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- Generalized differentiation (42)
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Articles 1 - 30 of 93
Full-Text Articles in Applied Mathematics
Second-Order Subdifferential Calculus With Applications To Tilt Stability In Optimization, Boris S. Mordukhovich, R T. Rockafellar
Second-Order Subdifferential Calculus With Applications To Tilt Stability In Optimization, Boris S. Mordukhovich, R T. Rockafellar
Mathematics Research Reports
The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of first-order sub differential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and …
Sensitivity Analysis For Two-Level Value Functions With Applications To Bilevel Programming, S Dempe, Boris S. Mordukhovich, B Zemkoho
Sensitivity Analysis For Two-Level Value Functions With Applications To Bilevel Programming, S Dempe, Boris S. Mordukhovich, B Zemkoho
Mathematics Research Reports
This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization spread under the name of the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that, although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do not to a large extent differ from those known for the conventional problem. It has been already well recognized in the literature that for optimality conditions of the usual optimistic bilevel program …
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.
Several Approaches For The Derivation Of Stationary Conditions For Elliptic Mpecs With Upper-Level Control Constraints, M Hintermüller, Boris S. Mordukhovich, T Surowiec
Several Approaches For The Derivation Of Stationary Conditions For Elliptic Mpecs With Upper-Level Control Constraints, M Hintermüller, Boris S. Mordukhovich, T Surowiec
Mathematics Research Reports
The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained …
Directional Subdifferentials And Optimality Conditions, Ivan Ginchev, Boris S. Mordukhovich
Directional Subdifferentials And Optimality Conditions, Ivan Ginchev, Boris S. Mordukhovich
Mathematics Research Reports
This paper is devoted to the introduction and development of new dual-space constructions of generalized differentiation in variational analysis, which combine certain features of subdifferentials for nonsmooth functions (resp. normal cones to sets) and directional derivatives (resp. tangents). We derive some basic properties of these constructions and apply them to optimality conditions in problems of unconstrained and constrained optimization.
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 …
Constraint Qualifications And Optimality Conditions For Nonconvex Semi-Infinite And Infinite Programs, Boris S. Mordukhovich, T T. A. Nghia
Constraint Qualifications And Optimality Conditions For Nonconvex Semi-Infinite And Infinite Programs, Boris S. Mordukhovich, T T. A. Nghia
Mathematics Research Reports
The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian-Fromovitz and Farkas-Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for efficient computing the appropriate normal cones to sets of feasible solutions for these programs by employing advanced …
Rated Extremal Principles For Finite And Infinite Systems, Hung M. Phan, Boris S. Mordukhovich
Rated Extremal Principles For Finite And Infinite Systems, Hung M. Phan, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them called here rated extremal principles. These developments are in the core geometric theory of variational analysis. We present their applications to calculus and optimality conditions for problems with infinitely many constraints.
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 …
Complete Characterizations Of Local Weak Sharp Minima With Applications To Semi-Infinite Optimization And Complementarity, Boris S. Mordukhovich, Naihua Xiu, Jinchuan Zhou
Complete Characterizations Of Local Weak Sharp Minima With Applications To Semi-Infinite Optimization And Complementarity, Boris S. Mordukhovich, Naihua Xiu, Jinchuan Zhou
Mathematics Research Reports
In this paper we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials, or tangent cones and subderivatives, or their mixture in finite-dimensional spaces. The results obtained not only significantly extend previous ones in the literature, but also allow us to provide new types of criteria for local weak sharpness. Applications of the developed theory are given to semi-infinite programming and to semi-infinite complementarity problems.
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.
Tangential Extremal Principles For Finite And Infinite Systems Of Sets, I: Basic Theory, Boris S. Mordukhovich, Hung M. Phan
Tangential Extremal Principles For Finite And Infinite Systems Of Sets, I: Basic Theory, Boris S. Mordukhovich, Hung M. Phan
Mathematics Research Reports
In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization.
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 …
Solving A Generalized Heron Problem By Means Of Convex Analysis, Boris S. Mordukhovich, Nguyen Mau Nam, Juan Salinas Jr
Solving A Generalized Heron Problem By Means Of Convex Analysis, Boris S. Mordukhovich, Nguyen Mau Nam, Juan Salinas Jr
Mathematics Research Reports
The classical Heron problem states: on a given straight line in the plane, find a point C such that the sum of the distances from C to the given points A and B is minimal. This problem can be solved using standard geometry or differential calculus. In the light of modern convex analysis, we are able to investigate more general versions of this problem. In this paper we propose and solve the following problem: on a given nonempty closed convex subset of IR!, find a point such that the sum of the distances from that point to n given nonempty …
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. …
Generalized Newton's Method Based On Graphical Derivatives, T Hoheisel, C Kanzow, Boris S. Mordukhovich, Hung M. Phan
Generalized Newton's Method Based On Graphical Derivatives, T Hoheisel, C Kanzow, Boris S. Mordukhovich, Hung M. Phan
Mathematics Research Reports
This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper …
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.
