Speakers 

Title: Welcome remarks

Boris Mordukhovich, Wayne State University, Detroit, Michigan

Title: Robust Stability and Optimality Conditions for Parametric Infinite and Semi-infinite Programs

Abstract. This talk concerns parametric problems of infinite and semi-infinite programming, where functional constraints are given by systems of infinitely many linear inequalities indexed by an arbitrary set, where decision variables run over Banach (infinite programming) or finite-dimensional (semi-infinite case) spaces, and where objectives are generally described by nonsmooth and nonconvex cost functions. The parameter space of admissible perturbations in such problems is formed by all bounded functions over the index set equipped with the standard supremum norm. Unless the index set is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable). By using advanced tools of variational analysis and generalized differentiation and largely exploiting underlying specific features of linear infinite constraints, we establish complete characterizations of robust Lipschitzian stability (with computing the exact bound of Lipschitzian moduli) for parametric maps of feasible solutions governed by linear infinite inequality systems and then derive verifiable necessary optimality conditions for the infinite and semi-infinite programs under consideration expressed in terms of their initial data. A crucial part of our analysis addresses the precise computation of coderivatives and their norms for infinite systems of parametric linear inequalities in general Banach spaces of decision variables. The results obtained are new in both frameworks of infinite and semi-infinite programming. Based on the joint work with M. J. Canovas, M. A. Lopez and J. Parra.

Binwu Wang, Eastern Michigan University, Michigan

Title: Some Remarks on the weak-star extensibility

Abstract. The weak-star extensibility of subspaces of Banach spaces was introduced by Mordukhovich and the speaker when studying the restrictive metric regularity. The talk will give a new point of view of the property in the frame work of Banach space geometry, and also propose the concept of "weark-star extensible Banach space." The talk will explore ome new results and some revised proofs of known results, as well as relations with some well-known notions such as injectivity of Banach spaces.

Mau Nam Nguyen,  University of Texas - Pan American

Title: Coderivatives of normal cone mappings and applications to robust stability

Dean Carlson, Mathematical Review, Ann Arbor, Michigan

Title: An Equivalent Problem Approach to Absolute Extrema for Calculus of
Variations Problems with Differential Constraints

Abstract. In this presentation we consider problems in the calculus of variations of the form
\begin{equation}\label{OBJ}
minimize J(x(\cdot))=\int_a^bf(t,x(t),\dot{x}(t))\, dt
\end{equation}
over the class of all piecewise smooth functions (i.e., continuous functions with piecewise continuous first derivatives) satisfying the constraint
\begin{equation}\label{EQ}
g(t,x(t),\dot{x}(t))=0,\quad a<t<b,
 \end{equation}
and the end conditions
\begin{equation}\label{EC}
x(a)=x_a\quad\mbox{and}\quad x(b)=x_b,
\end{equation}
where $(f,g)(\cdot,\cdot,\cdot):[a,b]\times \R^n\times \R^n\to \R\times \R^m$ are continuously differentiable functions with $g(t,x,z)\ge 0$ (componentwise) for all $(t,x,z)\in [a,b]\times \R^n \times \R^n$ and $m<n$.  Our approach to solving such problems is to combine a penalization method with Leitmann's direct sufficiency method.  More specifically we consider the family of unconstrained problems $(P_\lambda)$ of minimizing
\begin{equation}\label{POBJ}
J_\lambda(x(\cdot))\doteq \int_a^b f(t,x(t),\dot{x}(t))+ \lambda^{\ssf T} g(t,x(t),\dot{x}(t))\,dt
\end{equation}
over all piecewise smooth functions $x(\cdot)$ satisfying (\ref{EC}).  The parameter $\lambda\in \R^m$ is assumed to have positive components.  Our goal is to apply Leitmann's direct method to the penalized problem with sufficiently large $\lambda$.  An example will be presented to illustrate our technique.

This is a joint work with G. Leitmann

Morteza Seddighin, Indiana University East, Richmond, IN 

Title: Some techniques of matrix optimization

Abstract. A matrix optimization problem is stated in general as finding the minimizing and maximizing vectors for a function of the form $F((Af,f),||Af|| )$, $|| f|| =1$ . Matrix optimization problems have applications in Numerical Analysis, Statistics, and Quantum Mechanics. In this presentation we will focus on the following two particular techniques:

1) Converting matrix optimization problems into a convex programming problem

2) Use a dimension reduction property to reduce the problem into a twovariable optimization problem.

Both techniques are introduced and used extensively by the Author and Karl
Gustafson. 

Jeffrey Pang,  University of Waterloo, Waterloo, ON, Canada

Title: Generalized differentiation with positively homogeneous maps

Abstract. We propose a new concept of generalized differentiation of nonsmooth  single-valued and set-valued maps that captures the first order  information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and Lipschitz  continuity in single-valued maps, and the Aubin property and Lipschitz 
continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin property and coderivatives, and study how metric regularity and open covering can be refined to have a  directional property similar to our concept of generalized differentiation.

