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 ..."
Abstract

Cited by 88 (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.
ThreeDimensional Shape Representation via Shock Flows
 BROWN UNIVERSITY
, 2003
"... We address the problem of representing 3D shapes when partial and unorganized data is obtained as an input, such as clouds of point samples on the surface of a face, statue, solid, etc., of regular or arbitrary complexity (freeform), as is commonly produced by photogrammetry, laser scanners, comput ..."
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Cited by 7 (2 self)
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We address the problem of representing 3D shapes when partial and unorganized data is obtained as an input, such as clouds of point samples on the surface of a face, statue, solid, etc., of regular or arbitrary complexity (freeform), as is commonly produced by photogrammetry, laser scanners, computerized tomography, and so on. Our starting point is the medial axis (MA) representation which has been explored mainly for 2D problems since the 1960's in pattern recognition and image analysis. The MA makes explicit certain symmetries of an object, corresponding to the shocks of waves initiated at the input samples, but is itself difficult to directly use for recognition tasks and applications. Based on previous work on the 2D problem, we propose a new representation in 3D which is derived from the MA, producing a graph we call the shock scaffold. The nodes of this graph are defined to be certain singularities of the shock flow along the MA. This graph can represent exactly the MA  and the original inputs  or approximate it, leading to a hierarchical description of shapes. We develop accurate and efficient algorithms to compute for 3D unorganized clouds of points the shock scaffold, and thus the MA, as well as its close cousin the Voronoi diagram. One computational method relies on clustering and visibility constraints, while the other simulates wavefront propagation on a 3D grid. We then propose a method of splitting the shock scaffold in two subgraphs, one of which is related to the (a priori unknown) surface of the object under scrutiny. This allows us to simplify the shock scaffold making more explicit coarse scale object symmetries, while at the same time providing an original method for the surface interpolation of complex datasets. In the last part of this talk, we address extensions of the shock scaffold by studying the case where the inputs are given as collections of unorganized polygons. Keywords: 3D shape representation, medial axis, Voronoi diagram, maximal contact spheres and shocks, directed graphs (digraphs), shock scaffold hierarchy (5 levels), wave propagation, eikonal equation, Euclidean distance maps, Lagrangian versus Eulerian computations, deterministic celullar automata, Huygens versus Fermat's optical principles, visibility constraints, unorganized generators, point clouds, polygonal clouds, quadrics, quartics, octics, Groebner bases and hybrid elimination methods, surface interpolation and meshing, ribs and ridges.
Fast verification of convexity of piecewiselinear surfaces
 CoRR
"... Short Version We show that a realization of a closed connected PLmanifold of dimension n − 1 in R n (n> 3) is the boundary of a convex polyhedron if and only if the interior of each (n − 3)face has a point, which has a neighborhood lying on the boundary of a convex body. This result is derived ..."
Abstract

Cited by 1 (0 self)
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Short Version We show that a realization of a closed connected PLmanifold of dimension n − 1 in R n (n> 3) is the boundary of a convex polyhedron if and only if the interior of each (n − 3)face has a point, which has a neighborhood lying on the boundary of a convex body. This result is derived from a generalization of Van Heijenoort’s theorem on locally convex manifolds to spaces of constant curvature. Our convexity criterion for PLmanifolds imply an easy polynomialtime algorithm for checking convexity of a given PLsurface in R n. There is a number of theorems that infer global convexity from local convexity. The oldest one belongs to Jacque Hadamard (1897) and asserts that any compact smooth surface embedded in R 3 and having strictly positive Gaussian curvature is the boundary of a convex body. Local convexity can be defined in many different ways (see van Heijenoort (1952) for a survey). We will use van Heijenoort’s (1932) notion of local convexity. In this definition a surface M in the affine space R n is called locally convex at point p if p has a neighborhood
Convexity, Duality, and Lagrange Multipliers
"... Contents 1. Convex Analysis and Optimization 1.1. Linear Algebra and Analysis . . . . . . . . . . . . . . . . . 1.1.1. Vectors and Matrices . . . . . . . . . . . . . . . . . . 1.1.2. Topological Properties . . . . . . . . . . . . . . . . . . 1.1.3. Square Matrices . . . . . . . . . . . . . . . . . ..."
Abstract

