ABSTRACTS:
Tan Cao
Mathematics, Wayne State University
Optimal control of the perturbed sweeping process over
polyhedral controlled set
The paper addresses a new class of optimal control problems
governed by the dissipative non-Lipschitzian di fferential
inclusion of the perturbed sweeping/Moreau process over a moving
controlled polyhedral set. Besides the highly non-Lipschitzian
nature of the unbounded diff erential inclusion of the controlled
perturbed sweeping process, the optimal control problems under
consideration contain intrinsic state constraints of the
inequality and equality types. All of this creates serious
challenges for deriving necessary optimality conditions. We rst
establish the strong convergence of optimal solutions of discrete
approximations to a local minimizer of a continuous-time system
and obtain necessary conditions for discrete counterparts of the
controlled sweeping process under consideration. And then we
derive constructive necessary conditions for the original sweeping
process problem expressed entirely in terms of the data and the
reference trajectory. Our approach to necessary optimality
conditions is based on the method of discrete approximations and
generalized di fferential tools of variational analysis. The
established necessary optimality conditions for the perturbed
sweeping process are illustrated by some nontrivial examples.
This is joint work with Boris Mordukhovich
Dean Carlson
American Mathematical Society
Property (D) and the Lavrentiev Phenomenon.
In 1926 M.Lavrentiev gave an example of a free problem in the
calculus of variations for which the in mum over the class of
functions in W^{1,1}[t_1, t_2] satisfying prescribed end point
conditions was strictly less than the in mum over the dense subset
of admissible functions in W^{1,\infty}[t_1, t_2]. After
Lavrentiev's discovery L.Tonelli and B.Mania gave sucient
conditions under which this phenomenon does not arise. After these
results, the study of the Lavrentiev phenomenon lay dormant until
the 1980s when a series of papers by Ball and Mizel and by Clarke
and Vinter gave a number of new examples for which the Lavrentiev
phenomenon occurred. Also in 1979, T.S. Angell showed that the
Lavrentiev phenomenon did not occur if the integrands satisfy a
certain analytic property known as property (D). Since Angell's
result there have been several papers that have discussed the
nonoccurence of the Lavrentiev phenomenon for free problems in the
calculus of variations. The purpose of this paper is two-fold.
First to present a general approach to the proofs of these later
papers which uni es the results, and second to show that the extra
conditions imposed on the integrands insure property (D) holds
with respect to the relevant sequence.
Sien Deng
Mathematical Sciences, Northern Illinois University
Exact Regularization and Weak Sharp Minima in Optimization
This presentation will focus on inner connections among exact
regularization, exact penalization, and weak sharp minima in
optimization. We will discuss some new insights about these
connections. Along the way, we will illustrate with examples, how
to obtain both new results and reproduce many existing results
from a fresh perspective.
Darinka Dentcheva
Mathematical Sciences, Stevens Institute of Technology
Common Mathematical Foundations of the Expected Utility and the
Dual Utility Theory
We show that the main results of the expected utility and dual
utility theories can be derived in a uni ed way based on two
fundamental mathematical ideas: the separation principle of convex
analysis, and the integral representations of continuous linear
functionals from functional analysis. Our analysis reveals the
dual character of utility functions. Additionally, we derive new
integral representations of dual utility models.
This is joint work with Andrzej Ruszczynski
Kazimierz Goebel
Mathematics, University of Maria Curie Sklodowska, Lublin, Poland
An Example Connected to Convex Functions Theory
This short presentation, dedicated to Terry Rockafellar, contains
an example of a convex function on nonreflexive Banach space with
some special properties. Examples of continuous mappings having
exotic behavior are constructed with the use of this function.
Yurii Ledyaev
Mathematics, Western Michigan University
Optimal control under a fi nite set of evolution scenarios
We consider a problem of control under uncertainty. This
uncertainty is modeled by a nite set of scenarios of dynamic
evolution of some system parameter. We develop mathematical
techniques for derivation of optimality condition characterizing
optimal control quasi-strategy. We demonstrate how such optimality
conditions can be used for designing model
predicting feedback control.
