v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
20 CHAPTER 1. OVERVIEW Figure 1: Orion nebula. (astrophotography by Massimo Robberto) Euclidean distance geometry is, fundamentally, a determination of point conformation (configuration, relative position or location) by inference from interpoint distance information. By inference we mean: e.g., given only distance information, determine whether there corresponds a realizable conformation of points; a list of points in some dimension that attains the given interpoint distances. Each point may represent simply location or, abstractly, any entity expressible as a vector in finite-dimensional Euclidean space; e.g., distance geometry of music [91]. It is a common misconception to presume that some desired point conformation cannot be recovered in the absence of complete interpoint distance information. We might, for example, want to realize a constellation given only interstellar distance (or, equivalently, distance from Earth and relative angular measurement; the Earth as vertex to two stars). At first it may seem O(N 2 ) data is required, yet there are many circumstances where this can be reduced to O(N). If we agree that a set of points can have a shape (three points can form a triangle and its interior, for example, four points a tetrahedron), then
ˇx 4 ˇx 3 ˇx 2 Figure 2: Application of trilateration (5.4.2.2.4) is localization (determining position) of a radio signal source in 2 dimensions; more commonly known by radio engineers as the process “triangulation”. In this scenario, anchors ˇx 2 , ˇx 3 , ˇx 4 are illustrated as fixed antennae. [181] The radio signal source (a sensor • x 1 ) anywhere in affine hull of three antenna bases can be uniquely localized by measuring distance to each (dashed white arrowed line segments). Ambiguity of lone distance measurement to sensor is represented by circle about each antenna. So & Ye proved trilateration is expressible as a semidefinite program; hence, a convex optimization problem. [278] we can ascribe shape of a set of points to their convex hull. It should be apparent: from distance, these shapes can be determined only to within a rigid transformation (rotation, reflection, translation). Absolute position information is generally lost, given only distance information, but we can determine the smallest possible dimension in which an unknown list of points can exist; that attribute is their affine dimension (a triangle in any ambient space has affine dimension 2, for example). In circumstances where stationary reference points are also provided, it becomes possible to determine absolute position or location; e.g., Figure 2. 21
- Page 1 and 2: DATTORRO CONVEX OPTIMIZATION & EUCL
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- Page 5 and 6: for Jennie Columba ♦ Antonio ♦
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- Page 11 and 12: CONVEX OPTIMIZATION & EUCLIDEAN DIS
- Page 13 and 14: List of Figures 1 Overview 19 1 Ori
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ˇx 4<br />
ˇx 3<br />
ˇx 2<br />
Figure 2: Application of trilateration (5.4.2.2.4) is localization (determining<br />
position) of a radio signal source in 2 dimensions; more commonly known by<br />
radio engineers as the process “triangulation”. In this scenario, anchors<br />
ˇx 2 , ˇx 3 , ˇx 4 are illustrated as fixed antennae. [181] The radio signal source<br />
(a sensor • x 1 ) anywhere in affine hull of three antenna bases can be<br />
uniquely localized by measuring distance to each (dashed white arrowed line<br />
segments). Ambiguity of lone distance measurement to sensor is represented<br />
by circle about each antenna. So & Ye proved trilateration is expressible as<br />
a semidefinite program; hence, a convex optimization problem. [278]<br />
we can ascribe shape of a set of points to their convex hull. It should be<br />
apparent: from distance, these shapes can be determined only to within a<br />
rigid transformation (rotation, reflection, translation).<br />
Absolute position information is generally lost, given only distance<br />
information, but we can determine the smallest possible dimension in which<br />
an unknown list of points can exist; that attribute is their affine dimension<br />
(a triangle in any ambient space has affine dimension 2, for example). In<br />
circumstances where stationary reference points are also provided, it becomes<br />
possible to determine absolute position or location; e.g., Figure 2.<br />
21
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