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38
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 122 (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.
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 24 (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.
Ample parameterization of variational inclusions
 SIAM JOURNAL ON OPTIMIZATION
, 2001
"... For a general category of variational inclusions in finite dimensions, a class of parameterizations, called “ample” parameterizations, is identified that is rich enough to provide a full theory of Lipschitztype properties of solution mappings without the need for resorting to the auxiliary introdu ..."
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Cited by 18 (9 self)
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For a general category of variational inclusions in finite dimensions, a class of parameterizations, called “ample” parameterizations, is identified that is rich enough to provide a full theory of Lipschitztype properties of solution mappings without the need for resorting to the auxiliary introduction of canonical parameters. Ample parameterizations also support a detailed description of the graphical geometry that underlies generalized differentiation of solutions mappings. A theorem on protoderivatives is thereby obtained. The case of a variational inequality over a polyhedral convex set is given special treatment along with an application to minimizing a parameterized function over such a set.
Error bounds: Necessary and sufficient conditions
, 2010
"... The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivativelike objects both from the primal as well as from the dual space ..."
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Cited by 17 (6 self)
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The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivativelike objects both from the primal as well as from the dual space
Constantsign and signchanging solutions for nonlinear elliptic equations with Neumann boundary values
 Adv. Differential Equations
"... Abstract. We establish the existence of a smallest positive solution, a greatest negative solution, and a nontrivial signchanging solution when the parameter λ is greater than the second eigenvalue of the Steklov eigenvalue problem. Our approach is based on truncation techniques and comparison pr ..."
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Cited by 15 (14 self)
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Abstract. We establish the existence of a smallest positive solution, a greatest negative solution, and a nontrivial signchanging solution when the parameter λ is greater than the second eigenvalue of the Steklov eigenvalue problem. Our approach is based on truncation techniques and comparison principles for nonlinear elliptic differential inequalities. In particular, we make use of variational and topological tools, such as critical point theory, MountainPass Theorem, the Second Deformation Lemma and variational characterizations of the second eigenvalue of the Steklov eigenvalue problem. 1.
Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization
, 2000
"... We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the FritzJohn theorem, and we introduce new and general conditi ..."
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Cited by 14 (3 self)
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We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the FritzJohn theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new properties, pseudonormality and quasinormality, emerge as central within the taxonomy of interesting constraint characteristics. In the case where there is no abstract set constraint, these properties provide the connecting link between the classical constraint qualifications and two distinct pathways to the existence of Lagrange multipliers: one involving the notion of quasiregularity and Farkas' Lemma, and the other involving the use of exact penalty functions. The second pathway also applies in the general case where there is an abstract set constraint.
Robinson’s implicit function theorem and its extensions
 MATH. PROGRAM
, 2008
"... S. M. Robinson published in 1980 a powerful theorem about solutions to certain “generalized equations” corresponding to parameterized variational inequalities which could represent the firstorder optimality conditions in nonlinear programming, in particular. In fact, his result covered much of the ..."
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Cited by 11 (3 self)
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S. M. Robinson published in 1980 a powerful theorem about solutions to certain “generalized equations” corresponding to parameterized variational inequalities which could represent the firstorder optimality conditions in nonlinear programming, in particular. In fact, his result covered much of the classical implicit function theorem, if not quite all, but went far beyond that in ideas and format. Here, Robinson’s theorem is viewed from the perspective of more recent developments in variational analysis as well as some lesserknown results in the implicit function literature on equations, prior to the advent of generalized equations. Extensions are presented which fully cover such results, translating them at the same time to generalized equations broader than variational inequalities. Robinson’s notion of firstorder approximations in the absence of differentiability is utilized in part, but even looser forms of approximation are shown to furnish significant information about solutions.
Ostrover.: Bounds for Minkowski billiard trajectories in convex bodies
, 2012
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New Farkastype constraint qualifications in convex infinite programming, ESAIM: COCV 13
, 2007
"... Abstract. This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given cl ..."
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Cited by 10 (4 self)
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Abstract. This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily realvalued). The key result in the paper is the characterization of those reverseconvex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas ’ lemma we also characterize the containment of convex sets in reverseconvex sets. The main results in the paper are obtained under a suitable Farkastype constraint qualifications and/or a certain closedness assumption.
Second order optimality conditions for semilinear elliptic control problems with finitely . . .
 SIAM JOURNAL ON OPTIMIZATION
, 2002
"... This paper deals with necessary and sufficient optimality conditions for control problems governed by semilinear elliptic partial differential equations with finitely many equality and inequality state constraints. Some recent results on this topic for optimal control problems based upon results fo ..."
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Cited by 7 (3 self)
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This paper deals with necessary and sufficient optimality conditions for control problems governed by semilinear elliptic partial differential equations with finitely many equality and inequality state constraints. Some recent results on this topic for optimal control problems based upon results for abstract optimization problems are compared with some new results using methods adapted to the control problems. Meanwhile, the Lagrangian formulation is followed to provide the optimality conditions in the first case; the Lagrangian and Hamiltonian functions are used in the second statement. Finally, we prove the equivalence of both formulations.