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Articles 1 - 7 of 7
Full-Text Articles in Physical Sciences and Mathematics
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 …
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. …
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 …
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 …