Title: Response Surface Methods based on Radial Basis Function Models for Global Optimization
Speaker: Fusheng Bai,
Chongqing Normal University, China
Date and Time: Monday 9 December, 12:15
Abstract: Global optimization remains as a very challenging task for many years, especially for problems with computationally expensive objective functions. In response surface methods, surrogate models are built to approximate the original objective function incrementally aiming to find a satisfactory solution within a limited number of function evaluations of the original objective function. In this talk, several newly developed response surface methods based on radial basis function (RBF) models will be introduced, and some numerical examples on hyperparameter optimization problems of deep learning algorithms will be reported.
Title: Aggregate subgradient method for nonsmooth DC optimization
Speaker: Adil Bagirov,
Federation University Australia
Date and Time: Monday 9 December, 14:15
Abstract: In this talk, we present the aggregate subgradient method for solving unconstrained nonsmooth DC optimization problems. We discuss convergence results, demonstrate results of numerical experiments using some academic test problems and compare the aggregate subgradient method with several other nonsmooth DC optimization solvers.
Joint work with N. Karmitsa, K. Joki, S. Taheri and M. Makela
Title: On tangent cone to systems of inequalities and equations under relaxed constant rank condition
Speaker: Ewa Bednarczuk,
Systems Research Insttitute Polish Academy of Sciences
Date and Time: Monday 9 December, 9:15
Abstract: Under the relaxed constant rank condition, introduced by Minchenko and Stakhovski, we prove that the linearized cone is contained in the tangent cone (Abadie condition) for sets represented as solution sets to systems of finite numbers of inequalities and equations given by continuously differentiable functions defined on Hilbert spaces.
The talk is based on the joint works with Krzysztof Leśniewski, Krzysztof Rutkowski and Leonid Minchenko.
Title: Geometric non-intersection properties of finite collections of sets
Speaker: Hoa Thi Bui,
Curtin University
Date and Time: Monday 9 December, 9:45
Abstract: We introduce a terminology of geometric non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle.
We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality, stationarity and transversality.
As a result, some new (even in the linear setting) characterizations of the conventional extremality/transversality properties are obtained straightforwardly.
Title: A short, easy and constructive proof of the alternation Tchebicheff theorem
Speaker: Jean-Pierre Crouzeix
University Clermont-Auvergne
Date and Time: Tuesday 10 December, 14:45
Abstract: Based on a general result on continuous functions, we present a very short and easy proof on the celebrated Tchebycheff theorem on the uniform approximation of a continuous function by a polynomial function of degree fixed.
Joint research with Julien Ugon and Nadia Sukhorukova.
Title: Conically averaged nonexpansive operators
Speaker: Minh Dao
UNSW Sydney
Date and Time: Tuesday 10 December, 14:15
Abstract: The notion of an averaged nonexpansive operator plays an important role in the analysis and the numerical solution of optimisation problems. In this work, we consider a conical extension of averaged nonexpansive operators and its usefulness in the convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically studied, in particular, the stability under relaxations, convex combinations, and compositions. We then derive the conical averagedness of the resolvents of operators that possess certain types of generalised monotonicity. These properties are used to prove the global convergence and the rate of asymptotic regularity of the relaxed proximal point algorithm, the relaxed forward-backward algorithm, and the adaptive Douglas--Rachford algorithm when one operator is no longer monotone. An application to the strongly and weakly convex optimisation problem is also discussed.
Title: Semitransversality of collections of set-valued mappings
Speaker: Nguyen Duy Cuong
Federation University Australia
Date and Time: Monday 9 December, 14:45
Abstract: In this talk, we study primal and dual necessary and sufficient conditions for a new property so-called "semitransversality of collections of set-valued mappings," which can be seen as a direct counterpart of the local extremality of collections of set-valued mappings introduced by Mordukhovich et al. (SIAM J. Optim., 14(2):359–379, 2003) as well as a natural extension of semitransversality of collections of sets (Kruger, A.Y., Thao, N.H., J. Optim. Theory Appl., 164(1):41–67, 2015). Examples are provided to illustrate the property. Connections between the property and semiregularity of set-valued mappings are also discussed.
Title: Alternating conditional gradient method for feasibility problem
Speaker: Reinier Diaz Millan
Deakin University
Date and Time: Monday 10 December, 15:15
Abstract: In this paper, we deal with the classical convex feasibility problem in finite-dimensional Euclidean space. We are interested in two cases. In the first one, we assume one of the sets with an easy exact projection, and the other is compact such that the conditional gradient method (CondG) can be used for computing efficiently an inexact projection. In the second case, we assume that both sets are compacts then we can use the CondG for computing inexact projections onto them. We combine the alternating projection method with the CondG to design a new method which is an "inexact version" of the Alternating Projection Method. The proposed methods generate two different sequences converging to the solution sets, in both cases, when the intersection is empty and when the sets meetings.