Michael Malisoff
Mathematics, Louisiana State University
Tracking Control for Neuromuscular Electrical Stimulation
We present a new tracking controller for neuromuscular electrical
stimulation, which is an emerging technology that can artificially
stimulate skeletal muscles to help restore functionality to human
limbs. We use a musculoskeletal model for a human using a leg
extension machine. The novelty of our work is that we prove that
the tracking error globally asymptotically and locally
exponentially converges to zero for any positive input delay and
for a general class of possible reference trajectories that must
be tracked, coupled with our ability to satisfy a state
constraint. Our controller only requires sampled measurements of
the states instead of continuous measurements
and allows perturbed sampling schedules. Our work is based on a
new method for constructing predictor maps for a large class of
nonlinear time-varying systems. Prediction is a key method for
delay compensation that uses dynamic control to compensate for
arbitrarily long input delays.
This work is joint with Marcio de Queiroz, Iasson Karafyllis, and
Ruzhou Yang.
Aleksander Olshevsky
Industrial and Enterprise Systems Engineering, University of
Illinois Urbana-Champaign
Optimization over Directed Graphs
We consider the problem of optimizing a sum of convex functions in
a network where each node knows only one of the functions.
Further, we assume that the nodes can only communicate with
neighbors in some time-varying sequence of directed graphs. We
develop a version of the stochastic gradient method which is fully
decentralized and, up to logarithmic factors, achieves the optimal
error decay with number of iterations.
Terry Rockafellar
Mathematics, University of Washington and Industrial and Systems
Engineering, University of Florida
Applying Variational Analysis to the Stability of Economic
Equilibrium
A fundamental topic in economics is the existence of prices which
induce the agents who buy and sell goods to act in such a way that
supply will equal demand. Along with existence, however, there are
important questions of uniqueness, at least in some localized
sense, and whether the local equilibrium then responds stably to
perturbations of the resources. Economists got some very
discuraging news about whether those questions could ever have
reasonable answers, but this turns out to be due in part to an
inadequate view of what local should mean. Now, with the help of
powerful tools in variational analysis, brought to bear on
variational inequality models of equilibrium, surprisingly
positive results have been achieved.
Andrzej Ruszczynski
Department of Management Science and Information Systems, Rutgers
Risk Preferences on the Space of Quantile Functions
We propose a novel approach to quanti cation of risk preferences
on the space of nondecreasing functions. When applied to law
invariant risk preferences among random variables, it compares
their quantile functions. The axioms on quantile functions impose
relations among comonotonic random variables. We infer the
existence of a numerical representation of the preference relation
in the form of a quantile-based measure of risk. Using conjugate
duality theory by pairing the Banach space of bounded functions
with the space of fi nitely additive measures on a suitable
algebra, we develop a variational representation of the
quantile-based measures of risk. Furthermore, we introduce a
notion of risk aversion based on quantile functions, which enables
us to derive an analogue of Kusuoka representation of coherent
law-invariant measures of risk.
This is joint work with Darinka Dentcheva
Ebrahim Sarabi
Mathematics, Wayne State University
Second-order Analysis of Piecewise Linear Functions with
Applications to Stability
We present relationships between nondegeneracy and second-order
quali fication for fully amenable compositions involving piecewise
linear functions. Moreover, we consider new applications of the
developed second-order subdiff erentials for convex piecewise
linear functions to full stability in composite optimization and
constrained minimax problems.
This is joint work with Boris Mordukhovich
Nghia Tran
Mathematics and Statistics, Oakland University
On the Convergence of the Proximal Forward-Backward Splitting
Method with Line Searches
In this talk we focus on the convergence analysis of the proximal
forward-backward splitting method for solving nonsmooth
optimization problems in Hilbert spaces when the objective
function is the sum of two convex functions. Assuming that one of
the functions is Frechet di erentiable and using two new
linesearches, the weak convergence is established without any
Lipschitz continuity assumption on the gradient. Furthermore, we
obtain many complexity results of cost values at the iterates when
the stepsizes are bounded below by a positive constant.
This is joint work with J. Yunier Bello Cruz.
