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The University of Queensland
26 June - 8 July 2011
St Lucia Campus,
Brisbane, QLD
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Courses
The theme for the AMSI 2011 Australian Graduate Theme Program in
Mathematical Sciences is:
Global
Optimization: theory and applications.
.Within this theme there will be two courses
presented by internationally renowned researchers, who will also be available
for consultations and tutorials. The theme will comprise approximately 30
hours of lectures. Participants
are required to attend both courses.
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Moments,
Positive Polynomials and Their Applications
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Lecturer:
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Dr Jean Lasserre
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Duration:
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Two weeks (June 26 – 8 July)
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Content:
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Many
important applications in global optimization, algebra, probability and
statistics, applied mathematics, control theory, financial mathematics,
inverse problems, etc. can be modeled as a particular instance of the
Generalized Moment Problem (GMP). Particularly important applications are
to be found in "Global Optimization".
This course introduces a new general methodology to solve the GMP when its
data are polynomials and basic semi-algebraic sets. This methodology
combines semidefinite programming with recent results from real algebraic
geometry to provide a hierarchy of semidefinite relaxations converging to the
desired optimal value. Applied on appropriate cones, standard duality in
convex optimization nicely expresses the duality between moments and
positive polynomials. On the application side, a particular emphasis
is put on (global) polynomial optimization but if time permits, other
important applications (in e.g. probability, optimal control, mathematical
finance, multivariate integration, etc.) will be also described in some
details.
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Prerequisites:
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An understanding of Probability
Measure. Further, you should go through the following papers before coming
to Brisbane, both by Jean Lasserre:
A semidefinite programming approach to the generalized problem of
moments, Math. Program., Ser. B (2008), 65-92, and
Global optimization with polynomials and the problem of moments,
SIAM J. Optim. 11 (2001) 796-813.
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Notes:
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The course is based on the
book J.B. Lasserre. "Moments,
Positive Polynomials and Their Applications" Imperial College Press,
London, 2009.
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Competitive Markov Decision Processes
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Lecturer:
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Prof Jezry Filar
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Duration:
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Two weeks (June 26 – 8 July)
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Content:
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This
course is devoted to a unified treatment of both Markov Decision Processes
and Stochastic Games. It examines these processes from the
standpoints of modelling and of optimization, providing newcomers to the
field with an accessible account of algorithms, theory, and applications.
The treatment is self-contained, requiring only some knowledge of linear
algebra and real analysis. Material covered will include a selection of
topics from: (i) Mathematical programming: Markov decision processes (the
non-competitive case), and stochastic games via mathematical programming,
(ii) Existence, structure and applications: Summable stochastic games,
average reward stochastic games and applications and special classes of
stochastic games, and (iii) Necessary tools: Matrix games, bimatrix games
and nonlinear programming, a theorem of Hardy and Littlewood, Markov chains;
and complex varieties and the limit discount equation.
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Prerequisites:
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A
first course in Probability, Linear Algebra or
Matrix Analysis and some appreciation of basic concepts of
Analysis, such as convergence and limits.
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Notes:
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The course is based on the book Filar, Jerzy; Vrieze, Koos
“Competitive Markov decision processes” Springer Publishing, New York 1996
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*Please note that the information for the courses and timetabling may be
varied slightly. Any changes will be posted and highlighted.
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