When: 9:00 am, Fri, 16th Nov 2012
Room: V205, Mathematics Building
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Speaker: Dr Mau Nam Nguyen, Fariborz Maseeh Department of Mathematics and Statistics, Portland State University
Title: Subgradients of Minimal Time Functions and Applications to Set Facility Location
Abstract: In this talk, we present our ongoing efforts in solving a number of continuous facility location problems that involve sets using recently developed tools of variational analysis and generalized differentiation. Subgradients of a class of nonsmooth functions called minimal time functions are developed and employed to study these problems. Our approach advances the applications of variational analysis and optimization to a well-developed field of facility location, while shedding new light on well-known classical geometry problems such as the Fermat-Torricelli problem, the Sylvester smallest enclosing circle problem, and the problem of Apollonius.
When: 10:00 am, Fri, 5th Oct 2012
Room: V205, Mathematics Building
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Speaker: Robert Hesse, Institute for Numerical and Applied Mathematics, University of Goettingen
Title: Simple algorithms for nonconvex feasibility: analysis and some convergence results
Abstract: In this talk projection algorithms for solving (nonconvex) feasibility problems in Euclidian spaces are considered. Of special interest are the Method of Alternating Projections (MAP) and the Averaged Alternating Reflection Algorithm (AAR) which cover some of the state of the art algorithms for our intended application, the phase retrieval problem. In the case of convex feasibility, firm nonexpansiveness of projection mappings is a global property that yields global convergence of MAP, and, for consistent problems, AAR. Based on epsilon-delta-regularity of sets (Bauschke, Luke, Phan, Wang 2012) a relaxed local version of firm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems. This combined with a type of coercivity condition, which relates to the regularity of the intersection, yields local linear convergence of MAP for a wide class of nonconvex problems, and even local linear convergence of AAR in more limited nonconvex settings.
When: 10:00 am, Fri, 21st Sep 2012
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Dr Liangjin Yao, CARMA, The University of Newcastle
Title: Legendre-type integrands and convex integral functions
Abstract:

In this talk, we study the properties of integral functionals induced on the Banach space of integrable functions by closed convex functions on a Euclidean space.

We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We show that in this generality functions such as the Boltzmann-Shannon entropy and the Fermi-Dirac entropy are strongly rotund. We also study convergence in measure and give various limiting counter-example.

When: 3:00 pm, Mon, 27th Aug 2012
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Qiji Jim Zhu, Department of Mathematics, Western Michigan University
Title: Variational methods in the presence of symmetry
Abstract:

Variational methods have been used to derive symmetric solutions for many problems related to real world applications. To name a few we mention periodic solutions to ODEs related to N-body problems and electrical circuits, symmetric solutions to PDEs, and symmetry in derivatives of spectral functions. In this talk we examine the commonalities of using variational methods in the presence of symmetry.

This is an ongoing collaborative research project with Jon Borwein. So far our questions still outnumber our answers.

When: 3:00 pm, Mon, 13th Aug 2012
(Joint talk with Michael Rose.)
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
Title: Expectation integrals on fractal sets
Abstract:

(Joint speakers, Jon Borwein and Michael Rose)

p>Using fractal self-similarity and functional-expectation relations, the classical theory of box integrals is extended to encompass a new class of fractal “string-generated Cantor sets” (SCSs) embedded in unit hypercubes of arbitrary dimension. Motivated by laboratory studies on the distribution of brain synapses, these SCSs were designed for dimensional freedom: a suitable choice of generating string allows for fine-tuning the fractal dimension of the corresponding set. We also establish closed forms for certain statistical moments on SCSs and report various numerical results. The associated paper is at http://www.carma.newcastle.edu.au/jon/papers.html#PAPERS.

