ABSTRACTS:
Nick Barron
Mathematics & Statistics, Loyola University Chicago
Applications of Quasiconvex Duality to Hamilton-Jacobi
Equations and Optimal Transport
Quasiconvex functions are functions with convex level sets.
Hamilton Jacobi equations which are quasiconex in the gradient may
be written using quasiconvex duality as Bellman equations of
appropriate variational problems in L-infinity. This results in
representation formulas for the solution of a large class of
Hamilton-Jacobi equations. Furthermore, using quasiconvex duality
we may derive the dual of an optimal transport problem with
L-infinity cost functional.
The optimal transport result is joint with M. Bocea and R. Jensen.
Olga Brezhneva
Mathematics, Miami University
Optimality Conditions for Nonregular Inequality-Constrained Optimization
Problems
In this talk, we present necessary and sufficient optimality
conditions for some classes of nonregular inequality--constrained
optimization problems. First, we analyze the cases, when
optimality conditions of the Karush-Kuhn-Tucker-type (KKT) hold
for nonregular problems and prove geometric necessary conditions
and the KKT-type optimality conditions under some new regularity
assumptions. Then we continue with consideration of nonregular
problems, for which the KKT-type of conditions do not hold, and
propose some new necessary and sufficient optimality conditions.
Asen L. Dontchev
Mathematical Reviews (AMS) and Mathematics, University of Michigan
Around the Inverse Function Theorem
The classical inverse/implicit function theorems revolves around
solving an equation in terms of a parameter and tell us when
the solution mapping associated with this equation is a
differentiable function with respect to the parameter. Already in
1927 Hildebrandt and Graves observed that one can put aside
differentiability and focus on Lipschitz continuity only.
More sophisticated results may be obtained by employing
various concepts of generalized differentiability. As an
illustration I will present an unconventional implicit
function theorem for an optimal control problem.
Yuri Ledyaev
Mathematics, Western Michigan University
Robert Kipka
Mathematics, Queens University
Optimal Control on Infinite-dimensional Manifolds
Optimal control problems for systems described by partial
differential equations were studied intensively during last few
decades. But for systems with internal symmetries it is
natural to consider such optimal control problems on
infinite-dimensional manifolds. Surprisingly there is no
literature on such infinite-dimensional optimal control problems
on manifolds.
In this talk we discuss a mathematical framework for analysis of
optimal control problems on infinite-dimensional manifolds.
In particular, we demonstrate nonsmooth analysis methods and
Lagrangian charts techniques which can be used for study of global
variations of optimal trajectories of such control systems and
derivation of Pontryagin maximum principle for them.
Daniel Liberzon
Electrical and Computer Engineering and Coordinated Science Lab,
University of Illinois Urbana-Champaign
On Almost Lyapunov Functions.
In this talk we will discuss asymptotic stability properties of
nonlinear systems in the presence of ``almost Lyapunov" functions
which decrease along solutions in a given region not everywhere
but rather on the complement of a set of small volume. Nothing
specific about the structure of this set is assumed besides an
upper bound on its volume. We will show that solutions starting
inside the region approach a small set around the origin whose
volume depends on the volume of the set where the Lyapunov
function does not decrease, as well as on other system parameters.
The result is established by a perturbation argument which
compares a given system trajectory with nearby trajectories that
lie entirely in the set where the Lyapunov function is known to
decrease, and trades off convergence speed of these trajectories
against the expansion rate of the distance to them from the given
trajectory.
This is joint work with Charles Ying and Vadim Zharnitsky (UIUC
Math)
Mau Nam Nguyen
Mathematics, Portland State University
Nonsmooth Optimization Algorithms and Applications to Location
Problems Involving Sets
Traditional facility location studies location problems of
negligible sizes (points), but it is natural to consider location
problems of large sizes (sets). Besides the geometric beauty
reflected by their connections to many well-known computational
geometry problems, these problems have promising applications. In
this talk we present a number of nonsmooth optimization algorithms
for solving continuous location problems involving sets. We also
discuss further applications of these algorithms to other
nonsmooth optimization problems in computational geometry and
machine learning.
