Mathematics of Computation and Optimisation

Title: Tweaking Ramanujan’s Approximation of n!

Speaker: Sidney Morris (Federation University, and La Trobe University)

Date and Time: Wed Oct 20, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling’s approximation of n!. He gave a good formula which is asymptotic to n!. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave a nice proof of Ramanujan’s result and an error estimate for the approximation. 

This century there have been several improvements of Stirling’s formula including by Nemes, Windschitl, and Chen. In this presentation it is shown 

(i) how all these asymptotic results can be easily verified; 

(ii) how Hirschhorn and Villarino’s argument allows a tweaking of Ramanujan’s result to give a better approximation; 

(iii) that a new asymptotic formula can be obtained by further tweaking of Ramanujan’s result;

(iv) that Chen’s asymptotic formula is better than the others mentioned here, and the new asymptotic formula is comparable with Chen’s.

Title: A product space reformulation with reduced dimension

Speaker: Rubén Campoy (University of Valencia)

Date and Time: Wed Oct 13, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

The product space reformulation is a powerful trick when tackling monotone inclusions defined by finitely many operators with splitting algorithms. This technique constructs an equivalent two-operator problem, embedded in a product Hilbert space, that preserves computational tractability. Each operator in the original problem requires one dimension in the product space. In this talk, we propose a new reformulation with a reduction on the dimension of the outcoming product Hilbert space. We shall discuss the case of not necessarily convex feasibility problems. As an application, we obtain a new parallel variant of the Douglas-Rachford algorithm with a reduction in the number of variables. The computational advantage is illustrated through some numerical experiments.

Title: Alternating projections with applications to Gerchberg-Saxton error reduction

Speaker: Dominikus Noll (Institut de Mathématiques de Toulouse)

Date and Time: Wed Oct 6, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

We discuss alternating projections between closed non-convex sets A, B in R^n and obtain criteria for convergence when A, B do not intersect transversally. The infeasible case, A∩B=∅, is also addressed, and here we expect convergence toward a gap between A, B. For sub-analytic sets A, B sub-linear convergence rates depending on the Lojasiewicz exponent of the distance function can be computed. We then present applications to the Gerchberg-Saxton error reduction algorithm, to Cadzow’s denoising algorithm, and to instances of the Gaussian EM-algorithm.

Title: Randomized Douglas-Rachford Splitting Algorithms for Federated Composite Optimization

Speaker: Quoc Tran-Dinh (University of North Carolina)

Date and Time: Wed Sep 29, 11:00 AEST (Register here for remote connection via Zoom)

Abstract:

In this talk, we present two randomized Douglas-Rachford splitting algorithms to solve a class of composite nonconvex finite-sum optimization problems arising from federated learning. Our algorithms rely on a combination of three main techniques: Douglas-Rachford splitting scheme, randomised block-coordinate technique, and asynchronous strategy. We show that our algorithms achieve the best-known communication complexity bounds under standard assumptions in the nonconvex setting, while allow one to inexactly updating local models with only a subset of users each round, and handle nonsmooth convex regularizers. Our second algorithm can be implemented in an asynchronous mode using a general probabilistic model to capture different computational architectures. We illustrate our algorithms with many numerical examples and show that the new algorithms have a promising performance compared to common existing methods.

This talk is based on the collaboration with Nhan Pham (UNC), Lam M. Nguyen (IBM), and Dzung Phan (IBM).

Title: DC Semidefinite Programming

Speaker: Maxim Dolgopolik (Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences)

Date and Time: Wed Sep 22, 17:00 AEST (Register here for remote connection via Zoom)

Abstract: DC (Difference-of-Convex) optimization has been an active area of research in nonsmooth nonlinear optimization for over 30 years. The interest in this class of problems is based on the fact that one can efficiently utilize ideas and methods of convex analysis/optimization to solve DC optimization problems. The main results of DC optimization can be extended to the case of nonlinear semidefinite programming problems, i.e. problems with matrix-valued constraints, in several different ways. We will discuss two possible generalizations of the notion of DC function to the case of matrix-valued functions and show how these generalizations lead to two different DC optimization approaches to nonlinear semidefinite programming.