West Coast Optimization Meeting 24–25 April 1998 Abstracts

09:00–09:45 Jim Zhu An implicit function theorem

We establish an implicit function theorem for multifunctions by using the multidirectional mean value inequality. We then deduce from this result the open covering theorem and sufficient conditions for metric regularity and pseudo-Lipschitz property. (Joint work with Yu. S. Ledyaev.)

09:45–10:30 Paul Tseng On Global Tangency, Global Normality, and Metric Regularity Conditions for the Intersection of (two) Convex Sets

In a recent paper by Bauschke, Borwein and Li, it is discussed that various topics in convex optimization (including error bounds and constraint qualification for a system of convex inequalities, convergence rate analysis of certain projection methods, and constrained interpolation from a convex subset/cone) are closely related to one of the following three properties for two (or more) closed convex sets/cones C_{1} and C_{2} in Euclidean space and their intersection C:

P1. For points x in any bounded set, the distance from x to C is at most a constant multiple of the maximum of the distance from x to C_{1} and to C_{2}.

P2. The normal cone of C equals the sum of the normal cones

of C_{1} and C_{2} everywhere on C. P3. The tangent cone of C equals the intersection of the tan-

gent cones of C_{1} and C_{2} everywhere on C. In the same paper, it was proven that P1⇒P2⇒P3 and, while P3 was shown to imply P1 in a variety of circumstances, it was left open whether the converse implications are always true. In this talk, we show that the converse implications are not always true, alas. We also discuss additional conditions under which the converse implications would be true. (Joint work with Heinz Bauschke and Jon Borwein)

10:45–11:30 Jane Ye Necessary and Sufficient Optimality Conditions for Optimization Problems with Complementarity Constraints Optimization problems with complementarity constraints are a class of nonlinear programming problems where all or parts of the constraints are complementarity systems. These problems are in general nonconvex and do not satisfy the usual constraint qualifications. In this talk we give sharp necessary optimality conditions involving Mordukhovich and proximal coderivatives that may hold even when the usual KKT conditions do not hold. The necessary conditions involving the proximal coderivatives turn out to be sufficient when the problem is “convex”. The optimality conditions involving the proximal coderivatives are shown to be equivalent to a system of linear and nonlinear equations. Applications to bilevel programming problems will be also discussed.

11:30–12:30 K. O. Kortanek, V. G. Medvedev Models for Estimating the Structure of Interest Rates from Observations of Yield Curves

We present a dynamical systems approach for modelling the term structure of interest rates based on a linear differential equation under uncertainty. Impulse or point-impulse perturbations are introduced on either

• the spot (shortest-term, risk neutral) interest rate as the unknown function, or

• its integral, namely the yield function, or

• both simultaneously. Parameters are estimated by minimizing the maximum absolute value of the measurement errors, which is a nonlinear semi-infinite programming problem.

Beyond the learning period (the current time), the solved-for spot rate function becomes the forecast of the unobservable function in a future period, while its integral should approximate the yield function well.

Non-arbitrage is addressed by providing a sufficient condition under which non-arbitrage is guaranteed. The property of meanreversion is also preserved, and functional estimates are provided for the market price of risk.

Analogous concepts to ”drift” and ”volatility” are treated in a manner that provides a criterion for the choice of perturbation to employ in a given real situation.

We test the approach empirically with daily Treasury yield curve rates data, mainly for discount bonds having 3 to 6 month maturities over observation periods of up to one year. Computational results are reported for many numerical experiments together with

some financial interpretations: see https://www.biz.uiowa.edu/kwel/kwel/

14:15–14:55 Pepe Moreno On Norm Attaining Functionals

Consider a Banach space X and a bounded linear functional f. The norm of f is defined as the supremum of f over the unit ball of X. When this supremum is attained, we say that f attains its norm. We denote by NA the set of norm attaining functionals. The classical James’ Theorem asserts that X is reflexive if and only if every functionals attains its norm. On the other hand, the Bishop-Phelps Theorem tell us that NA is always dense in the dual space. It is our purpose to discuss some topological properties of NA implying reflexivity. More precisely, we are motivated by the following question: is a Banach space reflexive provided the set NA has nonempty interior? It can be easily seen that the answer is negative when the topology is the norm topology. This situation suggests, at least, two possibilities. First, to investigate which additional geometric conditions imply a positive answer. Second, to consider the same problem for the weak or the weak* topologies.

14:55–15:15 Pierre Marechal Maximization of a Measure of Surprise for the Surprise Examination Problem

In a recent paper on the ‘surprise examination’ or ‘unexpected hanging’ paradox, a new entropic formulation of this problem was proposed. We will discuss some aspects of the optimization problems arising from this formulation. This will provide a pleasing application of convex programming.

15:15–15:55 Jonathan Borwein Partially Smooth Variational Analysis and Applications

I will describe the basic ideas from Banach space theory that allow one to fruitfully apply notions from smooth analysis in spaces which do not admit smooth renorms (or “bumps”) and so to perform partially smooth variational analysis.

I also intend to survey some of the outstanding open functional analytic questions relating to best approximation, viscosity and variational analysis.