Bao Truong
Mathematics and Computer Science, Northern Michigan University
A Blended Proof of the Vectorial Ekeland Variational Principle
This talk discusses a blended proof of a vectorial Ekeland
variational principle. It bases on the nonlinear scalarization
functional in Tammer (Gerth) and Weidner's nonconvex separation
theorem widely used in the scalarization approach (Gerth (Tammer),
C., Weidner, P.: Nonconvex separation theorems and some
applications in vector optimization. J. Optim. Theory Appl. 67
(1990) 297{320)) and on an iterative scheme in (Bao T.Q.,
Mordukhovich B.S.: Relative Pareto minimizers for multiobjective
problems: existence and optimality conditions. Math. Progr. 122
(2010) 301{347) developed in the variational approach. It is
important to emphasize that this new proof works well even in the
case where the ordering cone of the partial ordering image space
has an empty interior. Illustrative examples are provided.
Bingwu Wang
Mathematics, Eastern Michigan University
On the Weak Diff erentiability in Variational Analysis
The talk involves the weak di fferentiability of functions between
Banach spaces, which corresponds to the diff erentiability of the
scalarizations of the functions. We will review basic results of
this notion and explore its applications to generalized
di fferentiation in variational analysis. In this way we
demonstrate that this notion and its variant, in contrast to the
usual diff erentiability, are more suitable for Frechet and
limiting/Mordukhovich constructions.
Stephen Wright
Computer Sciences, University of Wisconsin-Madison
A Proximal Method for Composite Minimization
We consider minimization of functions that are compositions of
convex or prox-regular functions (possibly extended-valued) with
smooth vector functions. A wide variety of important optimization
problems fall into this framework. We describe an algorithmic
framework based on a subproblem constructed from a linearized
approximation to the objective and a regularization term.
Properties of local solutions of this subproblem underlie both a
global convergence result and an identifi cation property of the
active manifold containing the solution of the original problem.
Preliminary computational results on both convex and nonconvex
examples are promising.
This is joint work with Adrian Lewis.
Ming Yan
Computational Mathematics, Science and Engineering, Michigan State
University
ARock: an Algorithmic Framework for Asynchronous Parallel
Coordinate Updates
The problem of fi nding a fi xed point to a nonexpansive operator
is an abstraction of many models in numerical linear algebra,
optimization, and other areas of scienti c computing. To solve
this problem, we propose ARock, an asynchronous parallel
algorithmic framework, in which a set of agents (machines,
processors, or cores) update randomly selected coordinates of the
unknown variable in an asynchronous parallel fashion. The
resulting algorithms are not a ffected by load imbalance. When the
coordinate updates are atomic, the algorithms are free of memory
locks.
We show that if the nonexpansive operator has a fi xed point, then
with probability one, the sequence of points generated by ARock
converges to a fi xed point of the operator. Stronger convergence
properties such as linear convergence are obtained under stronger
conditions. As special cases of ARock, novel algorithms for linear
systems, convex optimization, machine learning, distributed and
decentralized optimization are introduced with provable
convergence. Very promising numerical performance of ARock has
been observed. Considering the paper length, we present the
numerical results of solving linear equations and sparse logistic
regression problems.
This is joint work with Zhimin Peng, Yangyang Xu, and Wotao Yin.
Chao Zhu
Mathematical Sciences, University of Wisconsin Milwaukee
Optimal Inventory Control with Path-Dependent Cost Criteria
This work deals with a stochastic control problem arising from
inventory control, in which the cost structure depends on the
current position as well as the running maximum of the state
process. A control mechanism is introduced to control the growth
of the running maximum which represents the required storage
capacity. The in nite horizon discounted cost minimization problem
is addressed and it is used to derive a complete solution to the
long-run average cost minimization problem. An associated control
cost minimization problem subject to a storage capacity constraint
is also addressed. Finally, as an application of the above
results, this paper also formulates and solves an
in finite-horizon discounted control problem with a
regime-switching inventory model.
This is a joint work with Ananda Weerasinghe.
Jim Zhu
Mathematics, Western Michigan University
Convex Duality and the Fundamental Theorem of Asset Pricing
The Fundamental Theorem of Asset Pricing (FTAP) relates
arbitrage-free prices of fi nancial assets to martingale measures
and plays an important role in mathematical finance.
Traditionally, this theorem is proven using a separation argument.
We observe that by considering an individual agent's portfolio
maximization problem and its convex dual, one can arrive at a more
precise version of the FTAP which in a (realistic) incomplete
market reveals how the agent's risk aversion relates to a
corresponding martingale measure.
This talk is adapted from lectures of a special topic course in
Courant Institute on convex duality and mathematical finance by
Peter Carr and Qiji Zhu.