
CARMASponsored Seminar Series: Colloquia, Seminars and More.

[Note: events are listed by descending date.]
 SIGMAOPT SEMINAR/OCANA SEMINAR
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Dates: Tue, 28^{th} Jul 2015  Tue, 28^{th} Jul 2015
Jon's talk will be 60 minutes, followed by 30 minutes for Yves.
 Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
 Title: Monotone inclusions and Fitzpatrick functions
 Abstract for Monotone inclusions and Fitzpatrick functions:
We study maximal monotone inclusions from the perspective of (convex) gap functions.
We propose a very natural gap function and will demonstrate how this function arises from the Fitzpatrick function — a convex function used effectively to represent maximal monotone operators.
This approach allows us to use the powerful strong Fitzpatrick inequality to analyse solutions of the inclusion.
 We also study the special cases of a variational inequality and of a generalised variational inequality problem.
 The associated notion of a scalar gap is also considered.
 Corresponding local and global error bounds are developed for the maximal monotone inclusion.
This is joint work with Joydeep Dutta.
 Speaker: Yves Lucet, University of British Colombia
 Title: On the convexity of piecewisedefined functions
 Abstract for On the convexity of piecewisedefined functions:
Functions that are piecewise defined are a common sight in mathematics while convexity is a property
especially desired in optimization. Suppose now a piecewisedefined function is convex on each of its defining
components – when can we conclude that the entire function is convex? Our main result provides sufficient
conditions for a piecewisedefined function f to be convex. We also provide a sufficient condition for checking the
convexity of a piecewise linearquadratic function, which play an important role in computeraided convex analysis.
Based on joint work with Heinz H. Bauschke (Mathematics, UBC Okanagan) and Hung M. Phan (Mathematics,
University of Massachusetts Lowell).
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 SIGMAOPT/OCANA SEMINAR / CROSSPACIFIC WORKSHOP
 Speaker: Prof Dominikus Noll, Institut de Mathématiques , Université Paul Sabatier
 Title: Convergence of descent methods using the KurdykaLojasiewicz inequality
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 10:00 am, Thu, 20^{th} Feb 2014
 Abstract:
Without convexity the convergence of a descent algorithm can normally only be certified in the weak sense that every accumulation point of the sequence of iterates is critical. This does not at all correspond to what we observe in practice, where these optimization methods always converge to a single limit point, even though convergence may sometimes be slow.
Around 2006 it has been observed that convergence to a single limit can be proved for objective functions having certain analytic features. The property which is instrumental here is called the Lojasiewicz inequality, imported from analytic function theory. While this has been successfully applied to smooth functions, the case of nonsmooth functions turns out more difficult. In this talk we obtain some progress for upperC1 functions. Then we proceed to show that this is not just out of a theoretical sandpit, but has consequences for applications in several fields. We sketch an application in destructive testing of laminate materials.
 To be received via videoconferencing.
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: CARMA [ENQUIRIES]
 Time and Date: 9:30 am, Wed, 23^{rd} Oct 2013
 Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
 Title: DouglasRachford Feasibility Methods For Matrix Completion Problems
 Abstract for DouglasRachford Feasibility Methods For Matrix Completion Problems:
Many successful nonconvex applications of the DouglasRachford method can be viewed as the reconstruction of a matrix, with known properties, from a subset of its entries. In this talk we discuss recent successful applications of the method to a variety of (real) matrix reconstruction problems, both convex and nonconvex.
This is joint work with Fran Aragón and Matthew Tam.
 Speaker: Prof Heinz Bauschke, Mathematics and Statistics, UBC Okanagan
 Title: The Douglas–Rachford algorithm for two subspaces
 Abstract for The Douglas–Rachford algorithm for two subspaces:
I will report on recent joint work (with J.Y. Bello Cruz, H.M. Phan, and X. Wang) on the Douglas–Rachford algorithm for finding a point in the intersection of two subspaces. We prove that the method converges strongly to the projection of the starting point onto the intersection. Moreover, if the sum of the two subspaces is closed, then the convergence is linear with the rate being the cosine of the Friedrichs angle between the subspaces. Our results improve upon existing results in three ways: First, we identify the location of the limit and thus reveal the method as a best approximation algorithm; second, we quantify the rate of convergence, and third, we carry out our analysis in general (possibly infinitedimensional) Hilbert space. We also provide various examples as well as a comparison with the classical method of alternating projections.
