Physical Sciences and Mathematics Commons^™

Equilibrium Problems With Equilibrium Constraints Via Multiobjective Optimization, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns a new class of optimization-related problems called Equilibrium Problems with Equilibrium Constraints (EPECs). One may treat them as two level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of Mathematical Programs with Equilibrium Constraints (MPECs). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This paper presents necessary …

Necessary Conditions In Nonsmooth Minimization Via Lower And Upper Subgradients, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Frechetjregular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with …

Optimal Control Of Delayed Differential-Algebraic Inclusions, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper concerns constrained dynamic optimization problems governed by delayed differential-algebraic systems. Dynamic constraints in such systems, which are particularly important for engineering applications, are described by interconnected delay-differential inclusions and algebraic equations. We pursue a two-hold goal: to study variational stability of such control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. We are not familiar with any results in these directions for differential-algebraic inclusions even in the delay-free case. In the first part of the paper …

The Approximate Maxium Principle In Constrained Optimal Control, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

The paper concerns optimal control problems for dynamic systems governed by a parametric family of discrete approximations of control systems with continuous time. Discrete approximations play an important role in both qualitative and numerical aspects of optimal control and occupy an intermediate position between discrete-time and continuous-time control systems. The central result in optimal control of discrete approximations is the Approximate Maximum Principle (AMP), which is justified for smooth control problems with endpoint constraints under certain assumptions without imposing any convexity, in contrast to discrete systems with a fixed step. We show that these assumptions are essential for the validity …

Optimization And Feedback Control Of Constrained Parabolic Systems Under Uncertain Perturbations, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

This paper concerns a minimax control design problem for a class of parabolic systems with nonregular boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We deal with boundary controllers acting through Dirichlet boundary conditions that are the most challenging for the parabolic dynamics.

Neumann Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

We consider optimal control problems for hyperbolic systems with controls in Neumann boundary conditions with pointwise (hard) constraints on control and state functions. Focusing on hyperbolic dynamics governed by the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the pointwise form of the Pontryagin Maximum Principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allows us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional.

Dirichlet Boundary Control Of Hyperbolic Equations In The Presence Of State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.

Optimization And Equilibrium Problems With Equilibrium Constraints, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns optimization and equilibrium problems with the so-called equilibrium constraints (MPEC and EPEC), which frequently appear in applications to operations research. These classes of problems can be naturally unified in the framework of multiobjective optimization with constraints governed by parametric variational systems (generalized equations, variational inequalities, complementarity problems, etc.). We focus on necessary conditions for optimal solutions to MPECs and EPECs under general assumptions in finite-dimensional spaces. Since such problems are intrinsically nonsmooth, we use advanced tools of generalized differentiation to study optimal solutions by methods of modern variational analysis. The general results obtained are concretized for special …

Optimal Control Of Differential-Algebraic Inclusions, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

No abstract provided.

Natural Superconvergent Points Of Triangular Finite Elements, Zhimin Zhang, Runchang Lin

Mathematics Research Reports

In this work, we analytically identify natural superconvergent points of function values and gradients for triangular elements. Both the Poisson equation and the Laplace equation are discussed for polynomial finite element spaces (with degrees up to 8) under four different mesh patterns. Our results verify computer findings of [2], especially, we confirm that the computed data have 9 digits of accuracy with an exception of one pair (which has 8-7 digits of accuracy). In addition, we demonstrate that the function value superconvergent points predicted by the symmetry theory [14] are the only superconvergent points for the Poisson equation. Finally, we …

Validation Of The A Posteriori Error Estimator Based On Polynomial Preserving Recovery For Linear Elements, Zhimin Zhang, Ahmed Naga

Mathematics Research Reports

In this paper the quality of the error estimator based on the Polynomial Preserving Recovery (PPR) is investigated using the computer-based approach proposed by Babiiska et al. A comparison is made between the error estimator based on the PPR and the one based on the Superconvergence Patch Recovery (SPR). It was found that the PPR is at least as good as the SPR.

Subdifferential And Superdifferential Optimality Conditions In Nonsmooth Minimization, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called subdifferential and superdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Frechet superdifferentials provided that they are not empty. All the superdifferential and major subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with equilibrium constraints.

Pareto Optimal Allocations In Nonconvex Models Of Welfare Economics, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to applications of modern variational analysis to the study of Pareto (as well as weak and strong Pareto) optimal allocations in nonconvex models of welfare economics with infinite-dimensional commodity spaces. Our basic tool is the extremal principle of variational analysis that provides necessary conditions for set extremality and may be viewed as a variational extension of the classical convex separation principle to the case of nonconvex sets. In this way we obtain new versions of the generalized second welfare theorem for nonconvex economies in terms of appropriate concepts of normal cones.

The Cohomology Of The Steendrod Algebra And Representations Of The General Linear Groups, Nguyen H. V. Hu'ng

Mathematics Research Reports

Let Tr_k be the algebraic transfer that maps from the coinvariants of certain GL_k-representation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr_k : pi_*^S((B[doublestrike V]_k)_+) --> pi_*^S(S^0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Tr_k is an isomorphism for k = 1, 2, 3 and that T_r = ⊕_k(Tr_k) is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree d, and apply Sq^0 repeatedly at most (k- 2) …

Impulse Control Of Stochastic Navier-Stokes Equations, J. L. Menaldi, S. S. Sritharan

Mathematics Faculty Research Publications

In this paper we study stopping time and impulse control problems for stochastic Navier-Stokes equation. Exploiting a local monotonicity property of the nonlinearity, we establish existence and uniqueness of strong solutions in two dimensions which gives a Markov-Feller process. The variational inequality associated with the stopping time problem and the quasi-variational inequality associated with the impulse control problem are resolved in a weak sense, using semigroup approach with a convergence uniform over path.

Optimal Control Of Neutral Functional-Differential Inclusions, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper deals with optimal control problems for dynamical systems governed by constrained functional-differential inclusions of neutral type. Such control systems contain time-delays not only in state variables but also in velocity variables, which make them essentially more complicated than delay-differential (or differential-difference) inclusions. Our main goal is to derive necessary optimality conditions for general optimal control problems governed by neutral functional-differential inclusions with endpoint constraints. While some results are available for smooth control systems governed by neutral functional-differential equations, we are not familiar with any results for neutral functional-differential inclusions, even with smooth cost functionals in the absence of …