When: 9:30 am, Wed, 18th Jul 2012
Room: V205, Mathematics Building
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Speaker: Prof Dominikus Noll, Institut de Mathématiques , Université Paul Sabatier
Title: Nonconvex bundle method with inexact function and subgradient evaluations
Abstract: We present a nonconvex bundle technique where function and subgradient values are available only up to an error tolerance which remains unknown to the user. The challenge is to develop an algorithm which converges to an approximate solution which, despite the lack of information, is as good as one can hope for. For instance, if data are known up to the error $O(\epsilon)$, the solution should also be accurate up to $O(\epsilon)$. We show that the oracle of downshifted tangents is an excellent tool to deal with this difficult situation.
When: 3:30 pm, Thu, 26th Apr 2012
Room: V205, Mathematics Building
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Speaker: Jefferson Melo, Universidade Federal Goiais
Title: Strong Convergence in Hilbert spaces via Gamma-duality
Abstract:

In this talk, we consider a general convex feasibility problem in Hilbert space, and analyze a primal-dual pair of problems generated via a duality theory introduced by Svaiter. We present some algorithms and their convergence properties. The focus is a general primal-dual principle for strong convergence of some classes of algorithms. In particular, we give a different viewpoint for the weak-to-strong principle of Bauschke and Combettes. We also discuss how subgradient and proximal type methods fit in this primal-dual setting.

Joint work with Maicon Marques Alves (Universidade Federal de Santa Catarina-Brazil)

When: 9:30 am, Thu, 19th Apr 2012
(Rescheduled from 29 March.)
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Dr Liangjin Yao, CARMA, The University of Newcastle
Title: A structure theorem for maximally monotone operators with points of continuity
Abstract: In this talk, we consider the structure of maximally monotone operators in Banach space whose domains have nonempty interior and we present new and explicit structure formulas for such operators. Along the way, we provide new proofs of the norm-to-weakstar closedness and property (Q) of these operators (recently established by Voisei). Various applications and limiting examples are given. This is the joint work with Jon Borwein.
When: 3:00 pm, Wed, 11th Apr 2012
(Rescheduled from 10th April)
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Dr Jean Lasserre, LAAS-CNRS, Université de Toulouse
Title: Sublevel sets of positively homogeneous functions and non-Gaussian integrals
Abstract:

We investigate various properties of the sublevel set $\{x : g(x) \leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$ are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(- g)$ on the whole space $\mathbb{R}^n$ (a non-Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of its coefficients.

In fact, whenever $h$ is non-negative, the functional $\int \phi(g)h dx$ is a convex function of $g$ for a large class of functions $\phi:\mathbb{R}_{+} \to \mathbb{R}$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently, to approximate the associated non-Gaussian integral). We also show that finding the sublevel set $\{x : g(x) \leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non-Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.

When: 9:30 am, Thu, 5th Apr 2012
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
Title: Pathological maximal monotone operators
Abstract: In this paper, we construct maximally monotone operators that are not of Gossez's dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC-functions will not always be a BC-function. This provides a negative answer to a challenging question posed by Stephen Simons. Among other consequences, we deduce that every Banach space which contains an isomorphic copy of the James space J or its dual $J^*$, or of $c_0$ or its dual $l^1$ admits a non type (D) operator.
When: 3:00 pm, Tue, 3rd Apr 2012
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Dr Jean Lasserre, LAAS-CNRS, Université de Toulouse
Title: A new look at nonnegativity and polynomial optimization
When: 3:00 pm, Thu, 29th Mar 2012
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
Title: Selection theorems in optimization, Part II: Applications
Abstract: Selection theorems assert that one can pick a well behaved function from a corresponding multifunction. They play a very important role in modern optimization theory. In Part I, I will survey their structure and some applications before sketching some important applications and open research problems in Part II.
When: 4:00 pm, Thu, 22nd Mar 2012
Room: V205, Mathematics Building
Access Grid Venue: Andrew.Danson@newcastle.edu.au
Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
Title: Selection theorems in optimization
Abstract: Selection theorems assert that one can pick a well behaved function from a corresponding multifunction. They play a very important role in modern optimization theory. I will survey their structure and some applications before sketching some important open research problems.
When: 10:00 am, Thu, 9th Feb 2012
Room: V206, Mathematics Building
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Speaker: Mr Shoham Sabach, Technion, Israel Institute of Technology
Title: A first-order method for finding minimal norm-like solutions of convex optimization problems
Abstract: We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem. We introduce a gradient-based method, called the minimal norm gradient method, for solving this class of problems, and establish the convergence of the sequence generated by the algorithm as well as a rate of convergence of the sequence of function values. A portfolio optimization example is given in order to illustrate our results.