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 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
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: TBC
 Time and Date: 9:00 am, Fri, 16^{th} Nov 2012
 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 welldeveloped field of facility location, while shedding new light on wellknown classical geometry problems such as the FermatTorricelli problem, the Sylvester smallest enclosing circle problem, and the problem of Apollonius.
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 Speaker: Robert Hesse, Institute for Numerical and Applied Mathematics, University of Goettingen
 Title: Simple algorithms for nonconvex feasibility: analysis and some convergence results
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: WestGrid
 Time and Date: 10:00 am, Fri, 5^{th} Oct 2012
 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 epsilondeltaregularity 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.
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 Speaker: Dr Liangjin Yao, CARMA, The University of Newcastle
 Title: Legendretype integrands and convex integral functions
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 10:00 am, Fri, 21^{st} Sep 2012
 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 (wellposed). We show that in this generality functions such as the BoltzmannShannon entropy and the FermiDirac entropy are strongly rotund. We also study convergence in measure and give various limiting counterexample.
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 SIGMAOPT SEMINAR
 Speaker: Qiji Jim Zhu, Department of Mathematics, Western Michigan University
 Title: Variational methods in the presence of symmetry
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 3:00 pm, Mon, 27^{th} Aug 2012
 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 Nbody 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.
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 SIGMAOPT/CARMA SEMINAR
 Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
 Title: Expectation integrals on fractal sets
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 3:00 pm, Mon, 13^{th} Aug 2012
 Abstract:
(Joint speakers, Jon Borwein and Michael Rose)
p>Using fractal selfsimilarity and functionalexpectation relations, the classical theory of box integrals is extended to encompass a new class of fractal “stringgenerated 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 finetuning 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.
 (Joint talk with Michael Rose.)
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 Speaker: Prof Dominikus Noll, Institut de Mathématiques , Université Paul Sabatier
 Title: Nonconvex bundle method with inexact function and subgradient evaluations
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: WestGrid
 Time and Date: 9:30 am, Wed, 18^{th} Jul 2012
 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.
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 SIGMAOPT SEMINAR
 Speaker: Jefferson Melo, Universidade Federal Goiais
 Title: Strong Convergence in Hilbert spaces via Gammaduality
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UniSA
 Time and Date: 3:30 pm, Thu, 26^{th} Apr 2012
 Abstract:
In this talk, we consider a general convex feasibility problem in Hilbert space, and analyze a primaldual pair of problems generated via a duality theory introduced by Svaiter. We present some algorithms and their convergence properties. The focus is a general primaldual principle for strong convergence of some classes of algorithms. In particular, we give a different viewpoint for the weaktostrong principle of Bauschke and Combettes. We also discuss how subgradient and proximal type methods fit in this primaldual setting.
Joint work with Maicon Marques Alves (Universidade Federal de Santa CatarinaBrazil)
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 Speaker: Dr Liangjin Yao, CARMA, The University of Newcastle
 Title: A structure theorem for maximally monotone operators with points of continuity
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 9:30 am, Thu, 19^{th} Apr 2012
 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 normtoweakstar 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.
 (Rescheduled from 29 March.)
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 SIGMAOPT SEMINAR
 Speaker: Dr Jean Lasserre, LAASCNRS, Université de Toulouse
 Title: Sublevel sets of positively homogeneous functions and nonGaussian integrals
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 3:00 pm, Wed, 11^{th} Apr 2012
 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 nonGaussian 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 nonnegative, 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 nonGaussian 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 Gaussianlike property of nonGaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.