Cited by 1 (0 self)
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Contents 1. Convex Analysis and Optimization 1.1. Linear Algebra and Analysis . . . . . . . . . . . . . . . . . 1.1.1. Vectors and Matrices . . . . . . . . . . . . . . . . . . 1.1.2. Topological Properties . . . . . . . . . . . . . . . . . . 1.1.3. Square Matrices . . . . . . . . . . . . . . . . . . . . 1.1.4. Derivatives . . . . . . . . . . . . . . . . . . . . . . . 1.2. Convex Sets and Functions . . . . . . . . . . . . . . . . . . 1.2.1. Basic Properties . . . . . . . . . . . . . . . . . . . . 1.2.2. Convex and A#ne Hulls . . . . . . . . . . . . . . . . . 1.2.3. Closure, Relative Interior, and Continuity . . . . . . . . . 1.2.4. Recession Cones . . . . . . . . . . . . . . . . . . . . 1.3. Convexity and Optimization . . . . . . . . . . . . . . . . . 1.3.1. Global and Local Minima . . . . . . . . . . . . . . . . 1.3.2. The Projection Theorem . . . . . . . . . . . . . . . . . 1.3.3. Directions of Recession and Existence of Optimal Solutions . . . . . . . . . . . . 1.3.4
A CONTINUATION RESULT FOR FORCED OSCILLATIONS OF CONSTRAINED MOTION PROBLEMS WITH INFINITE DELAY
, 903
"... Abstract. We prove a global continuation result for Tperiodic solutions of a Tperiodic parametrized second order retarded functional differential equation on a boundaryless compact manifold with nonzero EulerPoincaré characteristic. The approach is based on the fixed point index theory for locall ..."
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Abstract. We prove a global continuation result for Tperiodic solutions of a Tperiodic parametrized second order retarded functional differential equation on a boundaryless compact manifold with nonzero EulerPoincaré characteristic. The approach is based on the fixed point index theory for locally compact maps on ANRs. As an application, we prove the existence of forced oscillations of retarded functional motion equations defined on topologically nontrivial compact constraints. This existence result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case. 1.
On locally convex PLmanifolds and fast verification of convexity
, 2008
"... Short Version We show that a PLrealization of a closed connected manifold of dimension n − 1 in R n (n ≥ 3) is the boundary of a convex polyhedron if and only if the interior of each (n − 3)face has a point, which has a neighborhood lying on the boundary of a convex ndimensional body. This result ..."
Abstract
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Short Version We show that a PLrealization of a closed connected manifold of dimension n − 1 in R n (n ≥ 3) is the boundary of a convex polyhedron if and only if the interior of each (n − 3)face has a point, which has a neighborhood lying on the boundary of a convex ndimensional body. This result is derived from a generalization of Van Heijenoort’s theorem on locally convex manifolds to the spherical case. Our convexity criterion for PLmanifolds implies an easy polynomialtime algorithm for checking convexity of a given PLsurface in R n. There is a number of theorems that infer global convexity from local convexity. The oldest one belongs to Jacque Hadamard (1897) and asserts that any compact smooth surface embedded in R 3, with strictly positive Gaussian curvature, is the boundary of a convex body. Local convexity can be defined in many different ways (see van Heijenoort (1952) for a survey). We will use Bouligand’s (1932) notion of local convexity. In this definition a surface M in the affine space R n is called locally convex at point p if p has a neighborhood
RESTRICTIONS OF SMOOTH FUNCTIONS TO A CLOSED SUBSET
, 2004
"... We first provide an approach to the recent conjecture of BierstoneMilmanPaw̷lucki on Whitney’s old problem on C d extendability of functions defined on a closed subset of a Euclidean space, using the higher order paratangent bundle they introduced. For example, the conjecture is affirmative for cl ..."
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We first provide an approach to the recent conjecture of BierstoneMilmanPaw̷lucki on Whitney’s old problem on C d extendability of functions defined on a closed subset of a Euclidean space, using the higher order paratangent bundle they introduced. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on controllability of flatness by the values on a closed set. The multidimensional Vandermonde matrix plays an important role in both cases.
Acknowledgments
, 2008
"... 1 This is the first preliminary draft of lecture notes of a minicourse at University of California at Davis in 1999. Any critical comment and correction will be gratefully acknowledged. i ..."
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1 This is the first preliminary draft of lecture notes of a minicourse at University of California at Davis in 1999. Any critical comment and correction will be gratefully acknowledged. i
RESTRICTIONS OF SMOOTH FUNCTIONS TO A CLOSED SUBSET
, 2003
"... Abstract. We first provide an approach to the recent conjecture of BierstoneMilmanPaw̷lucki on Whitney’s old problem on C d extendability of functions defined on a closed subset of a Euclidean space, using the higher order paratangent bundle they introduced. For example, the conjecture is affirmat ..."
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Abstract. We first provide an approach to the recent conjecture of BierstoneMilmanPaw̷lucki on Whitney’s old problem on C d extendability of functions defined on a closed subset of a Euclidean space, using the higher order paratangent bundle they introduced. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on controllability of flatness by the values on a closed set. The multidimensional Vandermonde matrix plays an important role in both cases.