Optimization and Control Seminar, Spring 2009
| Thursday, April 2, 2009, 12:00-1:00 pm |
SOND 406 |
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Convexification and Robustification in Data Fitting
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| - Dr. James T.-H. Lo |
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| Abstract:
Two theorems will be discussed concerning the use of a type of
risk-averting error (RAE) criterion for fitting parametric or nonparametric
models to data. Theorem 1 states that the convexity region of the RAE
criterion expands monotonically as its risk-sensitivity index $B&K(B increases.
Theorem 2 states that as $B&K(B increases to infinity, the RAE criterion
approaches the minimax error criterion.
The RAE criterion is easily seen to converge to the mean squared error (MSE)
criterion as its $B&K(B goes to zero. Theorem 1 suggests that the RAE criterion
can be used to convexify the MSE criterion to avoid poor local minima. By
increasing $B&K(B gradually, tunnels (or worm holes) are created for a
local-search minimization procedure (e.g., quasi-Newton and conjugate
gradient) to travel through to a better and better local minimum.
The minimax error criterion is widely used for robustifying system
identifiers, controllers and filters. It comes from the game theory and is
known to be too pessimistic in most applications. Moreover, it is difficulty
to use. Theorem 2 shows that the RAE criterion with a very large $B&K(B acts like
the minimax error criterion, but is easier to use. More important, the RAE
criterion induces a continuous range of robustness indexed by $B&K(B ranging from
0 to $B!g(B. The induced range of robustness ranges from the MSE (or zero)
robustness at one end to the minimax (or infinite) robustness at the other
end. Depending on the application, a suitable degree of robustness can be
selected.
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| Thursday, April 9, 2009, 12:00-1:00 |
SOND 406 |
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A Gershgorin type theorem, spectral inequalities, and simultaneous
stability in Euclidean Jordan algebras
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| - Melania Moldovan |
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| Abstract:
For complex square matrices, the Levy-Desplanques theorem asserts that
a strictly diagonally dominant matrix is invertible. The well-known
Gershgorin theorem on the location of eigenvalues is equivalent to
this. In the first part of the talk, we present an extension of the
Levy-Desplanques theorem to an object in a Euclidean Jordan algebra.
As a consequence, we prove a Gershgorin type theorem for the spectral
eigenvalues of an object in a Euclidean Jordan algebra.
In matrix theory, the well known Cauchy's interlacing theorem states
that if a row-column pair is deleted from a real symmetric matrix,
then the eigenvalues of the resulting principal matrix interlace those
of the original one. This result was generalized recently to the
setting of simple Euclidean Jordan algebras by Gowda and Tao. In the
second part of the talk, we present some consequences of this
generalization and prove the dual form of the min-max theorem of
Hirzebruch.
Using duality and complementarity ideas, and {\bf Z}-transformations,
in the third part of the talk, we discuss ways of describing
simultaneous stability of linear transformations on Hilbert spaces. As
a consequence, we discuss the existence of common linear/quadratic
Lyapunov functions for switched linear systems. In particular, we
extend a recent result of Mason-Shorten on positive switched system
with two constituent linear time-invariant systems to an arbitrary
finite system.
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| Thursday, April 23, 2009, 12:00-1:00 |
SOND 406 |
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A homogeneous model for monotone mixed linear complementarity
problems
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| - Cosmin Petra |
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| Abstract:
In optimization, a homogeneous model is an artificial transformation of a
given problem. The transformation is done such that the homogeneous problem
has always a solution even if the original problem does not. Moreover,
without having any assumption on the feasibility of the original problem,
the homogeneous model is able to provide the solution if it exists, or a
certificate of infeasibility otherwise. In this work we introduce a
homogeneous model for mixed linear complementarity problems which represent
a generalization of the standard linear complementarity problems. We also
show that the homogeneous problem can be efficiently solved using
interior-point path-following methods.
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| Friday, May 1, 2009, 12:00-1:00 |
MP 401 |
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Hybrid inclusions: modeling and analysis of dynamical systems
with continuous-time and discrete-time features
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| - Dr. Rafal Goebel |
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| Abstract:
Various dynamical systems exhibit behaviors usually attributed to
continuous-time systems and behaviors usually attributed to discrete-time
systems. For example, currents in a circuit can change continuously, according to
Kirchoff's laws, and instantly, due to switches opening and closing.
Similarly, control algorithms designed for
continuous-time control systems may involve timers that are reset, modes
of operation that switch from one to another, etc. Hybrid inclusions
provide a simple framework for modeling and analysis of such systems,
through a combination of differential inclusions, difference
inclusions, and constraints on the motions resulting from these inclusions.
After motivating examples and an overview of some other mathematical
approaches to dynamical systems that combine continuous-time and
discrete-time dynamics, the talk will present the framework of and basic
results for differential inclusions. The use of generalized time domains
for the parameterization of solutions to differential inclusions will be
motivated. Elements of asymptotic stability theory for differential
inclusions will be presented. In particular, a technique of approximating a hybrid inclusion with a simpler one, far generalizing the concept of linearization, will be described. Then,
asymptotic stability for a hybrid inclusion will be deduced from
asymptotic stability for the
approximation.
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| Thursday, May 7, 2009, 12:00-1:00 |
SOND 406 |
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Infeasible high-order interior-point methods for linear
complementarity problems
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| - Adrian Vancea
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| Abstract:
We present three high order infeasible interior point algorithms
for sufficient linear complementarity problems. The first one is a
corrector-predictor one, which acts in the small neighborhood of the
central path and doesn't depend on the handicap \kappa of the LCP. It is
globally convergent for general positive starting points and it has
O((1+\kappa)\sqrt{n}L)-iteration complexity for feasible or "almost
feasible" starting points and O((1+\kappa)^2nL)-iteration complexity for
"sufficiently large" infeasible starting points. The second and the third
algorithm, both act in the wide and symmetric neighborhood of the central
path. The second is a predictor-corrector one which depends on the
handicap \kappa of the LCP, but we have a version of it which doesn't
depend on the handicap. The third is a corrector-predictor one which
doesn't depend on \kappa. Both algorithms have the best known iteration
complexity. We also mention that all these algorithms are superlinearly
convergent even in the absence of strict complementarity.
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