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Polynomial Algebraic Developments in Optimisation and Computation
The solution of polynomial algebraic equations is common in the design of efficient numerical and optimisation techniques. Examples include highly accurate quadrature methods in numerical analysis and semi-definite programming relaxation in optimisation.The feasibility of solution techniques is often due to the special structure of these problems. For more general problems, direct solvers used for polynomial systems are based on Buchberger's algorithm. These techniques yield exact answers, yet have prohibitively high computational cost even for moderate-sized problems. They are also affected by numerical instabilities and floating point errors.
Topics of the workshop include: computational algebraic geometry and floating point arithmetic, convex algebraic geometry, polynomial systems of equations and optimisation problems, semialgebraic problems (i.e., polynomials systems with polynomial constraints).
Speakers
Professor Andrew Eberhard (RMIT, Australia)
Professor Markus Hegland (ANU, Australia)
Dr Scott Lindstrom (Hong Kong Polytechnic)
Dr Vera Roschina (UNSW, Australia)
Sessions
| Time | Session | |
|---|---|---|
| 10am | Higher order Voronoi cells Dr Vera Roshchina, UNSW The classic Voronoi cells can be generalized to a higher-order version by k-element closest subsets of sites. We study the structure of the k-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher-order Voronoi cells on four sites. | |
| 11am | Morning tea | |
| 11:30am | Numerical methods for computing the GCD of univariate polynomials using floating point arithmetic Prof. Markus Hegland, ANU Computing the greatest common divisor (GCD) for two polynomials in floating point arithmetic is known to be computationally challenging and even in standard library software the methods used there might return the result GCD=1 for the case where the polynomials have a nontrivial GCD. Here we review Euclid's algorithm and test a variant for a class of random polynomials. We find that our variant of Euclid's method often produces an acceptable result. However, close monitoring of the values of the norm of the vector of coefficients of the intermediate polynomials arising in the computations is required. | |
| 12:15pm | Lunch | |
| 2pm | On the approximation of monotone operators in a general Banach space Prof. Andrew Eberhard, RMIT As the analysis of monotone operators with nonempty interior in their domain is much more tractable, the question arises as to whether one can approximate the harder FPV operators with ones with a nicer domain. Domain regularisation of maximal monotone operators exists for type-(NI) operators and general operators in reflexive spaces. In this talk we consider what can be said about approximation of this wider class of operators in the Attouch-Wets topology. We also show that convergence of maximal monotone operators in this sense preserves representativeness of the associated Fitzpatrick function and so we are able to deduce a sum theorem for FPV and FP(=NI) operators. Along the way some technical machinery of independent interest is discussed. | |
| 3:30pm | Afternoon tea | |
| 4pm | Using polynomials and complex analysis to check for numerical error in the experimental study of an optimisation algorithm Dr Scott Lindstrom, Hong Kong Polytechnic Splitting methods are special algorithms that are commonly used to solve optimisation problems for engineering and machine learning. One of these algorithms, the Douglas--Rachford method, works in many cases where the problems are not convex. In an effort to learn about the algorithm's special properties, we use the dynamical geometry software Cinderella to visualise its behaviour when it is used to find a point in the intersection of an ellipse and a line. In order to check the sensitivity of our Cinderella code to compounding numerical error, we use the polynomial structure of an ellipse and borrow a convenient tool from complex analysis: the Schwarz function. |
Registration closes 10 April, 2019
Room 4.41, Building #145, Science Road, The Australian National University
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Taxi
If you are catching a taxi or Uber to the ANU Mathematical Sciences Institute, ask to be taken to Building #145, Science Road, ANU. We are located close to the Ian Ross Building and the ANU gym. A Taxi will generally cost around $40 and will take roughly 15 minutes. Pricing and time may vary depending on traffic.
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