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Full-Text Articles in Physical Sciences and Mathematics
Optimal Control Of A Perturbed Sweeping Process With Applications To The Crowd Motion Model, Tan Hoang Cao
Optimal Control Of A Perturbed Sweeping Process With Applications To The Crowd Motion Model, Tan Hoang Cao
Wayne State University Dissertations
The dissertation is devoted to the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded dif- ferential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive neces- sary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to …
Variational Analysis Of Marginal Functions With Applications To Bilevel Programming, Boris S. Mordukhovich, Nguyen Mau Nam, Hung M. Phan
Variational Analysis Of Marginal Functions With Applications To Bilevel Programming, Boris S. Mordukhovich, Nguyen Mau Nam, Hung M. Phan
Mathematics Research Reports
This paper pursues a twofold goal. First to derive new results on generalized differentiation in variational analysis focusing mainly on a broad class of intrinsically nondifferentiable marginal/value functions. Then the results established in this direction apply to deriving necessary optimality conditions for the optimistic version of bilevel programs that occupy a remarkable place in optimization theory and its various applications. We obtain new sets of optimality conditions in both smooth and smooth settings of finite-dimensional and infinite-dimensional spaces.
Applications Of Variational Analysis To A Generalized Heron Problem, Boris S. Mordukhovich, Nguyen Mau Nam, Juan Salinas Jr
Applications Of Variational Analysis To A Generalized Heron Problem, Boris S. Mordukhovich, Nguyen Mau Nam, Juan Salinas Jr
Mathematics Research Reports
This paper is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to n given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point …
Quantitative Stability Of Linear Infinite Inequality Systems Under Block Perturbations With Applications To Convex Systems, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Quantitative Stability Of Linear Infinite Inequality Systems Under Block Perturbations With Applications To Convex Systems, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Mathematics Research Reports
The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is loo(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn …
Tangential Extremal Principles For Finite And Infinite Systems Of Sets, Ii: Applications To Semi-Infinite And Multiobjective Optimization, Boris S. Mordukhovich, Hung M. Phan
Tangential Extremal Principles For Finite And Infinite Systems Of Sets, Ii: Applications To Semi-Infinite And Multiobjective Optimization, Boris S. Mordukhovich, Hung M. Phan
Mathematics Research Reports
This paper contains selected applications of the new tangential extremal principles and related results developed in [20] to calculus rules for infinite intersections of sets and optimality conditions for problems of semi-infinite programming and multiobjective optimization with countable constraints.
Variational Analysis And Optimal Control Of The Sweeping Process, Hoang Dinh Nguyen
Variational Analysis And Optimal Control Of The Sweeping Process, Hoang Dinh Nguyen
Wayne State University Dissertations
We formulate and study an optimal control problem for the sweeping(Moreau) process, where control functions enter the moving sweeping
set. To the best of our knowledge, this is the first study in the literature devoted to optimal control of the sweeping process. We first establish an existence theorem of optimal solutions and then derive necessary optimality conditions for this optimal control problem of a new type, where the dynamics is governed by discontinuous differential inclusions with variable right-hand sides. Our approach to necessary optimality conditions is based on the method of discrete approximations and advanced tools of variational analysis and …
Quantitative Stability And Optimality Conditions In Convex Semi-Infinite And Infinite Programming, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Quantitative Stability And Optimality Conditions In Convex Semi-Infinite And Infinite Programming, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Mathematics Research Reports
This paper concerns parameterized convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional Banach (resp. finite-dimensional) spaces and that are indexed by an arbitrary fixed set T. Parameter perturbations on the right-hand side of the inequalities are measurable and bounded, and thus the natural parameter space is loo(T). Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map, which involves only the system data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. On one hand, in this …
Lipchitzian Stability Of Parametric Variational Inequalities Over Generalized Polyhedra In Banach Spaces, Liqun Ban, Boris S. Mordukhovich, Wen Song
Lipchitzian Stability Of Parametric Variational Inequalities Over Generalized Polyhedra In Banach Spaces, Liqun Ban, Boris S. Mordukhovich, Wen Song
Mathematics Research Reports
This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solutions maps entirely via their initial data. This is done on the base of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. …
Extended Second Welfare Theorem For Nonconvex Economies With Infinite Commodities And Public Goods, Aychiluhim Habte, Boris S. Mordukhovich
Extended Second Welfare Theorem For Nonconvex Economies With Infinite Commodities And Public Goods, Aychiluhim Habte, Boris S. Mordukhovich
Mathematics Research Reports
This paper is devoted to the study of nonconvex models of welfare economics with public goods and infinite-dimensional commodity spaces. Our main attention is paid to new extensions of the fundamental second welfare theorem to the models under consideration. Based on advanced tools of variational analysis and generalized differentiation, we establish appropriate approximate and exact versions of the extended second welfare theorem for Pareto, weak Pareto, and strong Pareto optimal allocations in both marginal price and decentralized price forms.
