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ODEs and the calculus of variations
Published in Alfio Borzì, Modelling with Ordinary Differential Equations, 2020
Next, to discuss the first-order optimality condition for a minimum, we define the so-called tangent cone. A local tangent cone T(V, y) of V at y ∈ V is the set of all w ∈ H such that there exist sequences yn ∈ V, tn∈R, tn > 0, n=1,2,…, for which it holds yn → y in B and tn(yn−y)⇀w in H.
New perspective on some classical results in analysis and optimization
Published in Optimization Methods and Software, 2021
Olga Brezhneva, Yuri G. Evtushenko, Alexey A. Tret'yakov
We call a tangent vector to a nonempty set at if there exist and a function with the property that for all , we have and The collection of all tangent vectors at is called the tangent cone to M at and is denoted by .
Fast computation of local minimal distances between CAD models for dynamics simulation
Published in Computer-Aided Design and Applications, 2018
Sébastien Crozet, Jean-Claude Léon, Xavier Merlhiot
Our method is based on the characterization of contact points between two solids and as stated by [6, 19]. Indeed, in addition to Eqn. (3.2) expressing the LMD between and , the direction of the LMD appears also as a key feature that can be characterized using the concepts of tangents, tangent cones, and cone polars, as defined in the field of convex analysis [3]. We recall some definitions from [3]. Let designates points of either or . Given a point (which can be either on the interior of the boundary of ), its tangents are the vectors such that there exists a sequence of vectors , and: The set of all tangents at form a cone (in the sense of convex analysis) called tangent cone. The tangent cone polar: is the set of vectors opposite to all the tangents at . Some tangent cones and their polars are depicted in Fig. 3 to illustrate common configurations. For sake of simplicity, a 2D domain is used rather than a 3D one. The tangent cone in Fig. 3(c) is larger than a half-space, therefore its polar degenerates to the singleton .