Computational Techniques and Applications Conference (CTAC2018)

27-30 November, 2018
Newcastle, NSW
https://carma.newcastle.edu.au/meetings/ctac2018/

CTAC is a biennial event organised by the special interest group in computational techniques and applications of ANZIAM. The meetings provide an interactive forum for researchers interested in the development and use of computational methods applied to engineering, scientific and other problems.

A counterexample to De Pierro's conjecture

The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, Alvaro de Pierro conjectured that the limit cycles generated by the $\varepsilon$-under-relaxed cyclic projection method converge when $\varepsilon\downarrow 0$ towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that the property is false in general by constructing a system of three compact convex sets in $\mathbb{R}^3$ for which the $\varepsilon$-under-relaxed cycles do not converge.

A key feature of this construction is that for any chosen value of $\varepsilon\in (0,1]$ the under-relaxed projections between $C_1$, $C_2$ and $C_3$ converge to the same unique $\varepsilon$-cycle, no matter what starting point is chosen. This cycle is located in a horisontal plane, with the z-coordinate depending on $\varepsilon$. As $\varepsilon\downarrow 0$ these limit cycles oscillate between the planes $z=-1$ and $z=1$, and hence contradict the conjecture, as the relevant limit does not exist.

This is joint work with Roberto Cominetti and Andrew Williamson, see arXiv:1801.03216.

Slides from a talk in Ballarat at VADU2018 (19-21 February 2018).

On lexicographic tangents

Title: On lexicographic tangents
Time and Venue: 15:00, 25 September 2017
Noah's on the beach, Jon Borwein commemorative conference

Abstract: Lexicographic differentiation was introduced by Yurii Nesterov in 1987. Lexicographic derivatives have some useful properties such as exact calculus rules. Lexicographic tangents are the geometric counterpart of this construction that allows to characterise some desirable properties of convex sets.

I will introduce the notions of lexicographic differentiation and tangents, and focus on characterisations of facial dual completeness property of convex cones. The talk is based on joint work with Prof. Levent Tuncel, University of Waterloo.

Talk in Alicante on outer limits of subdifferentials

Title: Outer limits of subdifferentials for min-max type functions

Time and Venue: 1pm, Friday 7 July 2017, University of Alicante
Abstract: Outer limits of subdifferentials is a limiting construction that can be used to estimate the error bound modulus in a range of problems. We present some new result related to the evaluation of such limiting subdifferentials for max-type and min-max functions.

This talk is based on joint work with Andrew Eberhard and Tian Sang.

Talk on open problems at AMSI Optimise

Title: Open problems in convex optimisation
Time and Venue: 16:10pm, Monash Conference Centre
AMSI Optimise

Abstract: Solving an optimisation problem usually involves choosing an appropriate algorithm or designing a new one if none exists. This may be a difficult task as some problems are notoriously hard, for instance NP-hard problems such as the famous travelling salesman problem. However “P versus NP” is not the only challenge faced by the researchers in optimisation theory. There are numerous open questions whose resolution will lead to breakthroughs in our understanding of optimisation.

Two applications of lexicographic differentiation

Lexicographic differentiation was introduced by Yurii Nesterov in 1987. A recent and more accessible overview is given in his Mathematical Programming paper. At the CIAO workshop on 27 April I talk about lexicographic differentiation and mention two applications: the construction of directed subdifferential and geometric conditions for facial dual completeness of closed convex cones.

The function shown in the Mathematica rendering is lexicographically smooth, but is neither quasidifferentiable nor tame. The idea of this example was suggested by Jeffrey Pang (NUS).

Open problems in convex geometry

I am giving a talk at Monash ACM seminar series on Friday, 10th of March, 2-3pm in Room 340.

Title: Open problems in convex geometry

Abstract: A convex model is the second best thing after a closed-form solution. Convex optimisation problems are often highly tractable, with a variety of numerical methods producing reliable approximations or exact solutions. The choice of the algorithms is vast, and includes general techniques such as subgradient descent or alternating projections, as well as highly specialised simplex and interior point methods. The major factors in the choice of the particular technique are the structure of the problem and the trade-offs between the resources available, the desired accuracy and the reliability of solutions.

Talk at the Workshop on Variational Analysis with Applications

Title: Outer limits of subdifferentials for min-max type functions
Date and Place: Workshop on Variational Analysis with Applications, 13–14 December 2016, PolyU (Hong Kong)
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.