This page contains historical information about a workshop held in the past.

Workshop on Fixed Points and Applications

Workshop Program

Scroll down for Abstracts

Time: Speaker: Title:
9.30 – 10.15am Hong-Kun Xu Convergence of Nesterov’s Acceleration Method for Convex Composite Optimization
10.15 – 10.45am Markus Hegland Solving convex problems in approximation spaces defined by fixed point constraints
10.45–11.15am Morning tea
11.15—11.45am Nadia Sukhorukova Signal approximation and clustering based on Chebyshev and Least Squares approximation
11.45—12.15pm Hoa Bui Robustly quasiconvex functions
12.15 – 2.00pm Lunch
2.00—2.30pm Scott Lindstrom Dynamical Tools for Studying Fixed Point Methods
2.30—3.00pm Alex Kruger Intrinsic transversality and alternating projections
3.00—3.30pm Afternoon Tea
3.30—4.00pm Janosch Rieger Solution sets of relaxed one-sided Lipschitz multifunctions
4.00—4.30pm Vera Roshchina Counterexample to De Pierro’s conjecture
4.30—5.00pm Andrew Eberhard A fixed point operator in discrete (& continuous) optimisation duality
5.00—5.15pm Andrew Eberhard + guests Wrap up and thanks.

Abstracts

Speaker: Hong-Kun Xu (Hangzhou Dianzi University)

Title: Convergence of Nesterov’s Acceleration Method for Convex Composite Optimization

Abstract: Convex composite optimization models many practical problems arising from various areas such as compressed sensing and machine learning. It is commonly needed to solve a problem where the objective function is a sum of two functions, one of which may have a simple structure and plays the role of regularization. To solve such a composite optimization problem, the proximal algorithm is prevail-ingly employed. This algorithm has however a slow sublinear convergence rate. To speed up, Yu. Nesterov (1983) invented an acceleration method which can speed up the convergence rate of the gradient-projection algorithm, in terms of the val-ues at the iterates of the objective function, from O(1/k) to O(1/k^2). This was extended to composite optimization by Beck and Teboulle in 2009. However, it remains an open question whether or not the sequence of the iterates, generated in the Nesterov’s way, converges (weakly). We will in this talk discuss recent results on this open question.

Speaker: Markus Hegland (ANU)

Title: Solving convex problems in approximation spaces defined by fixed point constraints

Abstract: The solution of convex optimisation problems in Hilbert spaces requires approximation in finite dimensional spaces. The commonly used Ritz method solves the problem in a finite dimensional subspace. It is typically quasi optimal. However, when the approximation space is defined as the set of fixed points of a family of contractive operators, then the Ritz method is very expensive. We will discuss a new approach which originated in fractal compression. It can be used in fractal approximation spaces including wavelets.

Speaker: Nadia Sukhorukova (Swinburne University)

Title: Signal approximation and clustering based on Chebyshev and Least Squares approximation

Abstract: There have been a number of attempts to extend a well-known K-means method to curve (signal) clustering. This method is based on repeated recomputations of cluster prototypes (centres, approximations) when one or more curves are moving from one cluster to another. In this talk I will talk about possible ways to accelerate the procedure. I will consider two types of problems: Least Squares and Chebyshev approximation.

Speaker: Hoa Bui (Federation University)

Title: Robustly quasiconvex functionse

Abstract: Some important generalised convexity properties like quasiconvexity, explicit quasiconvexity and psedoconvexity are not stable when the function is perturbed by a linear function with sufficiently small norm. In this talk, we will consider a class of generalised convex functions so-called robustly quasiconvex function that some expected optimisation properties remain stable under a small linear perturbation. We will recall some known results for continuously differentiable functions, and discuss on how to deal with non-smooth functions using Fréchet subdifferentials

Speaker: Scott Lindstrom

Title: Dynamical Tools for Studying Fixed Point Methods

Abstract: I will introduce some tools which I have found useful for studying fixed point algorithms, and I will share some methods I have used. Particular emphasis will be given to the Douglas-Rachford method.

Speaker: Alex Kruger

Title: Intrinsic transversality and alternating projectionse

Abstract: Several kinds of ‘regular’ arrangement of a pair of sets near a point in their intersection will be discussed: transversality, subtransversality and intrinsic transversality with the emphasis on the last one. Such regular intersection properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. Dual characterizations of the properties in terms of Fréchet and limiting normals will be provided.

References

  1. Drusvyatskiy, A. D. Ioffe and A. S. Lewis, Transversality and alternating projections for nonconvex sets, Found. Comput. Math. 15 (2015),1637–1651
  2. Y. Kruger and N. H. Thao, Regularity of collections of sets and convergence of inexact alternating projections, J. Convex Anal. 23 (2016), no. 3, 823–847.
  3. Y. Kruger, D. R. Luke and N. H. Thao, About subtransversality of collections of sets, Set-Valued Var. Anal. 25 (2017), 701–729
  4. Y. Kruger, About intrinsic transversality of pairs of sets, Set-Valued Var. Anal. 26 (2018), 111–142
  5. Y. Kruger, D. R. Luke and N. H. Thao, Set regularities and feasibility problems, Math. Program., Ser. B 168 (2018), 279–311

Speaker: Janosch Rieger (Monash University)

Title: Solution sets of relaxed one-sided Lipschitz multifunctionse

Abstract: Relaxed one-sided Lipschitz multifunctions arise in a natural way, for example in control theory and deterministic uncertainty quantification. This talk is concerned with the set of all solutions of the algebraic inclusion 0 ϵ F(x), where F is l-relaxed one-sided Lipschitz with l < 0. The analysis of this solution set is based on Kakutani’s fixed point theorem.

Speaker: Vera Roshchina (RMIT University)

Title: Counterexample to De Pierro’s conjecture

Abstract: I will illustrate the impact of facial structure on the performance of numerical methods using the recent counter-example to De Pierro’s conjecture about the convergence of under-related cyclic projections.

Speaker: Andrew Eberhard (RMIT University)

Title: A fixed point operator in discrete (and continuous) optimisation duality

Abstract: I will discuss some duality structures that have appeared in discrete optimisation duality in conjunction with studies of discrete proximal point algorithm, augmented Lagrangian duality, supporting theory for the “Feasibility Pump” and more recently with regard to Stochastic Integer programming. A common theme appears involving a fixed point operator.