Title: The Bartle-Graves Theorem
Speaker: Asen Dontchev, The University of Michigan
Date and Time: Tuesday 10 December, 9:15
Abstract: In the talk we will take a fresh look at the Bartle-Graves theorem pointing out the main differences with the standard implicit function theorem. We then will present a set-valued version of this theorem which generalizes some recent results. Applications to variational inequalities and differential inclusions will be also given.
Title: A Constructive Proof of the Existence of a Utility in Revealed Preference Theory
Speaker: Andrew Eberhard
RMIT University
Date and Time: Tuesday 10 December, 12:45
Abstract: The long-standing problem of revealed preferences in economics asks whether any demand relation can be characterised by a utility function.
We answer this affirmatively, under mild and standard assumptions including the General Axiom of Revealed Preferences (GARP), by showing that a dense sampling of a demand relation is sufficient to construct a pseudo-concave utility function that rationalises the relation.
We construct such as utility by a specific limiting process that makes use of existence, first demonstrated by Afriat, of concave utilities that rationalise any finite cardinality sample of a demand relation.
The technical challenge is that our asymptotic analysis must allow for convergence to a nonconcave function, which nevertheless retains convexity of its upper level sets, hence requires variational analysis as well as some techniques beyond convex analysis.
A perplexing point is that two utilities may rationalise the same demand relation (in the sense that they generating the same damand) but may be inconsistent with each other in the sense that the two utilities can leads to conflicting preference relations.
The existence of such examples means that constructive approximation is sensitive to scaling of input data.
In particular, GARP is not sufficient in general to ensure uniqueness of the total ordering that a rationalising utility yields.
Joint work with Danny Ralph and Jean-Pierre Crouzeix
Title: Regularity and Existence Problems
Speaker: Alexander Ioffe
Technion
Date and Time: Tuesday 10 December, 9:45
Abstract: The talk will survey some applications of the regularity theory to analysis. Common to them is the possibility to get much simpler and more transparent proofs for many known results and occasionally new and/or stronger results even in fairly classical situations. The simplest (and well known but still very impressive) example is that the qualification condition for the inclusion 
offered by the regularity theory for arbitrary closed sets in Banach spaces
(subtransversality of 
and 
at 
) is, even for convex sets in 
, strictly more general than its classical counterpart (relative interiors of and have a common point).
In the talk I shall mainly concentrate on various existence results (e.g. differential equations, implicit functions etc.).
Title: Semiregularity of Mappings
Speaker: Alex Kruger
Federation University Australia
Date and Time: Monday 9 December, 15:45
Abstract: There are two basic ways of weakening the definition of the celebrated metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which called semiregularity can be found under several names and the corresponding results are scattered in the literature. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point. In particular cases, we provide necessary and/or sufficient conditions of both primal and dual type.
References
Cibulka, R., Fabian, M. and Kruger, A. Y. On semiregularity of mappings. J. Math. Anal. Appl. 473, 2 (2019), 811–836
The research was supported by the Australian Research Council, project DP160100854
Title: Error Bounds for the Exposed Faces of the Exponential Cone
Speaker: Scott B Lindstrom, Hong Kong Polytechnic University
Date and Time: Sunday 8 December, 15:45
Abstract: Exponential cone programming has recently been introduced in Mosek. We show that the exposed faces of the exponential cone satisfy a generalised regularity condition, and we explain how this condition affords a reasonable error bound. The analysis relies upon properties of the Lambert W function.
This is research is part of a joint work with B Lourenço (The University of Tokyo) and TK Pong (The Hong Kong Polytechnic University).
Title: Lipschitz modulus of linear and convex systems with the Hausdorff metric
Speaker: Marco López
University of Alicante and Federation University Australia
Date and Time: Tuesday 10 December, 11:15
Abstract: In this talk we analyze the Lipschitz behavior of the feasible set in two parametric settings, associated with linear and convex systems in the Euclidean space. More precisely, we deal with the parameter space of linear (finite and semi-infinite) systems identified with the corresponding sets of coefficient vectors. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent 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 first part of the talk we present 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 of existing results in the Chebyshev setting for transferring them to the Hausdorff one. The second part of the presentation establishes some new results on the Lipschitz behavior of convex systems via linearization techniques. This talk is based on a recent manuscript co-authored by J. Beer, M.J. Cánovas and J. Parra.
