Speaker: Marco A. López, Alicante University
Title: Some glimpses on convex subdifferential calculus
Abstract: In this talk we present some glimpses on convex subdifferential calculus. In particular, we provide a general formula for the optimal set of the so-called relaxed minimization problem in terms of the approximate minima of the data function. More precisely, if 
is a (real) Hausdorff locally convex space and 
is an extended real valued
function on , the relaxed problem associated with
is defined as
where 
denotes the Legendre-Fenchel bi-conjugate of . It is well-known that the optimal values of both problems coincide:
Our main formula provides the optimal set of 
, i.e. 
, in terms of the approximate solutions of 
, i.e. 
.
Other related results presented in this talk are specific formulas for the
following objects:
1) The subdifferential of the conjugate of an extended-real-valued function,
, in terms of the data function.
2) The subdifferential of the supremum of an arbitrary family of functions, 
, yielding well-known formulas (Volle, Brøndsted, etc.) as simple consequences.
3) Alternative approaches for deriving the subdifferential of the supremum function via the involvement of finite subfamilies.
4) The subdifferential of the sum of two convex functions under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula, involving the approximate subdifferential, and the stronger assumption in the well-known Moreau-Rockafellar formula, which only uses the exact subdifferential.
Speaker: Alexander Kruger, Federation University Australia
Title: Stationarity and Transversality of Finite and Infinite Collections of Sets
Abstract: Extremality, stationarity and transversality properties of finite and infinite collections of sets in Banach spaces as well as qualitative and quantitative dual characterizations of these properties in terms of Fréchet normals will be discussed.
References.
A. Y. Kruger, About regularity of collections of sets, Set-Valued Anal. 14 (2006), 187-206.
A. Y. Kruger and M. A. López, Stationarity and regularity of infinite collections of sets, J. Optim. Theory Appl. 154 (2012), 339-369.
A. Y. Kruger and M. A. López, Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization, J. Optim. Theory Appl. 155 (2012), 390-416.
A. Y. Kruger, D. R. Luke and N. H. Thao, Set regularities and feasibility problems, Math. Program., Ser. B (2016), DOI 10.1007/s10107-016-1039-x.
Speaker: Adil Bagirov, Federation University Australia
Title: Bundle methods for Nonsmooth DC Optimization
Abstract: A version of the bundle method to locally solve unconstrained difference of convex (DC) programming problems is presented. It is assumed that a DC representation of the objective function is available. The main idea is to use subgradients of both the first and second components in the DC representation. The so-called nonconvex cutting plane model of the original objective function is given. We design the proximal bundle method for DC programming based on this approach and prove the convergence of the method to an ε-critical point. This algorithm is tested using some academic test problems.
Speaker: Nadia Sukhorukova, Swinburne University of Technology
Title: A generalisation of Vallée-Poussin procedure to multivariate approximations
Abstract: The theory of Chebyshev approximation is very elegant. In most cases, the optimality conditions are based on the notion of alternance (that is, maximal deviation points with alternating deviation signs). There are a number of approximation methods for polynomial and polynomial spline approximation. Some of them are based on the classical Vallée-Poussin procedure. This paper targets two important issues. First of all, we give a geometrical interpretation of multivariate alternance where the corresponding basis functions are not restricted to monomials. Second, we demonstrate that under certain assumptions the classical Vallée-Poussin procedure, developed for univariate polynomial approximation, can be extended to the case of multivariate approximation, where the basis functions are not restricted to be monomials.
Joint work with Julien Ugon
Speaker: Julien Ugon, Federation University Australia
Title: Alternation conditions for multivariate approximation
Abstract: Conditions for best uniform approximation of functions of one variable are generally formulated in terms of a finite sequence of points where the sign of the deviation alternates—so-called alternation conditions. These conditions are typically easy to check and allow for the development of numerical algorithms for finding best approximations. However they do not generalise easily to the case of approximation of functions of multiple variables. In this talk we report on our preliminary investigations of the generalisation of the notion of alternation to the multivariate settings.
