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129
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 120 (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.
State agreement for continuoustime coupled nonlinear systems
 SIAM Journal on Control and Optimization
, 2007
"... Abstract. Two related problems are treated in continuous time. First, the state agreement problem is studied for coupled nonlinear differential equations. The vector fields can switch within a finite family. Associated to each vector field is a directed graph based in a natural way on the interactio ..."
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Cited by 43 (2 self)
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Abstract. Two related problems are treated in continuous time. First, the state agreement problem is studied for coupled nonlinear differential equations. The vector fields can switch within a finite family. Associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems. Generalizing the work of Moreau, under the assumption that the vector fields satisfy a certain subtangentiality condition, it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time. The proof uses nonsmooth analysis. Secondly, the rendezvous problem for kinematic pointmass mobile robots is studied when the robots ’ fields of view have a fixed radius. The circumcenter control law of Ando et al. [1] is shown to solve the problem. The rendezvous problem is a kind of state agreement problem, but the interaction structure is state dependent.
Stabilized sequential quadratic programming
 Computational optimization and Applications
, 1999
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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.
A.: On the string averaging method for sparse common fixed points problems
 Int. Trans. Oper. Res
, 2009
"... We study the common fixed points problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We propose a definition of sparseness of a family of operators and investigate a stringaveraging algorithmic sche ..."
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Cited by 18 (14 self)
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We study the common fixed points problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We propose a definition of sparseness of a family of operators and investigate a stringaveraging algorithmic scheme that favorably handles the common fixed points problem when the family of operators is sparse. The convex feasibility problem is treated as a special case and a new subgradient projections algorithmic scheme is obtained. 1
Visualizing the tightening of knots
 In VIS ’05: Proceedings of the conference on Visualization ’05
, 2005
"... The study of physical models for knots has recently received much interest in the mathematics community. In this paper, we consider the ropelength model, which considers knots tied in an idealized rope. This model is interesting in pure mathematics, and has been applied to the study of a variety of ..."
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Cited by 17 (1 self)
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The study of physical models for knots has recently received much interest in the mathematics community. In this paper, we consider the ropelength model, which considers knots tied in an idealized rope. This model is interesting in pure mathematics, and has been applied to the study of a variety of problems in the natural sciences as well. Modeling and visualizing the tightening of knots in this idealized rope poses some interesting challenges in computer graphics. In particular, selfcontact in a deformable rope model is a difficult problem which cannot be handled by standard techniques. In this paper, we describe a solution based on reformulating the contact problem and using constrainedgradient techniques from nonlinear optimization. The resulting animations reveal new properties of the tightening flow and provide new insights into the geometric structure of tight knots and links.
Generalized Hessian properties of regularized nonsmooth functions
 SIAM Journal on Optimization
, 1996
"... Abstract. The question of secondorder expansions is taken up for a class of functions of importance in optimization, namely Moreau envelope regularizations of nonsmooth functions f. It is shown that when f is proxregular, which includes convex functions and the extendedrealvalued functions repre ..."
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Cited by 16 (5 self)
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Abstract. The question of secondorder expansions is taken up for a class of functions of importance in optimization, namely Moreau envelope regularizations of nonsmooth functions f. It is shown that when f is proxregular, which includes convex functions and the extendedrealvalued functions representing problems of nonlinear programming, the many secondorder properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself. This clarifies the circumstances conducive to developing computational methods based on envelope functions, such as secondorder approximations in nonsmooth optimization and variants of the proximal point algorithm. The results establish that generalized secondorder expansions of Moreau envelopes, at least, can be counted on in most situations of interest in finitedimensional optimization. Keywords. Proxregularity, amenable functions, primallowernice functions, Hessians, first and secondorder expansions, strict protoderivatives, proximal mappings, Moreau envelopes, regularization, subgradient mappings, nonsmooth analysis, variational analysis, protoderivatives, secondorder epiderivatives, Attouch’s theorem.
Euler Lagrange and Hamiltonian formalisms in dynamic optimization
 Trans. Amer. Math. Soc
"... Abstract. We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler– Lagrange and Hamiltonian formalisms. The principal new results are: an exte ..."
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Cited by 16 (1 self)
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Abstract. We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler– Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler–Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set–valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler–Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one. 1.
A sampling theory for compacts in Euclidean space
 In Proc. 22nd Annu. ACM Sympos. Comput. Geom
, 2006
"... We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size, and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidea ..."
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Cited by 15 (7 self)
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We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size, and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points and critical values of distance functions. 1
Equivalent subgradient versions of Hamiltonian and EulerLagrange equations in variational analysis
 SIAM J. Control and Optimization
, 1996
"... Abstract. Much effort in recent years has gone into generalizing the classical Hamiltonian and EulerLagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints. These generalizations, in which nonsmoothness abou ..."
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Cited by 14 (3 self)
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Abstract. Much effort in recent years has gone into generalizing the classical Hamiltonian and EulerLagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints. These generalizations, in which nonsmoothness abounds and gradients are systematically replaced by subgradients, have succeeded in furnishing necessary conditions for optimality which reduce to the classical ones in the classical setting, but important issues have remained unsettled, especially concerning the exact relationship of the subgradient versions of the Hamiltonian equations versus those of the EulerLagrange equations. Here it is shown that new, tighter subgradient versions of these equations are actually equivalent to each other. The theory of epiconvergence of convex functions provides the technical basis for this development. Key words. EulerLagrange equations, Hamiltonian equations, variational analysis, nonsmooth analysis, subgradients, optimality.