Applying Metric Regularity To Compute Condition Measure Of Smoothing Algorithm For Matrix Games, Boris S. Mordukhovich, Javier Peña, Vera Roshchina
Applying Metric Regularity To Compute Condition Measure Of Smoothing Algorithm For Matrix Games, Boris S. Mordukhovich, Javier Peña, Vera Roshchina
Mathematics Research Reports
Abstract. We develop an approach of variational analysis and generalized differentiation to conditioning issues for two-person zero-sum matrix games. Our major results establish precise relationships between a certain condition measure of the smoothing first-order algorithm proposed in (4] and the exact bound of metric regularity for an associated set-valued mapping. In this way we compute the aforementioned condition measure in terms of the initial matrix game data.
First-Order And Second-Order Optimality Conditions For Nonsmooth Constrained Problems Via Convolution Smoothing, Andrew C. Eberhard, Boris S. Mordukhovich
First-Order And Second-Order Optimality Conditions For Nonsmooth Constrained Problems Via Convolution Smoothing, Andrew C. Eberhard, Boris S. Mordukhovich
Mathematics Research Reports
This paper mainly concerns deriving first-order and second-order necessary (and partly sufficient) optimality conditions for a general class of constrained optimization problems via smoothing regularization procedures based on infimal-like convolutions/envelopes. In this way we obtain first-order optimality conditions of both lower subdifferential and upper subdifferential types and then second-order conditions of three kinds involving, respectively, generalized second-order directional derivatives, graphical derivatives of first-order subdifferentials, and secondorder subdifferentials defined via coderivatives of first-order constructions.
Well-Posedness Of Minimal Time Problem With Constant Dynamics In Banach Spaces, Giovanni Colombo, Vladimir V. Goncharov, Boris S. Mordukhovich
Well-Posedness Of Minimal Time Problem With Constant Dynamics In Banach Spaces, Giovanni Colombo, Vladimir V. Goncharov, Boris S. Mordukhovich
Mathematics Research Reports
This paper concerns the study of a general minimal time problem with a convex constant dynamic and a closed target set in Banach spaces. We pay the main attention to deriving efficient conditions for the major well-posedness properties that include the existence and uniqueness of optimal solutions as well as certain regularity of the optimal value function with respect to state variables. Most of the results obtained are new even in finite-dimensional spaces. Our approach is based on advanced tools of variational analysis and generalized differentiation.