Title: Stability of optimal trajectories for time delay systems and its applications to biomedicine
Speaker: Musa Mammadov
Deakin University, School of IT
Date and Time: Tuesday 10 December, 11:45
Abstract: Optimal control theory provides a powerful tool to understand some aspects of normal and diseased circulatory systems. In this talk, we consider a special class of optimal control problems described by nonlinear differential equations with time delay. The existence of optimal solutions as well as asymptotic stability of optimal trajectories (that is, the turnpike property) are established under some quite mild assumptions. The results obtained are applied to optimal control problems involving the blood cell production process.
Title: Rational approximation and EEG signal approximation
Speaker: Vinesha Peiris
Swinburne University of Technology
Date and Time: Sunday 8 December, 15:15
Abstract: In Chebyshev approximation problem, the goal is to minimise the supremum of linear functions. In the case of polynomials and piecewise polynomial approximation with fixed knots, the corresponding problems are convex. Consider the rational approximation where the parameters of both polynomials must be optimised. This problem is not convex anymore, instead it is quasiconvex. In this talk I will first introduce rational approximation and then move on to EEG signal approximation using rational approximation.
EEG signals are known as the electric brain activity and approximating EEG signals is very beneficial for medical doctors to identify specific wave patterns. We approximate the EEG signals by optimising the amplitude using the rational approximation where we find the minimum objective value for specific frequency and shift value.
Title: On the isolated calmness property for conic programming
Speaker: Héctor Ramírez
University of Chile
Date and Time: Monday 9 December, 11:45
Abstract: This talk is devoted to the study of the isolated calmness property for critical maps of parameterized conic programs. Our aim is characterizing this property via the computation of second order generalized derivatives. For this, we need to assume that the constraint set is defined over a convex cone satisfying a reducibility assumption and some weak qualification constraint, and to establish some connections between second order derivatives and well-known conic tools, such as the sigma term. Our approach covers seminal examples such as (nonlinear) SDP and SOCP.
Title: An explicit formula for preimages of relaxed one-sided Lipschitz mappings with negative Lipschitz constants
Speaker: Janosch Rieger
Monash University
CANCELLED
Abstract: Relaxed one-sided Lipschitz mappings with negative Lipschitz constant are possess a localization property that is stronger than uniform metrical regularity. This talk complements this fact by providing an explicit formula for entire preimages of such a mapping.
Title: More on higher-order Voronoi cells
Speaker: Vera Roshchina
UNSW Sydney
Date and Time: Monday 9 December, 11:15
Abstract: Higher-order Voronoi cells generalise the classic notion of Voronoi cells. Given a set of sites 
(points in the Euclidean space), the higher-order cell is the set of points for which the longest distance to the points on a prescribed subset 
of is not larger than the distance to the closest point in 
.
In this talk I will continue from the last year, introducing new general results on the structure of higher-order cells, including the relations between cells of different order and the restrictions on the dimension of cells.
The talk is based on joint work with Ryan McKewen (UNSW Sydney)
Title: Extensions and applications of the Perron-Frobenius Theorem
Speaker: Björn Rüffer
University of Newcastle
Date and Time: Tuesday 10 December, 10:15
Abstract: The Perron Theorem states that the spectral radius of a matrix with positive entries is itself a simple, real eigenvalue and there is a corresponding positive eigenvector. This result has seen various extensions, e.g., by Frobenius to matrices that have some zero entries, and later by others to matrices that even allow some negative entries, as well as to nonlinear mappings that enjoy properties such as concavity, non-expansiveness, or subhomogeneity. Applications of this important results are many and include, e.g., the existence of stationary distributions in Markov chains and the Google PageRank. We discuss nonlinear extensions, connections to fixed point theory, and applications of this result in the stability theory of networks of dynamical systems.
Title: Orbit recovery in a noisy environment
Speaker: Nir Sharon
Tel Aviv University
Date and Time: Sunday 8 December, 16:15
Abstract: We discuss the problem of estimating an object from it noisy samples, generated by applying group action. Such models appear in several important applications and introduce both theoretical and algorithmic challenges. In the talk, we present an approach which based on low order statistics. As time permits, we will demonstrate the strength of our solution, its relation to group invariants, and some results on the application of estimating the 3D structure of a molecule from its (very) noisy 2D projection images, as appear in single-particle cryo-electron microscopy.
