Open Access. Powered by Scholars. Published by Universities.®
Articles 31 - 42 of 42
Full-Text Articles in Applied Mathematics
Variational Analysis In Nonsmooth Optimization And Discrete Optimal Control, Boris S. Mordukhovich
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 …
Variational Analysis Of Evolution Inclusions, Boris S. Mordukhovich
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 …
Decentralized Convex-Type Equilibrium In Nonconvex Models Of Welfare Economics Via Nonlinear Prices, Boris S. Mordukhovich
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 …
Optimal Control Of Semilinear Evolution Inclusions Via Discrete Approximations, Boris S. Mordukhovich, Dong Wang
Optimal Control Of Semilinear Evolution Inclusions Via Discrete Approximations, Boris S. Mordukhovich, Dong Wang
Mathematics Research Reports
This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions …
Variational Stability And Marginal Functions Via Generalized Differentiation, Boris S. Mordukhovich, Nguyen Mau Nam
Variational Stability And Marginal Functions Via Generalized Differentiation, Boris S. Mordukhovich, Nguyen Mau Nam
Mathematics Research Reports
Robust Lipschitzian properties of set-valued mappings and marginal functions play a crucial role in many aspects of variational analysis and its applications, especially for issues related to variational stability and optimizatiou. We develop an approach to variational stability based on generalized differentiation. The principal achievements of this paper include new results on coderivative calculus for set-valued mappings and singular subdifferentials of marginal functions in infinite dimensions with their extended applications to Lipschitzian stability. In this way we derive efficient conditions ensuring the preservation of Lipschitzian and related properties for set-valued mappings under various operations, with the exact bound/modulus estimates, as …
Optimal Control Of Delay Systems With Differential And Algebraic Dynamic Constraints, Boris S. Mordukhovich, Lianwen Wang
Optimal Control Of Delay Systems With Differential And Algebraic Dynamic Constraints, Boris S. Mordukhovich, Lianwen Wang
Mathematics Research Reports
This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between "slow" and "fast" variables. We pursue a two-hold goal: to study variational stability for this class of control systems with respect to discrete …
Necessary Conditions In Nonsmooth Minimization Via Lower And Upper Subgradients, Boris S. Mordukhovich
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
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 …
Optimization And Equilibrium Problems With Equilibrium Constraints, Boris S. Mordukhovich
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 …
Subdifferential And Superdifferential Optimality Conditions In Nonsmooth Minimization, Boris S. Mordukhovich
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
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.
Optimal Control Of Neutral Functional-Differential Inclusions, Boris S. Mordukhovich, Lianwen Wang
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 …