Variational Analysis In Semi-Infinite And Infinite Programming, Ii: Necessary Optimality Conditions, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Variational Analysis In Semi-Infinite And Infinite Programming, Ii: Necessary Optimality Conditions, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Mathematics Research Reports
This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi-infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development [5] from our viewpoint of robust Lipschitzian stability. We present meaningful interpretations and practical examples of such models. The main results establish necessary optimality conditions for a broad class of semi-infinite and infinite programs, where objectives are generally described by nonsmooth and nonconvex functions on Banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets. …
Variational Analysis In Semi-Infinite And Infinite Programming, I: Stability Of Linear Inequality Systems Of Feasible Solutions, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Variational Analysis In Semi-Infinite And Infinite Programming, I: Stability Of Linear Inequality Systems Of Feasible Solutions, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Mathematics Research Reports
This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to parametric problems of semi-infinite and infinite programming, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Part I is primarily devoted to the study of robust Lipschitzian stability of feasible solutions maps for such problems described by parameterized systems of infinitely many linear inequalities in Banach spaces of decision variables indexed by an arbitrary set T. The parameter space of admissible perturbations under consideration is formed by all bounded functions on T equipped with the standard supremum norm. Unless the index set is finite, this …
Second-Order Analysis Of Polyhedral Systems In Finite And Infinite Dimensions With Applications To Robust Stability Of Variational Inequalities, René Henrion, Boris S. Mordukhovich, Nguyen Mau Nam
Second-Order Analysis Of Polyhedral Systems In Finite And Infinite Dimensions With Applications To Robust Stability Of Variational Inequalities, René Henrion, Boris S. Mordukhovich, Nguyen Mau Nam
Mathematics Research Reports
This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order …
Robust Stability And Optimality Conditions For Parametric Infinite And Semi-Infinite Programs, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Robust Stability And Optimality Conditions For Parametric Infinite And Semi-Infinite Programs, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra
Mathematics Research Reports
This paper primarily concerns the study of 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 T, 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 on T equipped with the standard supremum norm. Unless the index set T is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l(infinity)-type. By using …
Metric Regularity Of Mappings And Generalized Normals To Set Images, Boris S. Mordukhovich, Nguyen Mau Nam, Bingwu Wang
Metric Regularity Of Mappings And Generalized Normals To Set Images, Boris S. Mordukhovich, Nguyen Mau Nam, Bingwu Wang
Mathematics Research Reports
The primary goal of this paper is to study some notions of normals to nonconvex sets in finite-dimensional and infinite-dimensional spaces and their images under single-valued and set-valued mappings. The main motivation for our study comes from variational analysis and optimization, where the problems under consideration play a crucial role in many important aspects of generalized differential calculus and applications. Our major results provide precise equality formulas (sometimes just efficient upper estimates) allowing us to compute generalized normals in various senses to direct and inverse images of nonconvex sets under single-valued and set-valued mappings between Banach spaces. The main tools …
Optimization Of Delay-Differential Inclusions Of Infinite Dimensions, Boris S. Mordukhovich, Dong Wang, Lianwen Wang
Optimization Of Delay-Differential Inclusions Of Infinite Dimensions, Boris S. Mordukhovich, Dong Wang, Lianwen Wang
Mathematics Research Reports
No abstract provided.
Limiting Subgradients Of Minimal Time Functions In Banach Spaces, Boris S. Mordukhovich, Nguyen Mau Nam
Limiting Subgradients Of Minimal Time Functions In Banach Spaces, Boris S. Mordukhovich, Nguyen Mau Nam
Mathematics Research Reports
The paper mostly concerns the study of generalized differential properties of the so-called minimal time functions associated, in particular, with constant dynamics and arbitrary closed target sets in control theory. Functions of this type play a significant role in many aspects of optimization, control theory: and Hamilton-Jacobi partial differential equations. We pay the main attention to computing and estimating limiting subgradients of the minimal value functions and to deriving the corresponding relations for Frechet type epsilon-subgradients in arbitrary Banach spaces.
