Open Access. Powered by Scholars. Published by Universities.®

Physical Sciences and Mathematics Commons™

Open Access. Powered by Scholars. Published by Universities.®

Wayne State University

Mathematics

2006

Articles 1 - 17 of 17

Full-Text Articles in Physical Sciences and Mathematics

Asymmetric Games For Convolution Systems With Applications To Feedback Control Of Constrained Parabolic Equations, Boris S. Mordukhovich Dec 2006

Asymmetric Games For Convolution Systems With Applications To Feedback Control Of Constrained Parabolic Equations, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to the study of some classes of feedback control problems for linear parabolic equations subject to hard/pointwise constraints on both Dirichlet boundary controls and state dynamic/output functions in the presence of uncertain perturbations within given regions. The underlying problem under consideration, originally motivated by automatic control of the groundwater regime in irrigation networks, is formalized as a minimax problemof optimal control, where the control strategy is sought as a feedback law. Problems of this type are among the most important in control theory and applications - while most challenging and difficult. Based on the Maximum Principle …

Go to article

Suboptimal Feedback Control Design Of Constrained Parabolic Systems In Uncertainty Conditions, Boris S. Mordukhovich Dec 2006

Suboptimal Feedback Control Design Of Constrained Parabolic Systems In Uncertainty Conditions, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns minimax control problems forlinear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in the Dirichlet boundary conditions. The underlying parabolic control system is functioning under hard/pointwise constraints on control and state variables. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We develop an efficient approach to the minimax control design of constrained parabolic systems that is based on certain characteristic features of the parabolic dynamics including …

Go to article

Generalized Differentiation Of Parameter-Dependent Sets And Mappings, Boris S. Mordukhovich, Bingwu Wang Dec 2006

Generalized Differentiation Of Parameter-Dependent Sets And Mappings, Boris S. Mordukhovich, Bingwu Wang

Mathematics Research Reports

The paper concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization, and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings, and nonsmooth functions), which naturally appear, e.g., in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this paper.

Go to article

Optimal Control Of Nonconvex Differential Inclusions, Boris S. Mordukhovich Dec 2006

Optimal Control Of Nonconvex Differential Inclusions, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization, and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings, and nonsmooth functions), which naturally appear, e.g., in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this paper.

Go to article

Epi-Convergent Discretization Of The Generalizaed Bolza Problem In Dynamic Optimization, Boris S. Mordukhovich, Teemu Pennanen Dec 2006

Epi-Convergent Discretization Of The Generalizaed Bolza Problem In Dynamic Optimization, Boris S. Mordukhovich, Teemu Pennanen

Mathematics Research Reports

The paper is devoted to well-posed discrete approximations of the so-called generalized Bolza problem of minimizing variational functionals defined via extended-real-valued functions. This problem covers more conventional Bolza-type problems in the calculus of variations and optimal control of differential inclusions as well of parameterized differential equations. Our main goal is find efficient conditions ensuring an appropriate epi-convergence of discrete approximations, which plays a significant role in both the qualitative theory and numerical algorithms of optimization and optimal control. The paper seems to be the first attempt to study epi-convergent discretizations of the generalized Bolza problem; it establishes several rather general …

Go to article

Multiobjective Optimization Problems With Equilibrium Constraints, Boris S. Mordukhovich Oct 2006

Multiobjective Optimization Problems With Equilibrium Constraints, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to new applications of advanced tools of modern variational analysis and generalized differentiation to the study of broad classes of multiobjective optimization problems subject to equilibrium constraints in both finite-dimensional and infinite-dimensional settings. Performance criteria in multiobjectivejvector optimization are defined by general preference relationships satisfying natural requirements, while equilibrium constraints are described by parameterized generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and are handled in this paper via appropriate normal/coderivativejsubdifferential constructions that exhibit full calculi. Most of the results obtained are new even in finite dimensions, while the case of …

Go to article

Necessary Conditions In Multiobjective Optimization With Equilibrium Constraints, Truong Q. Bao, Panjak Gupta, Boris S. Mordukhovich Sep 2006

Necessary Conditions In Multiobjective Optimization With Equilibrium Constraints, Truong Q. Bao, Panjak Gupta, Boris S. Mordukhovich

Mathematics Research Reports

In this paper we study multiobjective optimization problems with equilibrium constraints (MOECs) described by generalized equations in the form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models particularly arise from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish verifiable necessary conditions for the general problems under consideration and for their important specifications using modern tools of variational analysis and generalized differentiation. The application of the obtained necessary optimality conditions is illustrated by a numerical example from bilevel programming with convex …

Go to article

Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang Aug 2006

Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang

Mathematics Research Reports

We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two dimensional Poisson equation. Results of this paper are two folds. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered.gradient by the Polynomial Preserving Recovery (PPR) is superconvergent. Secondly, we demonstrate by numerical examples that an adaptive procedure with a posteriori error estimator based on PPR does produce adaptive meshes satisfy our mesh density assumption, and the recovered gradient by PPR is indeed supercoveregent in the adaptive process.