On Directionally Dependent Subdifferentials, Ivan Ginchev, Boris S. Mordukhovich
On Directionally Dependent Subdifferentials, Ivan Ginchev, Boris S. Mordukhovich
Mathematics Research Reports
In this paper directionally contextual concepts of variational analysis, based on dual-space constructions similar to those in [4, 5], are introduced and studied. As an illustration of their usefulness, necessary and also sufficient optimality conditions in terms of directioual subdifferentials are established, and it is shown that they can be effective in the situations where known optimality conditions in terms of nondirectional subdifferentials fail.
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.
Enhanced Metric Regularity And Lipschitzian Properties Of Variational Systems, Francisco J. Aragón Artacho, Boris S. Mordukhovich
Enhanced Metric Regularity And Lipschitzian Properties Of Variational Systems, Francisco J. Aragón Artacho, Boris S. Mordukhovich
Mathematics Research Reports
This paper mainly concerns the study of a large class of variational systems governed by parametric generalized equations, which encompass variational and hemivariational inequalities, complementarity problems, first-order necessary optimality conditions, and other optimization-related models important for optimization theory and applications. An efficient approach to these issues has been developed in our preceding work [1] establishing qualitative and quantitative relationships between conventional metric regularity jsubregularity and Lipschitzian/calmness properties in the framework of parametric generalized equations in arbitrary Banach spaces. This paper provides, on one hand, significant extensions of the major results in [1] to new partial metric regularity and hemiregularity properties. …
Optimal Control And Feedback Design Of State-Constrained Parabolic Systems In Uncertainty Conditions, Boris S. Mordukhovich
Optimal Control And Feedback Design Of State-Constrained Parabolic Systems In Uncertainty Conditions, Boris S. Mordukhovich
Mathematics Research Reports
The paper concerns minimax control problems for linear 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 a constructive approach to the minimax control design of constrained parabolic systems that is based on certain characteristic features of the parabolic dynamics …
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.
Hybrid Proximal Methods For Equilibrium Problems, Boris S. Mordukhovich, Barbara Panicucci, Mauro Passacantando, Massimo Pappalardo
Hybrid Proximal Methods For Equilibrium Problems, Boris S. Mordukhovich, Barbara Panicucci, Mauro Passacantando, Massimo Pappalardo
Mathematics Research Reports
This paper concerns developing two hybrid proximal point methods (PPMs) for finding a common solution of some optimization-related problems. First we construct an algorithm to solve simultaneously an equilibrium problem and a variational inequality problem, combing the extragradient method for variational inequalities with an approximate PPM for equilibrium problems. Next we develop another algorithm based on an alternate approximate PPM for finding a common solution of two different equilibrium problems. We prove the global convergence of both algorithms under pseudomonotonicity assumptions.
Infimal Convolutions And Lipschitzian Properties Of Subdifferentials For Prox-Regular Functions In Hilbert Spaces, Miroslav Bačák, Jonathan M. Borwein, Andrew Eberhard, Boris S. Mordukhovich
Infimal Convolutions And Lipschitzian Properties Of Subdifferentials For Prox-Regular Functions In Hilbert Spaces, Miroslav Bačák, Jonathan M. Borwein, Andrew Eberhard, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we study infimal convolutions of extended-real-valued functions in Hilbert spaces paying a special attention to a rather broad and remarkable class of prox-regular functions. Such functions have been well recognized as highly important in many aspects of variational analysis and its applications in both finite-dimensional and infinite-dimensional settings. Based on advanced variational techniques, we discover some new sub differential properties of infima! convolutions and apply them to the study of Lipschitzian behavior of subdifferentials for prox-regular functions in Hilbert spaces. It is shown, in particular, that the fulfillment of a natural Lipschitz-like property for (set-valued) sub differentials …
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 …
Metric Regularity And Lipschitzian Stability Of Parametric Variational Systems, Francisco J. Aragón Artacho, Boris S. Mordukhovich
Metric Regularity And Lipschitzian Stability Of Parametric Variational Systems, Francisco J. Aragón Artacho, Boris S. Mordukhovich
Mathematics Research Reports
The paper concerns the study of variational systems described by parameterized generalized equations/variational conditions important for many aspects of nonlinear analysis, optimization, and their applications. Focusing on the fundamental properties of metric regularity and Lipschitzian stability, we establish various qualitative and quantitative relationships between these properties for multivalued parts/fields of parametric generalized equations and the corresponding solution maps for them in the framework of arbitrary Banach spaces of decision and parameter variables.
Weak Sharp Minima On Riemannian Manifolds, Chong Li, Boris S. Mordukhovich, Jinhua Wang, Jen-Chih Yao
Weak Sharp Minima On Riemannian Manifolds, Chong Li, Boris S. Mordukhovich, Jinhua Wang, Jen-Chih Yao
Mathematics Research Reports
This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian …
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 …