Results 1  10
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13
LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 89 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows blackandwhite constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.
Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity
, 1996
"... In Gateaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fr'echet subderivative: In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These r ..."
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Cited by 15 (9 self)
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In Gateaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fr'echet subderivative: In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These rules are applied to obtain comparison results for viscosity solutions of HamiltonJacobi equations in fismooth spaces. A unified treatment of metric regularity in smooth spaces completes the paper. This illustrates the flexibility of viscosity subderivatives as a tool for analysis.
A Survey of Subdifferential Calculus with Applications
 TMA
, 1998
"... This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might profitably use subdifferentials as tools. ..."
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Cited by 14 (6 self)
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This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might profitably use subdifferentials as tools.
Proximal analysis and the minimal time function
 SIAM J. CONTROL OPTIM
, 1998
"... Under general hypotheses on the target set S and the dynamics of the system, we show that the minimal time function TS(·) is a proximal solution to the Hamilton–Jacobi equation. Uniqueness results are obtained with two different kinds of boundary conditions. A new propagation result is proven, and ..."
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Cited by 6 (2 self)
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Under general hypotheses on the target set S and the dynamics of the system, we show that the minimal time function TS(·) is a proximal solution to the Hamilton–Jacobi equation. Uniqueness results are obtained with two different kinds of boundary conditions. A new propagation result is proven, and as an application, we give necessary and sufficient conditions for TS(·) tobe Lipschitz continuous near S. A Petrovtype modulus condition is also shown to be sufficient for continuity of TS(·) near S.
OPTIMAL CONTROL OF UNBOUNDED DIFFERENTIAL INCLUSIONS
, 1994
"... We consider a Mayer problem of optimal control, whose dynamic constraint is given by a convexvalued differential inclusion. Both state and endpoint constraints are involved. We prove necessary conditions incorporating the Hamiltonian inclusion, the EulerLagrange inclusion, and the WeierstrassPon ..."
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Cited by 4 (0 self)
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We consider a Mayer problem of optimal control, whose dynamic constraint is given by a convexvalued differential inclusion. Both state and endpoint constraints are involved. We prove necessary conditions incorporating the Hamiltonian inclusion, the EulerLagrange inclusion, and the WeierstrassPontryagin maximum condition. Our results weaken the hypotheses and strengthen the conclusions of earlier works. Their main focus is to allow the admissible velocity sets to be unbounded, provided they satisfy a certain continuity hypothesis. They also sharpen the assertion of the EulerLagrange inclusion by replacing Clarke’s subgradient of the essential Lagrangian with a subset formed by partial convexification of limiting subgradients. In cases where the velocity sets are compact, the traditional Lipschitz condition implies the continuity hypothesis mentioned above, the assumption of “integrable boundedness” is shown to be superfluous, and our refinement of the EulerLagrange inclusion remains a strict improvement on previous forms of this condition.
Dualization of subgradient conditions for optimality
"... Abstract. A basic relationship is derived between generalized subgradients of a given function, possibly nonsmooth and nonconvex, and those of a second function obtained from it by partial conjugation. Applications are made to the study of multiplier rules in finitedimensional optimization and to t ..."
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Cited by 4 (2 self)
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Abstract. A basic relationship is derived between generalized subgradients of a given function, possibly nonsmooth and nonconvex, and those of a second function obtained from it by partial conjugation. Applications are made to the study of multiplier rules in finitedimensional optimization and to the theory of the EulerLagrange condition and Hamiltonian condition in nonsmooth optimal control. Keywords. Subgradients, nonsmooth analysis, Lagrange multiplier rules, EulerLagrange
On Optimality Conditions for Some Nonsmooth Optimization Problems over L p Spaces 1
"... Abstract. The paper deals with the minimization of an integral functional over an Lp space subject to various types of constraints. For such optimization problems, new necessary optimality conditions are derived, based on several concepts of nonsmooth analysis. In particular, we employ the generaliz ..."
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Cited by 4 (0 self)
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Abstract. The paper deals with the minimization of an integral functional over an Lp space subject to various types of constraints. For such optimization problems, new necessary optimality conditions are derived, based on several concepts of nonsmooth analysis. In particular, we employ the generalized differential calculus of Mordukhovich and the fuzzy calculus of proximal subgradients. The results are specialized to nonsmooth twostage and multistage stochastic programs. Key Words. Normal integrands, integral functionals, normal cones, subdifferentials, fuzzy calculus, coderivatives, stochastic programming,
The Equivalence of Several Basic Theorems for Subdifferentials
 CECM Research Report
"... . Several different basic properties are used for developing a system of calculus for subdifferentials. They are a nonlocal fuzzy sum rule in [5, 25], a multidirectional mean value theorem in [7, 8], local fuzzy sum rules in [14, 15] and an extremal principle in [19, 21]. We show that all these basi ..."
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Cited by 3 (1 self)
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. Several different basic properties are used for developing a system of calculus for subdifferentials. They are a nonlocal fuzzy sum rule in [5, 25], a multidirectional mean value theorem in [7, 8], local fuzzy sum rules in [14, 15] and an extremal principle in [19, 21]. We show that all these basic results are equivalent and discuss some interesting consequences of this equivalence. Keywords: Subdifferentials, mean value inequalities, local fuzzy sum rules, nonlocal fuzzy sum rules, extremal principles and Asplund spaces. AMS (1991) subject classification: Primary 26B05. 1 Introduction Smooth subdifferentials play important roles in nonsmooth analysis for two reasons. They characterize many important generalized differentials and results in terms of smooth subdifferentials often require very little technical assumptions. Currently there are several ways of developing a set of basic theorems for subdifferentials so that they can be conveniently applied to a wide range of problems. ...
Partial Differential Inclusions Governing Feedback Controls
, 1995
"... this paper the problem of finding feedback (or closedloop) controls r : K 7! Y satisfying the constraint ..."
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Cited by 3 (2 self)
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this paper the problem of finding feedback (or closedloop) controls r : K 7! Y satisfying the constraint
On Equivalence Of Some Basic Principles In Variational Analysis
"... . The primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization. Considering a broad class of Banach spaces admitting smooth renorms with respect to some bornology, we establish an equival ..."
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. The primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization. Considering a broad class of Banach spaces admitting smooth renorms with respect to some bornology, we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions, an extremal principle for nonconvex sets, and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks. Further refinements of the equivalence result are obtained in the case of a Fr'echet differentiable renorm. Based on the new enhanced sum rule, we provide a simplified proof for the refined sequential description of approximate normals and subgradients in smooth spaces. 1991 Mathematical Subject Classification. Primary 49J52; Secondary 58C20, 46B20. Key words and phrases. Nonsmooth analysis, smooth Banach spaces, variational and extremal pri...