 (Rescheduled from 10th April)
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
 Title: Pathological maximal monotone operators
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 9:30 am, Thu, 5^{th} Apr 2012
 Abstract:
In this paper, we construct maximally monotone operators that are not of Gossez's densetype (D) in many nonreflexive spaces. Many of these operators also fail to possess the BrønstedRockafellar (BR) property. Using these operators, we show that the partial infconvolution of two BCfunctions will not always be a BCfunction. 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.
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 SIGMAOPT SEMINAR
 Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
 Title: Selection theorems in optimization, Part II: Applications
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 3:00 pm, Thu, 29^{th} Mar 2012
 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.
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 Speaker: Mr Shoham Sabach, Technion, Israel Institute of Technology
 Title: A firstorder method for finding minimal normlike solutions of convex optimization problems
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: WestGrid
 Time and Date: 10:00 am, Thu, 9^{th} Feb 2012
 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 gradientbased 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.
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 SIGMAOPT SEMINAR/OCANA SEMINAR
 Speaker: Stephen Simons, Department of Mathematics, University of California, Santa Barbara
 Title: The asymmetric sandwich theorem
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: WestGrid
 Time and Date: 10:00 am, Fri, 9^{th} Dec 2011
 Abstract:
We discuss the asymmetric sandwich theorem, a generalization of the Hahn–Banach
theorem. As applications, we derive various results on the existence of linear functionals
in functional analysis that include bivariate, trivariate and quadrivariate generalizations
of the Fenchel duality theorem. We consider both results that use a simple boundedness
hypothesis (as in Rockafellar’s version of the Fenchel duality theorem) and also results
that use Baire’s theorem (as in the Robinson–Attouch–Brezis version of the Fenchel
duality theorem).
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 SIGMAOPT SEMINAR
 Speaker: Markus Hegland, Mathematical Sciences Institute, Australian National University
 Title: A finite element method for density estimation with Gaussian process priors
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 2:30 pm, Mon, 21^{st} Nov 2011
 Abstract:
Probability densities are a major tool in exploratory statistics and stochastic modelling. I will talk about a numerical technique for the estimation of a probability distribution from scattered data using exponential families and a maximum aposteriori approach with Gaussian process priors.
Using CameronMartin theory, it can be seen that density estimation leads to a nonlinear variational problem with a functional defined on a reproducing kernel Hilbert space. This functional is strictly convex. A dual problem based on Fenchel duality will also be given. The (original) problem is solved using a NewtonGalerkin method with damping for global convergence. In this talk I will discuss some theoretical results relating to the numerical solution of the variational problem and the results of some computational experiments. A major challenge is of course the curse of dimensionality which appears when highdimensional probability distributions are estimated.
 Download: Talk slides (452 KB)
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 SIGMAOPT SEMINAR
 Speaker: Vladimir Ejov, School of Mathematics and Statistics, University of South Australia
 Title: Perturbed Determinants, Spectral Theory and Longest Cycles on Graphs
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 2:30 pm, Tue, 8^{th} Nov 2011
 Abstract:
We interpret the Hamiltonian Cycle problem (HCP) as a an optimisation problem with the determinant objective function, naturally arising from the embedding of HCP into a Markov decision process. We also exhibit a characteristic structure of the class of all cubic graphs that stems from the spectral properties of their adjacency matrices and provide an analytic explanation of this structure.
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 SIGMAOPT SEMINAR
 Speaker: Dr Francisco Aragón Artacho, CARMA, The University of Newcastle
 Title: Lipschitzian properties of a generalized proximal point algorithm
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 4:00 pm, Thu, 1^{st} Sep 2011
 Abstract:
Basically, a function is Lipschitz continuous if it has a
bounded slope. This notion can be extended to setvalued maps in
different ways. We will mainly focus on one of them: the socalled Aubin
(or Lipschitzlike) property. We will employ this property to analyze
the iterates generated by an iterative method known as the proximal
point algorithm. Specifically, we consider a generalized version of this
algorithm for solving a perturbed inclusion
$$y \in T(x),$$
where $y$ is a perturbation element near 0 and $T$ is a setvalued mapping.