Go to article

Charactarizations Of Linear Suboptimality For Mathematical Programs With Equilibrium Constraints, Boris S. Mordukhovich Jul 2006

Charactarizations Of Linear Suboptimality For Mathematical Programs With Equilibrium Constraints, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we …

Go to article

Methods Of Variational Analysis In Multiobjective Optimization, Boris S. Mordukhovich Jun 2006

Methods Of Variational Analysis In Multiobjective Optimization, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns new applications of advanced methods of variational analysis and generalized differentiation to constrained problems of multiobjective/vector optimization. We pay the main attention to general notions of optimal solutions for multiobjective problems that are induced by geometric concepts. of extremality in variational analysis while covering various notions of Pareto and other type of optimality/efficiency conventional in multiobjective optimization. Based on the extremal principles in variational analysis and on appropriate tools of generalized differentiation with well-developed calculus rules, we derive necessary optimality conditions for broad classes of constrained multiobjective problems in the framework of infinite-dimensional spaces. Applications of variational …

Go to article

Pietro Paoli, Italian Algebraist, John L. Cuzzocrea, Shlomo S. Sawilowsky May 2006

Pietro Paoli, Italian Algebraist, John L. Cuzzocrea, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

Pietro Paoli was a leading Italian mathematician in the late 18th century. His signed letter pertaining to the death of astronomer Giuseppe Antonio Slop is translated from Italian to flowing (American) English.

Go to article

Joseph Liouville’S ‘Mathematical Works Of Évariste Galois’, Shlomo S. Sawilowsky May 2006

Joseph Liouville’S ‘Mathematical Works Of Évariste Galois’, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

Liouville’s 1846 introduction to the mathematical works of Galois is translated from French to flowing (American) English. It gave an overview of the tragic circumstances of the undergraduate mathematician whose originality led to major advances in abstract Algebra.

Go to article

Variational Analysis In Nonsmooth Optimization And Discrete Optimal Control, Boris S. Mordukhovich May 2006

Variational Analysis In Nonsmooth Optimization And Discrete Optimal Control, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to applications of modern methods of variational· analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main attention is paid to the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and. operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the "lack of compactness" in infinite dimensions, which …

Go to article

Variational Analysis Of Evolution Inclusions, Boris S. Mordukhovich Apr 2006

Variational Analysis Of Evolution Inclusions, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities. with generally nonsmooth functions. We develop a variational analysis of such roblems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of Asplund (and hence any reflexive Banach) spaces. In this way we establish extended results on stability of discrete approximations (with the strong W^1,2-convergence of optimal solutions under consistent perturbations of …

Go to article

Optimization And Equilibrium Problems With Equilibrium Constraints In Infinite-Dimensional Spaces, Boris S. Mordukhovich Mar 2006

Optimization And Equilibrium Problems With Equilibrium Constraints In Infinite-Dimensional Spaces, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to applications of modern variational f).nalysis to the study of constrained optimization and equilibrium problems in infinite-dimensional spaces. We pay a particular attention to the remarkable classes of optimization and equilibrium problems identified as MPECs (mathematical programs with equilibrium constraints) and EPECs (equilibrium problems with equilibrium constraints) treated from the viewpoint of multiobjective optimization. Their underlying feature is that the major constraints are governed by parametric generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and can be handled by using an appropriate machinery of generalized differentiation exhibiting a rich/full calculus. The …

Go to article

Decentralized Convex-Type Equilibrium In Nonconvex Models Of Welfare Economics Via Nonlinear Prices, Boris S. Mordukhovich Jan 2006

Decentralized Convex-Type Equilibrium In Nonconvex Models Of Welfare Economics Via Nonlinear Prices, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to applications of modern tools of variational analysis to equilibrium models of welfare economics involving nonconvex economies with infinite-dimensional commodity spaces. The main results relate to generalized/ extended second welfare theorems ensuring an equilibrium price support at Pareto optimal allocations. Based on advanced tools of generalized differentiation, we establish refined results of this type with the novel usage of nonlinear prices at the three types to optimal allocations: weak Pareto, Pareto, and strong Pareto. The usage of nonlinear (vs. standard linear) prices allow us to decentralized price equilibria in fully nonconvex models similarly to linear prices …

Go to article

Notes For Mat 7500 – Winter '93, Revised Winter '06, David Handel Jan 2006

Notes For Mat 7500 – Winter '93, Revised Winter '06, David Handel

Mathematics Faculty Research Publications

These notes developed from a one semester course at Wayne State University, taught several times in the last three decades of the 1900s. The subject matter is analysis on manifolds, consisting of the theory of smooth manifolds, differential forms, integration of forms, the generalized Stokes' Theorem, de Rham cohomology, and some related topics. The course is intended for first or second year graduate students in Mathematics with a background in Advanced Calculus, General Topology, linear algebra (including quotient spaces), and a little elementary group theory (including some familiarity with the symmetric groups). Given the above background, the notes are self-contained. …

Go to article