstract: one can study the point-wise partial ordering of representative functions for a monotone operator with graph 
, where 
is bigger conjugate representative for an initial monotone set 
(ie. in 
) and 
the duality pairing. We study the problem of constructing a new representative function from 
when we wish to add an additional point 
, where the later set denote the set of all monotonically related points to 
. This construction allows us to prove that all representable monotone sets (i.e monotone set 
for which there exists 
such that 
) are monotonically closed in that 
. This enables the proof of the following sum theorem: given a maximal monotone operator and a closed, strictly convex set 
with 
then 
is maximal monotone iff is of type FPV, strengthening the known implication to an equaivalence. Moreover if the space 
admits a strictly convex re-norm need only be convex, thus obtaining the basic sum theorem of the Rockafellar type for a general FPV monotone operator and a normal cone to a convex set.
Joint work with Robert Wenczel, RMIT University
Title: Ancient transversality
Speaker: David Yost, Federation University Australia
Title: On differential variational inequalities associated with solution schemes for solving maximally monotone inclusion problems
Speaker: Ewa Bednarczuk, Polish Academy of Sciences and Warsaw University of Technology
Abstract: In this talk we discuss dynamical systems related best approximation schemes for solving maximally monotone inclusions and convex optimization problems. In particular, we discuss local lipschitzness of projections onto moving polyhedral sets and pseudo-lipschitzness of polyhedral set-valued mappings given as solutions to parametric systems of equations and inequalities.
Title: Non-polyhedral extensions of the Frank and Wolfe Theorem
Speaker: Juan Enrique Martínez-Legaz, Universitat Autònoma de Barcelona
Abstract: In 1956 Marguerite Frank and Paul Wolfe proved that a quadratic function which is bounded below on a polyhedron P attains its infimum on P. In this work larger classes of sets F with this Frank-and-Wolfe property will be identified. The existence of non-polyhedral Frank-and-Wolfe sets will be established, internal characterizations by way of asymptotic properties will be presented, and the stability of the Frank-and-Wolfe class under various operations will be discussed.
Title: Conditions for globally best piecewise polynomial approximation
Speaker: Julien Ugon, Deakin University
Abstract: We will report on our progress on the problem of constructing the best uniform approximation of a continuous function by a continuous piecewise polynomial on an interval. We will use semi-infinite formulations of the problem to derive necessary and sufficient conditions for a globally best approximation. Based on these conditions we will propose an algorithm for obtaining a best approximant.
Title: Calmness in convex semi-infinite optimization. Modulus estimates
Speaker: Marco Antonio López Cerdá, University of Alicante
Abstract: The talk presents an overview of some research results on calmness in convex semi-infinite optimization. The first part addresses the calmness of the feasible set and the optimal set mappings for the linear semi-infinite optimization problem in the setting of canonical perturbations, and also in the framework of full perturbations. While there exists a clear proportionality between the calmness moduli of the feasible set mappings in both contexts, the analysis of the relationship between the calmness moduli of the argmin mappings is much more complicated. Point-based expressions (only involving the nominal problem's data) for the calmness moduli are provided.
The second part focuses on convex semi-infinite optimization, and provides a characterization of the Hölder calmness of the optimal set mapping, by showing its equivalence with the Hölder calmness of a certain (lower) level set mapping. This result, which is crucial in providing estimates of the modulus of Hölder calmness of the argmin mapping under some particular conditions, can be found in the preprint "Hölder Error Bounds and Hölder Calmness with Application to Convex Semi-Infinite Optimization", coauthored by Alexander Y. Kruger, Xiao Qi Yang and Jiangxing Zhu.
Title: Union averaged operators with applications to proximal algorithms for min-convex functions
Speaker: Minh N. Dao, University of Newcastle
Abstract: We introduce and study a class of structured set-valued operators which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions where the first is convex and the second can be expressed as the minimum of finitely many convex functions.
Title: The Campanato nearness condition revisited
Speaker: Michel Théra, University of Limoges
Title: Solving clustering problems through Linear programming and applications
Speaker: Nadia Sukhorukova, Swinburne University of Technology
Abstract: In this study we propose a new efficient linear programming based approach
for multi-resource allocation in disaster management. Such problems require
an integer solution and therefore, in most cases, the computations rely on integer and mixed-integer linear programming solvers. In general, these solvers can not handle large scaled problem. In this study we demonstrate that there exists a large class of disaster management problems whose exact solutions can be obtained by applying the simplex method (linear programming). The results of numerical experiments are provided. Another important contribution of this study is related to general cluster analysis and allocation. Namely,
we demonstrate that the classical k-medoid clustering method can be implemented using linear programming techniques (simplex method) without relying on integer solvers.
Title: Directional Metric Regularity of Multifunctions
Speaker: Ngai Huynh Va, Quy Nhon University
Abstract: We present the notion of directional metric regularity and some characterizations of this property by using the strong slope. These slope characterizations allow us to obtain a coderivative type criteration as well as the robustness of the directional metric regularity. Some applications to sensitivity theory are given.
Title: Nonlinear Transversality Properties of Collections of Sets
Speaker: Nguyen Duy Cuong, Federation University Australia
Abstract: In this talk, we discuss nonlinear transversality properties of collections of sets in normed
linear spaces. Equivalent metric characterizations of these properties are established.
Dual sufficient conditions for the nonlinear subtransversality are discussed. Relationships
between nonlinear transversality properties of collections of sets and the corresponding
regularity properties of set-valued mappings are examined.
Title: On higher order Voronoi cells
Speaker: Vera Roshchina, UNSW Sydney
Abstract: The classic Voronoi cells can be generalized to a higher-order version by
considering the cells of points for which a given 
-element subset of the
set of sites consists of the closest sites. We study the structure of the -order Voronoi cells and illustrate our theoretical findings with a case
study of two-dimensional higher-order Voronoi cells for four points.
Joint work with Juan Enrique Martínez-Legaz and Maxim Todorov
Title: On linear convergence of fixed-point iterations and application to phase retrieval
Speaker: Thao Nguyen, Delft University of Technology
Abstract: Sufficient and/or necessary conditions for linear convergence of fixed-point iterations are presented. An application to projection methods for phase retrieval problem is discussed.