We will analyze the behavior of the convergent iterates generated by the
algorithm and we will show that they inherit the regularity properties
of $T$, and vice versa. We analyze the cases when the mapping $T$ is
metrically regular (the inverse map has the Aubin property) and strongly
regular (the inverse is locally a Lipschitz function). We will not
assume any type of monotonicity.
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 SIGMAOPT SEMINAR
 Speaker: Liangjin Yao, CARMA, The University of Newcastle
 Title: For maximally monotone linear relations, dense type, negativeinfimum type, and FitzpatrickPhelps type all coincide with monotonicity of the adjoint
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UniSA
 Time and Date: 3:00 pm, Tue, 16^{th} Aug 2011
 Abstract:
It is shown that, for maximally monotone linear relations defined on a general Banach space, the monotonicities of dense type, of negativeinfimum type, and of FitzpatrickPhelps type are the same and equivalent to monotonicity of the adjoint. This result also provides affirmative answers to two problems: one posed by Phelps and Simons, and the other by Simons.
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 SIGMAOPT/CARMA SEMINAR
 Speaker: Liangjin Yao, CARMA, The University of Newcastle
 Title: The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 4:00 pm, Thu, 11^{th} Aug 2011
 Abstract:
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this talk, we prove the maximal monotonicity of the sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function satisfying Rockafellar's constraint qualification. Moreover, we show that this sum operator is of type (FPV).
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 CARMA COLLOQUIUM AND SIGMAOPT SEMINAR
 Speaker: Qiji Jim Zhu, Department of Mathematics, Western Michigan University
 Title: Why Bankers Should Learn Convex Analysis. (Part 2)
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 11:00 am, Fri, 4^{th} Mar 2011
 Abstract:
Concave utility functions and convex risk measures play crucial roles in economic and financial problems. The use of concave utility function can at least be traced back to Bernoulli when he posed and solved the St. Petersburg wager problem. They have been the prevailing way to characterize rational market participants for a long period of time until the 1970’s when Black and Scholes introduced the replicating portfolio pricing method and Cox and Ross developed the risk neutral measure pricing formula. For the past several decades the `new paradigm’ became the main stream. We will show that, in fact, the `new paradigm’ is a special case of the traditional utility maximization and its dual problem. Moreover, the convex analysis perspective also highlights that overlooking sensitivity analysis in the `new paradigm’ is one of the main reason that leads to the recent financial crisis. It is perhaps time again for bankers to learn convex analysis.
The talk will be divided into two parts. In the first part we layout a discrete model for financial markets. We explain the concept of arbitrage and the no arbitrage principle. This is followed by the important fundamental theorem of asset pricing in which the no arbitrage condition is characterized by the existence of martingale (risk neutral) measures. The proof of this gives us a first taste of the importance of convex analysis tools. We then discuss how to use utility functions and risk measures to characterize the preference of market agents. The second part of the talk focuses on the issue of pricing financial derivatives. We use simple models to illustrate the idea of the prevailing Black Scholes replicating portfolio pricing method and related CoxRoss riskneutral pricing method for financial derivatives. Then, we show that the replicating portfolio pricing method is a special case of portfolio optimization and the risk neutral measure is a natural byproduct of solving the dual problem. Taking the convex analysis perspective of these methods h
 Download: Presentation (1.3 MB)
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 CARMA COLLOQUIUM AND SIGMAOPT SEMINAR
 Speaker: Qiji Jim Zhu, Department of Mathematics, Western Michigan University
 Title: Why Bankers Should Learn Convex Analysis (Part 1)
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 11:00 am, Thu, 3^{rd} Mar 2011
 Abstract:
Concave utility functions and convex risk measures play crucial roles in economic and financial problems. The use of concave utility function can at least be traced back to Bernoulli when he posed and solved the St. Petersburg wager problem. They have been the prevailing way to characterize rational market participants for a long period of time until the 1970’s when Black and Scholes introduced the replicating portfolio pricing method and Cox and Ross developed the risk neutral measure pricing formula. For the past several decades the `new paradigm’ became the main stream. We will show that, in fact, the `new paradigm’ is a special case of the traditional utility maximization and its dual problem. Moreover, the convex analysis perspective also highlights that overlooking sensitivity analysis in the `new paradigm’ is one of the main reason that leads to the recent financial crisis. It is perhaps time again for bankers to learn convex analysis.
