MOM 2016Dean A. Carlson (Mathematical Reviews)
Recently, A. Greco utilized convex rearrangements to present some new and interesting existence results for noncoercive functionals in the calculus of variations. Moreover, the integrands were not necessarily convex. In particular, using convex rearrangements permitted him to establish the existence of convex minimizers essentially considering the uniform convergence of the minimizing sequence of trajectories and the pointwise convergence of their derivatives. The desired lower semicontinuity property is now a consequence of Fatou's lemma. In this paper we point out that such an approach was considered in the late 1930's in a series of papers by E. J. McShane for problems satisfying the usual coercivity condition. Our goal is to survey some of McShane‘s results and compare them with Greco's work. In addition, we will update some hypotheses that McShane made by making use of a result due to T. S. Angell on the avoidance of the Lavrentiev phenomenon. Yuri S. Ledyaev (Western Michigan University)
Matrix Riccati or Riccati-like equations appear in numerous applications in control engineering, in particular in stabilization. In this talk we demonstrate how to use methods of harmonic analysis, optimal control and differential games to derive analytic formulas for representation of exact solutions of algebraic and differential matrix Riccati equations. The important aspect of these formulas is their representation in terms of matrix transfer functions of linear dynamical control systems. We discuss application of this approach to problem of stabilization of some classes of linear control systems. Ekaterina Merkurjev (Michigan State University)
Graph-based variational methods have recently shown to be highly competitive for various classification problems of high-dimensional data, but are inherently difficult to handle from an optimization perspective. We propose a convex relaxation for a certain subset of graph-based multi-class data segmentation problems, featuring region homogeneity terms, supervised information and/or certain constraints or penalty terms acting on the class sizes. Particular applications include semi-supervised classification of high-dimensional data and unsupervised segmentation of unstructured 3D point clouds. Theoretical analysis indicates that the convex relaxation closely approximates the original NP-hard problems, and these observations are also confirmed experimentally. An efficient duality based algorithm is developed that handles all constraints on the labeling function implicitly. Experiments on semi-supervised classification indicate consistently higher accuracies than related local minimization approaches, and considerably so when the training data are not uniformly distributed among the data set. The accuracies are also highly competitive against a wide range of other established methods on three benchmark datasets. Experiments on 3D point clouds acquired by a LaDAR in outdoor scenes, demonstrate that the scenes can accurately be segmented into object classes such as vegetation, the ground plane and human-made structures. Boris Mordukhovich (Wayne State University)
We introduce the notions of critical and noncritical multipliers for subdifferential variational systems extending to a general framework the corresponding notions by Izmailov and Solodov developed for classical KKT systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal-dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain error bound for a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness. Dat Pham (Wayne State University)
The paper studies the notions of Lipschitzian and Holderian full stability of solutions to general parametric variational systems described via partial subdifferential and normal cone mappings acting in Hilbert spaces. We derive sufficient conditions of those full stability notions without assuming differentiability and computable formula for the modulus of prox-regularity of lower semicontinuous functions. The obtained results combining with related results are specified for important classes of variational inequalities and variational conditions in both finite and infinite dimensions. Hassan Saoud (Lebanese University and Fulbright Fellow at Michigan State University)
Stability analysis of dynamical systems constitutes a very important topic in mathematics and engineering. This is the case of mechanical systems subject to unilateral constraints and/or Coulomb friction and/or impacts or electrical circuits with switches, diodes and many other problems. So, it is not surprising that the unilateral dynamical system has played a central role in the understanding of mechanical processes. The mathematical formulation of the unilateral dynamical system involved inequality constraints and necessarily contains natural non-smoothness. The non-smoothness could originate from the discontinuous control term, or from the environment (non-smooth impact), or from the dry friction. In this talk, we are interested in the study of Evolution Variational Inequalities. We introduce some recent stability results. Indeed, Lyapunov’s theory will be used to study the stability, the LaSalle’s invariance principle in order to prove the asymptotical stability, the finite time stability and the semistability. Ebrahim Sarabi (Miami University)
In this talk we present the new developments about the the Newton method via secondorder generalized differentiation. We first provide the Newton method for Bingwu Wang (Eastern Michigan University)
This talk concerns the study of the directional versions of Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings, and subdifferentials of extended-real-valued functions in the framework of general Banach spaces. We will explore characterizations, basic properties of these constructions, and calculus such as sum rules and chain rules, as well as the upper estimates of the directional subdifferentials and singular subdifferentials of marginal functions. The talk is based on the joint work with Pujun Long and Xinmin Yang. Jerome Weston (Louisiana State University)
We provide globally asymptotically stabilizing backstepping controls for time-varying systems in a partially linear form, under constant feedback delays. Instead of measuring the full current states of the systems, our main feedback uses an output that consists of several delayed values of a suitable function of the states of the nonlinear subsystems of the original systems, and our main feedback design has no distributed terms. Other advantages are our controller bounds, the fact that we do not require differentiability of the available nominal controls for the nonlinear subsystems, and the fact that our controls do not contain Lie derivatives. This improves on a recent work in Automatica on special cases where the linear subsystem has one integrator, since we now allow an arbitrary number of integrators. Our main assumption is a delayed converging-input-converging-state condition. We can provide sufficient conditions for our converging-input-converging-state assumption to hold, in terms of the existence of certain Lyapunov functions. Although our converging-input-converging-state assumption does not include upper bounds on the delays and so can be stated for arbitrarily long feedback delays, our Lyapunov-based sufficient conditions put restrictions on the delay lengths. Therefore, we formulate an optimization problem, whose goal is to find the largest possible constant delays under which our converging-input-converging-state assumption can be satisfied. This work is joint with Prof. Michael Malisoff and Dr. Frederic Mazenc. Qiji Jim Zhu (Western Michigan University)
As an extension of the principle of maximum entropy statistical mechanics, entropy maximization method has wide range of applications in diverse fields. The structure of the solution to an entropy maximization problem often offers crucial insight into the solutions of a particular application. In this talk we summarize the role of such structures in several financial problems. They are the two fund theorem for Markowitz efficient portfolios, the existence and uniqueness of a market portfolio in capital asset pricing model, the fundamental theorem of asset pricing and the selection of a martingale measure for derivative pricing in an incomplete market. The connection of diverse important results in finance with the method of entropy maximization is a strong indication of the influence of physical science in financial research. However, the behavior of financial markets and financial systems are the results of human psychology and behaviors. The heavy influence of physical sciences should perhaps be viewed as a reason for caution in applying these financial theories. |
functions and then compare it with the well-known semismooth Newton method. Finally, we show that how these results can be extended for prox-regular functions.