Title: Approximation by Rational functions in Chebyshev norm
Speaker: Nadia Sukhorukova
Swinburne University of Technology
Date and Time: Monday 9 December, 10:15
Abstract: In this talk, we will talk about rational approximation. This approximation is more flexible than the classical polynomial approximation approach, but the corresponding optimisation problems are more complex. We formally introduce the problem and highlight the challenges, in particular the extension of the Remez algorithm to this type of functions.
Title: Spatio-temporal instability analysis as a non-smooth optimisation problem
Speaker: Sergey Suslov
Swinburne University of Technology
Date and Time: Tuesday 10 December, 12:15
Abstract: The concept of spatio-temporal instability deals with the evolution of initially localised perturbations over extended domains and is commonly found in the fluid and plasma dynamics contexts. To illustrate it, imagine a water surface. Throwing a stone in the middle of it will generate a localised perturbation seen as waves. In a quiescent pond such waves will propagate in all directions while in a fast flowing creek waves will only be seen downstream of the impact point. The first case corresponds to the so-called absolute instability while the second is an example of convective instability. Distinguishing these two regimes is practically very important: the dynamics of an absolutely unstable system is fully self-sustained while the response of a convectively unstable system can be externally controlled (e.g. by changing the background stream speed in the above example). Despite the relative simplicity of a physical interpretation of the concept, its mathematical treatment is rather complicated and involves asymptotic analysis of the Fourier/Laplace type integrals describing the evolution of perturbation wave packets. In this talk we will demonstrate that the problem of finding the parametric boundary between the convective and absolute instability regimes can be formulated as an optimisation problem with a non-smooth objective function that can be evaluated only numerically.
Title: Adaptive piecewise linear support vector regression
Speaker: Sona Taheri
Federation University Australia
Date and Time: Monday 9 December, 15:15
Abstract: In this talk, a new regression method, called the adaptive piecewise linear support vector regression method, is considered. The 
-risk function is used to define regression errors and the support vector machine approach in combination with the piecewise linear regression is applied to develop a new model for regression problems. We formulate the regression problem with this model as an unconstrained nonconvex nonsmooth optimization problem, where the objective function is presented as a difference of two convex (DC) functions. To address the nonconvexity of the problem we propose to build piecewise linear estimates in an adaptive way using a novel incremental approach. The use of such approach allows us to select starting points being rough approximations to the solution. The double bundle method for nonsmooth DC optimization is applied to solve the underlying optimization problems. The proposed adaptive piecewise linear support vector regression method is evaluated using several synthetic and real-world data sets for regression and compared with various mainstream machine learning regression methods.
Title: Projected Subgradient Methods in Infinite Dimensional Spaces
Speaker: Hong-Kun Xu
Hangzhou Dianzi University
Date and Time: Sunday 8 December, 14:45
Abstract:
Subgradient methods, introduced by Shor and developed by Albert, Iusem, Nesterov, Polyak, Soloov, and many others, are used to solve nondifferentiable optimization problems. The major differences from the gradient descent methods (or projection-gradient methods) for differentiable optimization problems lie in the selection manners of the step-sizes.
For instance, constant step-sizes for differentiable objective functions no longer work for nondifferentiable objective functions; for the latter case, diminishing step-sizes must however be adopted.
In this talk, we will first review some existing projected subgradient methods and the main purpose is to discuss weak and strong convergence of projected subgradient methods in an infinite-dimensional Hilbert space.
Some regularization technique for strong convergence of projected subgradient methods will particularly be presented.
Title: Applications of Banach spaces to data compression
Speaker: David Yost
Federation University Australia
Date and Time: Monday 9 December, 12:45
Abstract: Extension problems (when can a mapping be extended to a superset of its domain, in a way that preserves its important properties?) are fundamental in analysis. Recall Tietze's Theorem (continuous functions), the Hahn-Banach Theorem (linear functions), Kirszbraun's Theorem (Lipschitz functions) and Whitney-type Theorems (smooth functions).
In 1984, William B. Johnson and Joram Lindenstrauss published a profound improvement of Kirszbraun's Theorem. For some years, their work remained of interest mostly to functional analysts. A key technical result in their argument was that a finite subset of a high-dimensional euclidean space can be embedded into a space of significantly lower dimension, in such a way that distances between points obey good metric bounds.
Much of the electronic data in use today is stored in the form of points in a vector space. Algorithms applied to it become bogged down quickly if the dimension of the space is too high, so it is desirable to reduce this dimension, whilst preserving the metric structure of the data. In 1997, it was realised that the Johnson–Lindenstrauss lemma provides such low-distortion representations.
We will briefly survey the history and applications of this result.