The talk will be divided into two parts. In the first part we layout a discrete model for financial markets. We explain the concept of arbitrage and the no arbitrage principle. This is followed by the important fundamental theorem of asset pricing in which the no arbitrage condition is characterized by the existence of martingale (risk neutral) measures. The proof of this gives us a first taste of the importance of convex analysis tools. We then discuss how to use utility functions and risk measures to characterize the preference of market agents. The second part of the talk focuses on the issue of pricing financial derivatives. We use simple models to illustrate the idea of the prevailing Black Scholes replicating portfolio pricing method and related CoxRoss riskneutral pricing method for financial derivatives. Then, we show that the replicating portfolio pricing method is a special case of portfolio optimization and the risk neutral measure is a natural byproduct of solving the dual problem. Taking the convex analysis perspective of these methods h
 Download: Presentation (1.3 MB)
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 CROSSPACIFIC WORKSHOP
 Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle
 Title: Cross Pacific Workshop
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 10:00 am, Tue, 15^{th} Feb 2011
 Abstract:
CARMA is currently engaged in several shared projects with the IRMACS Centre and the OCANA Group UBCO, both in British Columbia, Canada. This workshop will be an opportunity to learn about irmacs, Centres and to experience the issues in collaborating for research and teaching across the Pacific.
This will be followed by discussion and illustrations of collaboration, technology, teaching and funding etc.
Cross Pacific Collaboration pages at irmacs.
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 CARMA COLLOQUIUM AND SIGMAOPT SEMINAR
 Speaker: Angelos Tsoukalas , School of Mathematical and Geospatial Sciences, RMIT University
 Title: Using Cutting Planes in the Feasibility Pump
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: RMIT
 Time and Date: 3:30 pm, Wed, 27^{th} Oct 2010
 Abstract:
We discuss the feasibility pump heuristic and we interpret it as a multistart, global optimization algorithm that utilizes a fast local minimizer. The function that is minimized has many local minima, some of which correspond to feasible integral solutions. This interpretation suggests alternative ways of incorporating restarts one of which is the use of cutting planes to eliminate local optima that do not correspond to feasible integral solutions. Numerical experiments show encouraging results on standard test libraries.
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 SIGMAOPT SEMINAR
 Speaker: Dr Victoria MartínMárquez, Department of Mathematical Analysis, Universidad de Sevilla
 Title: Recent advances in nonexpansive and monotone operator theory in Hadamard manifolds
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UniSA
 Time and Date: 11:30 am, Fri, 10^{th} Sep 2010
 Abstract:
Riemannian manifolds constitute a broad and fruitful framework for the development of different fields in mathematic, such as convex analysis, dynamical systems, optimization or mathematical programming, among other scientific areas, where some of its approaches and methods have successfully been extended from Euclidean spaces. The nonpositive sectional curvature is an important property enjoyed by a large class of differential manifolds, so Hadamard manifolds, which are complete simply connected Riemannian manifolds of nonpositive sectional curvature, have worked out a suitable setting for diverse disciplines. On the other hand, the study of the class of nonexpansive mappings has become an active research area in nonlinear analysis. This is due to the connection with the geometry of Banach spaces along with the relevance of these mappings in the theory of monotone and accretive operators. We study the problems that arise in the interface between the fixed point theory for nonexpansive type mappings and the theory of monotone operators in the setting of Hadamard manifolds. Different classes of monotone and accretive setvalued vector fields and the relationship between them will be presented, followed by the study of the existence and approximation of singularities for such vector fields. Then we analyze the problem of finding fixed points of nonexpansive type mappings and the connection with monotonicity. As a consequence, variational inequality and minimization problems in this setting will be discussed.
