Fitzpatrick functions and continuous linear monotone operators
The notion of a maximal monotone operator is crucial in
optimization as it captures both the subdifferential operator
of a convex, lower semicontinuous, and proper function and any
(not necessarily symmetric) continuous linear positive operator.
It was recently discovered that most fundamental results on
maximal monotone operators allow simpler proofs utilizing
Fitzpatrick functions.
In this talk, I will report on recent joint work
with Jon Borwein (Dalhousie) and Xianfu Wang (UBC Okanagan).
We study Fitzpatrick functions of continuous linear
monotone operators defined on a Hilbert space. A novel
characterization of skew operators is presented. We investigate
the Fitzpatrick function of the sum of two operators, and we show
that a known upper bound is actually exact in finite-dimensional and
more general settings. Cyclic monotonicity properties are also
analyzed, and closed forms of the Fitzpatrick functions of all orders
are provided for all rotators in the Euclidean plane.
Projecting on the Cone of Vectors with Autocorrelated Components
In signal processing and communications, some optimization problems
naturally involve the cone of vectors with autocorrelated components
(also called finite autocorrelation sequences ). A vector
x=(x_0,x_1,
...,x_n) in Rn+1 is said to have autocorrelated components if
there exists a vector y=(y_0,y_1,...,y_n) in Rn+1 such that
x_k=sumi=0n-k yi yi+k : for all k=0,1,...,n.
Surprisingly enough, this cone turns out to be convex (even if there is
no direct simple proof of this property). In the present work, we
pursue the works by L. Vandenberghe et al. in studying more
thoroughly this
cone and proposing an algorithm to project a vector on it; One first
effective approach to compute the projection uses interior point methods involving
the polar cone. We propose here a non-convex relaxation to overcome the
problem of dimensionality inherent to interior point methods and enable us
to treat problems of larger size. We will also breafly evocate the two-variables case, which can have some
applications in image processing
Mathematics involved in such an approach are : convex analysis
and optimization, Toeplitz matrices, Riesz-Fejer representation
theorem and trigonometric polynomials.
Also see
abstract (PDF)
A rapidly convergent VU-algorithm for general convex minimization
This talk describes a globally and superlinearly convergent algorithm for
minimizing a convex function. The method is useful for minimizing outer
functions resulting from decomposition, separation and/or penalty techniques
applied to large structured problems. It can be viewed as a generalization
of an SQP method (but with very different quadratic programming subproblems)
for a class of functions which includes maximum eigenvalue functions and
other "infinitely-defined" functions resulting from inner maximization or
integration.
The algorithm exploits natural (VU-decomposition) structure of a function
without having to know it explicitly and without adding extraneous structure
from barrier or smoothing functions. When a function has a "primal-dual track"
leading to a minimizing point and zero subgradient pair the algorithm follows
the track using a combination of cutting-plane and (quasi-) Newton techniques.
These two techniques are connected via proximal point and bundle algorithm
theories, since proximal points are on the primal track and they can be
approximated arbitrarily well by a bundle algorithm subprocedure. This
subroutine collects subgradients for constructing a V(or cutting-plane)-model
of the function and a corresponding U-gradient expression needed for the
method's U-Newton-predictor-steps that alternate with its V-corrector-steps.
Superlinear convergence is illustrated with some numerical test examples.
This is joint work with Claudia Sagastizabal
Primal-Dual reconstruction of a convex function
Given a finite number of subgradients, we present a method to build a convex function while preserving the symmetry of the primal-dual relationship. The method first solves a quadratic program to obtain a convex interpolant, which is smoothed and combined with the conjugate of the dual using the proximal average.
On the complexity of stochastic programming
This talk presents a complexity result for a class of stochastic programming problems where uncertainty is described by continuous probability distributions.
Lipschitz functions with maximal Clarke Subdifferentials are staunch
In a recent paper Borwein and I have shown that most non-expansive Lipschitz functions (in the sense of Baire's category) have a maximal Clarke subdifferential. In this talk, we will show that in a separable Banach space the set of non-expansive Lipschitz functions with a maximal Clarke subdifferential is not only of generic, but also staunch.
Term and volatility structures
All valuations (discounted cash flow, instrument pricing, option pricing) and other financial calculations require an estimate of the evolution of the risk-free rates as implied by the term and volatility structures. This presumes that one has, if not perfect knowledge, at least very good estimates of these market term structures. In this lecture I shall (partially) review and compare the existing methodologies for deriving zero-curves (spot rates, forward rates and discount factors) and volatility estimates.
Stochastic Semidefinite Programming with Recourse
We introduce an apparently new class of optimization problems which may be termed stochastic semidefinite programs with recourse (SSDP's). (Deterministic) semidefinite programs (SDP's) have been the focus of intense research during the past 15 years, and on one hand SSDP's extend SDP's by allowing uncertainty in data defining SDP's. On the other hand, SSDP's extend stochastic linear programs with recourse by allowing positive semidefinite matrix variables in place of nonnegative vector variables. We indicate applications of SSDP's and a volumetric center decomposition algorithm for their solution.