Speaker: Reinier Díaz Millán, Federal Institute of Goiás and UniSA
Title: Conditional extragradient algorithms for variational inequalities
Abstract: In this paper, we generalize the classical extragradient algorithm for solving variational inequality problems using non-null normal vectors of the feasible set. In particular, two conceptual algorithms are proposed and each of them has three different variants related to modified extragradient algorithms. Our analysis contains two main parts: The first part contains two different linesearches, one on the boundary of the feasible set and the other along the feasible direction. The linesearches allow us to find suitable halfspaces containing the solution set of the problem. By using non-null normal vectors of the feasible set, these linesearches can potentially accelerate the convergence. If all normal vectors are chosen as zero, then some of these variants reduce to several well-known projection methods proposed in the literature for solving the variational inequality problem. The second part consists of three special projection steps, generating three sequences with different interesting features.
Convergence analysis of both conceptual algorithms is established assuming the existence of solutions, continuity, and a weaker condition than pseudo-monotonicity on the operator. Examples, on each variant, show that the modifications proposed here perform better than previous classical variants. These results suggest that our scheme may significantly improve the extragradient variants.
Speaker: Scott Linstrom, University of Newcastle
Title: Douglas-Rachford Method for Solving Differential Equations
Abstract: The Douglas-Rachford method of iterated projections (also known as: reflect, reflect, average) is used to solve two set feasibility problems, and it can be extended - via the splitting method - to solve feasibility problems of many sets. It has gained popularity in recent years because of its surprising success in cases where the sets involved are not convex. We investigate its convergence properties by applying it to finding numerical solutions for boundary-valued differential equations. This project is a collaboration with Brailey Sims and Bishnu Lamichhane.
Speaker: David Yost, Federation University Australia
Title: Lipschitz selections for set-valued functions
Abstract: Nature throws multifunctions (set-valued functions) at us. They arise in diverse areas of mathematics, including differential inclusions, approximation theory, calculus of variations, interpolation theory, and others.
The general selection problem is this: given a set-valued mapping 
, can we find a selection 
with 
for all 
? If we only care about functions and sets, the answer is obviously yes, by the axiom of choice. But , 
and 
will usually have some additional structure (algebraic and/or topological), and we want to know whether some relevant properties of 
are preserved by the selection. The literature on finding linear, continuous or measurable selections is now vast. We will attempt to survey the special case when is Lipschitz, and ask whether a Lipschitz 
can be found.
Speaker: Musa Mammadov, Federation University Australia
Title: Turnpike theorems for terminal functionals in infinite horizon
Speaker: Guoyin Li, University of New South Wales
Title: Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rate
Abstract: In this talk, we consider parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish an extension of the Lojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (Holder type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semi-algebraic settings.
Speaker: Andrew Eberhard, RMIT University
Title: Tilt Stability Revisited
Abstract: There have been several new studies of tilt stability over the last 5 years and each has been able to establish new and interesting results about the interaction of various conditions and assumptions. Initially the role of pseudo-Lipschitz behaviour of a localization of the inverse of the subdifferential was seen to be central to this development. Later the role of the strongly related notion of strong metric regularity was emphasised. More recently the role of sub-regularity and implicit convexification has surfaced as important element. The notion of stable strong local minimizer and the quadratic growth condition has also emerged as important concepts.
We will discuss the relationship between these elements and raise some open problems that still exist. We will also discuss how differentiability can arise from these theories providing a new direction of research.
Speaker: Vera Roshchina, RMIT University
Title: Outer limits of subdifferentials for min-max type functions
Abstract: Outer limits of subdifferentials is a limiting construction that can be used to estimate the error bound modulus. We present some new result related to the evaluation of such limiting subdifferentials for max-type and min-max functions.
The talk is based on joint work with Andrew Eberhard and Tian Sang (RMIT University).