Yuri Ledyaev, Western Michigan University, Kalamazoo, Michigan

Urszula Ledzewicz, Southern Illinois University, Illinois

Urszula Ledzewicz,    [Speaker] Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, USA

Heinz Schättler, Department of Electrical and Systems Engineering,Washington University, St. Louis, Missouri, USA

Helmut Maurer, Institut für Numerische und Angewandte Mathematik,
Westfälische Wilhelms Universität Münster, Germany

Abstract. Combination therapy in cancer is a new approach pursued in clinical trials that combines different treatments, especially novel and traditional ones, in the hope of achieving synergistic effects. In this talk various versions of a mathematical model for combination of anti-angiogenic inhibitors (vessel disruptive agents) with chemotherapy (cytotoxic agents) are formulated. The dynamics in the model describes the growth of the tumor volume and its vascularization under the effects of two controls representing the dosages of both agents with constraints on the total amount of each of the drugs given imposed. The proposed dynamics is based on a well-know model for anti-angiogenesis introduced by a group of researchers then at Harvard School of Medicine (Hahnfeldt et al., 1999) and its modifications.

These models are analyzed as optimal control problems with the objective to minimize the tumor volume. The presence of two agents makes this a multi-control problem which even with simplified dynamical equations becomes mathematically challenging. Using geometric methods in optimal control (Lie-derivatives, high-order conditions for optimality), an analysis of controls and trajectories resulting from the maximum principle is carried out that sufficiently restricts the class of candidates. Then, using optimization methods based on these theoretical results, effective numerical computations allow the determination of the optimal controls for the problem. It turns out that the optimal control representing the anti-angiogenic agent follows the theoretical solution obtained earlier for the monotherapy treatment where optimal controls are concatenations of bang-bang controls (representing therapies of full dose with rest periods) and singular controls (therapies with specific, time-varying partial doses). The cytotoxic agents on the other hand follow a strict bang-bang regimen switching from the value 0 in the initial period (no drug given) to the maximal value in the second part (drug given at the maximum dose). Medically speaking, this means that it is optimal to start the two-drug treatment with first applying anti-angiogenic agents and at one point start giving chemotherapy and apply it at the maximum dose in one session. These numerical results seem to find confirmation in recent clinical trials.

Sien Deng, Northern Illinois University, Chicago, Illinois

Title: Optimization Methods for Finite Element Model Updating in
Structural Dynamics

Jim Zhu, Western Michigan University, Kalamazoo, Michigan

Title: Term Structure of Interest Rates with Consumption Commitments

Abstract. We study the term structure of interest rates in the presence of consumption commitments using an equilibrium model. Under reasonable assumptions we prove the existence and uniqueness of the equilibrium and develop computation methods. Examples are analyzed to illustrate the effect of consumption commitments on the term structure and its manifestations. This is a joint working with J.C. Duan, Risk Management Institute, National University of Singapore

Nguyen Dihn Hoang, Wayne State University, Detroit, Michigan

Title:Necessary and sufficient conditions of one-sided Lipschitz mappings and applications

Abstract. This talk discusses the necessary and sufficient conditions of one-sided Lipschitz mappings defined on Hilbert spaces. The second part of the talk, we will discuss about the necessary optimality conditions of Bolza problem for one-sided Lipschitzian differential inclusions.

* This talk is based on the joint work with Prof. Boris Mordukhovich

T.T.A. Nghia, Wayne State University, Detroit, Michigan

Title: Robust stability of parametric infinite and sem-infinite systems

Abstract. This paper concerns the study of the parametric infinite and semi-infinite systems that  contain equalities and infinitely many inequalities indexed by an arbitrary set. The parameter space of admissible pertubations in such problems is infinite dimensional Banach space $Y\times l_\infty(T)$ endowed with the supremum norm, where $Y$ is a Banach space and $T$ is the infinite index set. In this framework, we provide some necessary and sufficient conditions for the metric regularity of the inverse of the parametric feasible set mapping. These conditions consist of the Slater constraint qualification for  linear systems and the extended Mangasarian-Fromovitz constraint qualification for nonlinear systems. Under these conditions, we establish the precise coumputations of coderivative norms, exact metric regularity bound and the radius of metrical regularity of the inverse of the parametric feasible set mapping based on advanced tools of variational analysis and generalized differentiation. The results obtained in the paper are new not only for the classes of inÞnite systems under consideration but also for their semi-inÞnite counterparts. 

This talk is based on the joint work with Prof. Mordukhovich

Nhi Nguyen,  Wayne State University, Detroit, Michigan

Title: Coderivatives in Parametric Optimization in Asplund Spaces

Abstract. In this talk, we first develop some calculus rules for second-order partial subdifferentials of extended real-valued functions in the framework of Asplund spaces. We then apply these rules in the study of a family of parameterized optimization problems in which both cost function and constraint function are nonsmooth extended real-valued, and conduct local sensitivity analysis for the stationary point and stationary point-multiplier multifunctions."

This is a joint work with Prof. Boris Mordukhovich and Prof. N.M. Nam.

Hung Phan,  Wayne State University, Detroit, Michigan

Title: Extremal principle for sub-extremal systems

Abstract. Extremal principle, first introduced by Prof.Mordukhovich, has played an important role in variational analysis. By many authors, it is considered a counterpart of the classical separation theorem for convex sets in functional analysis. For that reason, extremal principle has been widely used in many applications. In this work, we study the extremal principle for sub-extremal systems, an extension of the extremal systems. Later on, we will apply it to some problems of optimizations.

This talk is based on the joint work with Prof. Boris Mordukhovich

Mehari Gebregziabher,  Wayne State University, Detroit, Michigan

Title: Subdifferential to the suprimum and infinimum of infinite functions