Aleksander Olshevsky
Industrial and Enterprise Systems Engineering, University of
Illinois Urbana-Champaign
Convergence Rates in Decentralized Optimization
The widespread availability of copious amounts of data has created
a pressing need to develop optimization algorithms which can work
in parallel when input data is unavailable at a single place but
rather spread throughout multiple locations. In this talk, we
consider the problem of optimizing a sum of convex functions in a
network where each node knows only one of the functions; this is a
common model which includes as particular cases a number of
distributed regression and classification problems. We develop a
stochastic gradient method which is fully decentralized and robust
to unpredictable node and link failures. Our main results yield
convergence time bounds which simultaneously achieve the currently
best scalings with time and network size for this problem.
Ebrahim Sarabi
Boris Mordukchovich
Mathematics, Wayne State University
Full Stability of Optimal Solutions to Constrained and Minimax
Problems
This talk presents new developments and applications of advanced
tools of second-order variational analysis and generalized
differentiation to the fundamental notion of full stability of
local minimizers in general classes of constrained optimization
and minimax problems. In particular, we derive second-order
characterizations of full stability and investigate its
relationships with other notions of stability for parameterized
conic programs and minimax problems.
Victor Zavala
Mathematics and Computer Science Division, Argonne National
Laboratory
An Interior Point Framework for Structured Nonconvex
Optimization
We present a filter line-search interior point framework for
nonconvex NLPs (PIPS-NLP) capable of exploiting embedded
structures in the linear algebra system without requiring inertia
information. We focus on the issue of inertia because we argue
that this hinders the use of modular linear algebra
implementations. The proposed framework relies on a test for
curvature along the tangential direction is sufficient to
guarantee global convergence. We provide numerical evidence that
the inertia-free test is as effective as an inertia detection
strategy based on LBL^T factorizations. We also provide
scalability results for our linear algebra implementation using
NLPs arising from stochastic optimal control of natural gas
networks.
Chao Zhu
Mathematical Sciences, University of Wisconsin Milwaukee
A Measure Approach for Continuous Inventory Models: Discounted
Cost Criterion
This work develops a new approach to the solution of impulse
control problems for continuous inventory models under a
discounted cost criterion. The analysis imbeds this
stochastic problem in two different infinite-dimensional linear
programs, parametrized by the initial inventory level $x_0$, by
concentrating on particular functions and capturing the expected
(discounted) behavior of the inventory level process and ordering
decisions as measures. The first imbedding then naturally
leads to the minimization of a nonlinear function representing the
cost associated with an $(s,S)$ ordering policy and an optimizing
pair determines optimal levels $(s^*,S^*)$. The lower bound
arising from this imbedding is tight when $x_0 \geq s^*$ but is a
strict lower bound when $x_0 < s^*$. Solving the first
linear program determines the value function in the ``no order''
region and is critical to the formulation of the second linear
program. The dual of the second linear program is then
solved to provide a tight lower bound for all $x_0$, in particular
for $x_0 < s^*$, and thereby completely determines the value
function. Existence of an optimal $(s,S)$ policy in the
general class of ordering policies (and its characterization) is a
consequence of the method, not an a priori assumption.
No smoothness of the value function is required; instead, the
level of smoothness results from its construction using the
particular functions from which the linear programs are derived.
This work places minimal assumptions on a general stochastic
differential equation model for the inventory level.
This is a joint work with Kurt Helmes (Humboldt University of
Berlin) and Richard H. Stockbridge (University of
Wisconsin-Milwaukee).
Jim Zhu
Mathematics, Western Michigan University
Variational Approach to Lagrange Multipliers
Despite an extensive literature on various Lagrange multiplier
rules, several fine points related to this important result, are
still worthy further attention. First, Lagrange multipliers are
intrinsically related to the derivative or derivative like objects
of the optimal value function. Moreover, complementary slackness
conditions are nice additional information to have when the
optimal solution exists but is not intrinsic to the existence and
application of the Lagrange multiplier rule. Finally, computing
Lagrange multipliers often rely on decoupling information in term
of each individual constraint. Sufficient conditions are often
needed for this purpose but they are often mixed with the core
condition. We approach the Lagrange multiplier rule from a
variational perspective and, time permits will illustrate the
issues alluded to above with examples.
This talk is based on a collaborative survey project with Jon
Borwein.