Title: Error bounds: stability
Speaker: Alex Kruger
Federation University Australia
Abstract: We extend the results in [1,2] to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We consider families of arbitrary, convex and linear perturbations of the given function defining the system under consideration. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the 'radius of error bounds'. The definitions and characterizations are illustrated by examples.
The presentation follows [3].
References.
- Kruger, A. Y., Ngai, H. V., Théra, M.: Stability of error bounds for semi-infinite convex constraint systems. SIAM J. Optim. 20(4), 2080–2096 (2010).
- Kruger, A. Y., Ngai, H. V., Théra, M.: Stability of error bounds for convex constraint systems in Banach spaces. SIAM J. Optim. 20(6), 3280–3296 (2010).
- Kruger, A. Y., López, M. A., Théra, M.: Perturbation of error bounds. Math. Program., Ser. B, DOI 10.1007/s10107-017-1129-4.
Title: Radius Theorems for Monotone Mappings
Speaker: Andrew Eberhard
RMIT University
Abstract: For a Hilbert space 
and a mapping 
(potentially set-valued) that is maximal monotone locally around a pair 
in its graph, we obtain a radius theorem of the following kind: the infimum of the norm of a linear and bounded single-valued mapping 
such that 
is not locally monotone around 
equals the monotonicity modulus of 
. Moreover, the infimum is not changed if taken with respect to $B$ symmetric, negative semidefinite and of rank one, and also not changed if taken with respect to all functions 
that are Lipschitz continuous around 
and 
is replaced by the Lipschitz modulus of 
at . As applications, a radius theorem is obtained for the strong second-order sufficient optimality condition of an optimization problem, which in turn yields a radius theorem for quadratic convergence of the Newton method applied to that problem. A radius theorem is also derived for mappings that are merely hypomonotone.
This is joint work with A. Donchev and R.T.Rockafellar.
Title: Constructing a counter example to the De Pierro Conjecture
Speaker: Andrew Williamson
RMIT University
Title: Stability Analysis for Parameterized Conic Programs
Speaker: Héctor Ramírez
University of Chile
Title: About Extremality and Stationarity for a collection of sets
Speaker: Bui Thi Hoa
Federation University Australia
Title: Galerkin polytope spaces for set optimisation
Speaker: Janosch Rieger
Monash University
Title: Lagrangian Duality in Stochastic Integer Programming
Speaker: Jeffrey Christiansen
RMIT University
Abstract: Stochastic Integer Programs (SIPs) are a powerful modelling tool for representing decision problems which incorporate elements of uncertainty. The structure of the SIPs we will consider are based on outcome scenarios, which allows us to relax and decompose the problem into smaller subproblems using tools such as Lagrangian relaxation.
This talk will explore the strength of the Lagrangian dual bounds which can be obtained for SIPs, as well as algorithms for calculating these bounds.
Title: Generalising alternation results for multivariate approximation
Speaker: Julien Ugon
CIAO, Federation University Australia
Abstract: Chebyshev’s alternation theorem is a celebrated result in approximation theory, which provides easy-to-check optimality condition and paves the way for algorithms for constructing best approximation of functions. The results only apply for univariate approximation and there are no straightfoward generalisation to the case of multivariate approximation. In this talk we propose a discrete geometrical view of alternation, which applies to higher dimensions.
Title: Optimal Transport Mass Theory in Bilevel Optimization Models
Speaker: Lina Mallozzi
University of Naples Federico II, Italy
Abstract: Optimal transport theory is widely used to solve problems in mathematics and different areas of the sciences. We present and discuss two-stage optimization models corresponding to economic equilibrium problems. A distribution of citizens in an urban area, where a given number of services must be located, is given. Citizens are partitioned in service regions such that each facility serves the costumer demand in one of the service regions. At first, it is assumed that the demand is totally satisfied and in the spirit of a market survey, a social planner divides the market region into a set of service regions in order to minimize the total cost: the objective is to find the optimal location of the services in the urban area and the related costumers partition. Existence results are obtained by using optimal transport mass tools. Similar models where the customers have the option of not purchasing the good are also considered and the existence problem is solved via partial transport mass theory.
Title: New perspectives in the theory of BVP of PDEs
Speaker: Maria Alessandra Ragusa
Università di Catania, Italy; RUDN University, Moscow, Russia
Title: Alternance and its modifications
Speaker: Nadia Sukhorukova
Swinburne University of Technology
Abdtract: A sequence of maximal deviation points whose deviation signs are alternating is called an alternating sequence (also known as alternance). The notion of alternance is used to characterise best Chebyshev polynomial approximations. In this talk we will illustrate how the geometry of alternance transforms when other types of functions are used to construct approximations.
Title: On the splitting optimization problem with enlargement
Speaker: Reinier Díaz Millán
Federal Institute of Goias, Brazil
Title: Comparing and verifying calmness conditions for MPECs
Speaker: René Henrion
Weierstrass Institute, Germany
Title: Douglas-Rachford Method for Boundary Valued ODEs
Speaker: Scott Lindstrom
University of Newcastle
Abstract: Notwithstanding the absence of theoretical justification, the Douglas-Rachford method has been used to solve many nonconvex feasibility problems. We explore an application to solving boundary valued ordinary differential equations by considering an approximate solution obtained by finding the intersection of hypersurfaces.
Title: A DC optimization algorithm for piecewise linear regression
Speaker: Sona Taheri
Federation University Australia
Abstract: The problem of finding a continuous piecewise linear function approximating a regression function is considered. This problem is formulated as a nonconvex nonsmooth optimization problem where the objective function is represented as a difference of convex (DC) functions. Subdifferentials of DC components are computed and an algorithm is designed using these subdifferentials to find piecewise linear functions. The algorithm is tested using some synthetic and real world data sets and compared with other regression algorithms.
Title: Chebyshev Sets on the Sphere
Speaker: Theo Bendit, University of Newcastle
Abstract: The Chebyshev Conjecture states that Chebyshev subsets of a Hilbert Space are convex. Ficken, Klee and Asplund provided an equivalent formulation involving uniquely remotal sets. We provide another equivalent formulation, looking at Chebyshev subsets of the unit sphere. We also provide a characterisation of such sets in terms of their stereographic projections, and exploit the Radon-Nikodym property for some local structure.
Title: Transversality in facial structure of convex sets
Speaker: Vera Roshchina
RMIT University