Necessary Conditions For Nonsmooth Optimization Problems With Operator Constraints In Metric Spaces, Boris S. Mordukhovich, Libin Mou
Necessary Conditions For Nonsmooth Optimization Problems With Operator Constraints In Metric Spaces, Boris S. Mordukhovich, Libin Mou
Mathematics Research Reports
This paper concerns nonsmooth optimization problems involving operator constraints given by mappings on complete metric spaces with values in nonconvcx subsets of Banach spaces. We derive general first-order necessary optimality conditions for such problems expressed via certain constructions of generalized derivatives for mappings on metric spaces and axiomatically defined subdifferentials for the distance function to nonconvex sets in Banach spaces. Our proofs arc based on variational principles and perturbation/approximation techniques of modern variational analysis. The general necessary conditions obtained are specified in the case of optimization problems with operator constraints dDScribcd by mappings taking values in approximately convex subsets of …
Failure Of Metric Regularity For Major Classes Of Variational Systems, Boris S. Mordukhovich
Failure Of Metric Regularity For Major Classes Of Variational Systems, Boris S. Mordukhovich
Mathematics Research Reports
The paper is devoted to the study of metric regularity, which is a remarkable property of set-valued mappings playing an important role in many aspects of nonlinear analysis and its applications. We pay the main attention to metric regularity of the so- called parametric variational systems that contain, in particular, various classes of parameterized/perturbed variational and hemivariational inequalities, complementarity systems, sets of optimal solutions and corresponding Lagrange multipliers in problems of parametric optimization and equilibria, etc. Based on the advanced machinery of generalized differentiation1 we surprisingly reveal that metric regularity fails for certain major classes of parametric variational systems, which …
Optimal Control Of Delay-Differential Inclusions With Multivalued Initial Conditions In Infinite Dimensions, Boris S. Mordukhovich, Dong Wang, Lianwen Wang
Optimal Control Of Delay-Differential Inclusions With Multivalued Initial Conditions In Infinite Dimensions, Boris S. Mordukhovich, Dong Wang, Lianwen Wang
Mathematics Research Reports
This paper is devoted to the study of a general class of optimal control problems described by delay-differential inclusions with infinite-dimensional state spaces, endpoints constraints, and multivalued initial conditions. To the best of our knowledge, problems of this type have not been considered in the literature, except some particular cases when either the state space is finite-dimensional or there is no delay in the dynamics. We develop the method of discrete approximations to derive necessary optimality conditions in the extended Euler-Lagrange form by using advanced tools of variational analysis and generalized differentiation in infinite dimensions. This method consists of the …
Relative Pareto Minimizers To Multiobjective Problems: Existence And Optimality Conditions, Truong Q. Bao, Boris S. Mordukhovich
Relative Pareto Minimizers To Multiobjective Problems: Existence And Optimality Conditions, Truong Q. Bao, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we introduce and study enhanced notions of relative Pareto minimizers to constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers to general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient …
Necessary Conditions For Super Minimizers In Constrained Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich
Necessary Conditions For Super Minimizers In Constrained Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich
Mathematics Research Reports
This paper concerns the study of the so-called super minimizers related to the concept of super efficiency in constrained problems of multiobjective optimization, where cost mappings are generally set-valued. We derive necessary conditions for super minimizers on the base of advanced tools of variational analysis and generalized differentiation that are new in both finite-dimensional and infinite-dimensional settings for problems with single-valued and set-valued objectives.
Suboptimality Conditions For Mathematical Programs With Equilibrium Constraints, Truong Q. Bao, Panjak Gupta, Boris S. Mordukhovich
Suboptimality Conditions For Mathematical Programs With Equilibrium Constraints, Truong Q. Bao, Panjak Gupta, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended 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 arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal …
Variational Principles For Set-Valued Mappings With Applications To Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich
Variational Principles For Set-Valued Mappings With Applications To Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich
Mathematics Research Reports
This paper primarily concerns the study of general classes of constrained multiobjective optimization problems (including those described via set-valued and vector-valued cost mappings) from the viewpoint of modern variational analysis and generalized differentiation. To proceed, we first establish two variational principles for set-valued mappings, which~being certainly of independent interest are mainly motivated by applications to multiobjective optimization problems considered in this paper. The first variational principle is a set-valued counterpart of the seminal derivative-free Ekeland variational principle, while the second one is a set-valued extension of the subdifferential principle by Mordukhovich and Wang formulated via an appropriate subdifferential notion for …
Generalized Differentiation Of Parameter-Dependent Sets And Mappings, Boris S. Mordukhovich, Bingwu Wang
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.
Optimal Control Of Nonconvex Differential Inclusions, Boris S. Mordukhovich
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.
Multiobjective Optimization Problems With Equilibrium Constraints, Boris S. Mordukhovich
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 …
Necessary Conditions In Multiobjective Optimization With Equilibrium Constraints, Truong Q. Bao, Panjak Gupta, Boris S. Mordukhovich
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 …