 Download: Talk slides (1.75MB)
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 SIGMAOPT SEMINAR
 Speaker: Andreas Hamel, Department of Operations Research and Financial Engineering, Princeton University
 Title: The Fundamental Duality Formula in SetValued Optimization
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UniSA
 Time and Date: 11:30 am, Fri, 27^{th} Aug 2010
 Abstract:
The fundamental duality formula (see Zalinescu ”Convex Analysis in General Vector Spaces”, Theorem 2.7.1) is extended to functions mapping into the power set of a topological linear space with a convex cone which is not necessarily pointed. Pairs of linear functionals are used as dual variables instead of linear operators. The talk will consist of three parts. First, motivations and explanations are given for the infimum approach to setvalued optimization. It deviates from other approaches, and it seems to be the only way to obtain a theory which completely resembles the scalar case. In the second part, the main results are presented, namely the fundamental duality formula and several conclusions. The third part deals with duality formulas for setvalued risk measures, a cutting edge development in mathematical finance. It turns out that the proposed duality theory for setvalued functions provides a satisfying framework not only for setvalued risk measures, but also for noarbitrage and superhedging theorems in conical market models.
 Download: Lecture notes (110 KB)
 (11:00 am Adelaide time.)
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 SIGMAOPT SEMINAR
 Speaker: Stephen Simons, Department of Mathematics, University of California, Santa Barbara
 Title: The HahnBanachLagrange Theorem
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UniSA
 Time and Date: 2:30 pm, Wed, 18^{th} Aug 2010
 Abstract:
We discuss the HahnBanachLagrange theorem, a generalized form of the HahnBanach theorem. As applications, we derive various results on the existence of linear functionals in functional analysis, on the existence of Lagrange multipliers for convex optimization problems, with an explicit sharp lower bound on the norm of the solutions (multipliers), on finite families of convex functions (leading rapidly to a minimax theorem), on the existence of subgradients of convex functions, and on the Fenchel conjugate of a convex function. We give a complete proof of Rockafellar's version of the Fenchel duality theorem, and an explicit sharp lower bound for the norm of the solutions of the Fenchel duality theorem in terms of elementary geometric concepts.
 Download: Seminar slides (309 KB)
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 CARMA COLLOQUIUM AND SIGMAOPT SEMINAR
 Speaker: Conjoint Prof Steve Wright, Computer Sciences Department and Wisconsin Institute for Discovery, University of WisconsinMadison
 Title: More Tools and Applications of Sparse Optimization
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UNewcastle [ENQUIRIES]
 Time and Date: 4:00 pm, Thu, 24^{th} Jun 2010
 Abstract:
Machine learning problems are a particularly rich source of applications for sparse optimization, giving rise to a number of formulations that require specialized solvers and structured, approximate solutions. As case studies, we discuss two such applications  sparse SVM classification and sparse logistic regression  and present algorithms that are assembled from different components, including stochastic gradient methods, random approximate matrix factorizations, block coordinate descent, and projected Newton methods. We also describe a third (distantly related) application to selection of captive breeding populations for endangered species using binary quadratic programming, a project started during a visit to Newcastle in June 2009.
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 SIGMAOPT SEMINAR
 Speaker: Dr C. Yalcin Kaya, University of South Australia
 Title: Numerical Methods For Convex Multiobjective Control Problems
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UniSA
 Time and Date: 2:00 pm, Wed, 9^{th} Jun 2010
 Abstract:
We consider multiobjective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multiobjective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach. This is joint work with Henri Bonnel (University of New Caledonia).
 Download: Seminar slides by C. Yalcin Kaya
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