v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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152 CHAPTER 2. CONVEX GEOMETRYS = {s | s ≽ 0, 1 T s ≤ 1}1Figure 53: Unit simplex S in R 3 is a unique solid tetrahedron but not regular.

2.13. DUAL CONE & GENERALIZED INEQUALITY 1532.12.3.1 SimplexThe unit simplex comes from a class of general polyhedra called simplex,having vertex-description: [94] [307] [371] [113]conv{x l ∈ R n } | l = 1... k+1, dim aff{x l } = k , n ≥ k (296)So defined, a simplex is a closed bounded convex set possibly notfull-dimensional. Examples of simplices, by increasing affine dimension, are:a point, any line segment, any triangle and its relative interior, a generaltetrahedron, any five-vertex polychoron, and so on.2.12.3.1.1 Definition. Simplicial cone.A proper polyhedral (2.7.2.2.1) cone K in R n is called simplicial iff K hasexactly n extreme directions; [20,II.A] equivalently, iff proper K has exactlyn linearly independent generators contained in any given set of generators.△simplicial cone ⇒ proper polyhedral coneThere are an infinite variety of simplicial cones in R n ; e.g., Figure 24,Figure 54, Figure 64. Any orthant is simplicial, as is any rotation thereof.2.12.4 Converting between descriptionsConversion between halfspace- (286) (287) and vertex-description (86) (289)is nontrivial, in general, [15] [113,2.2] but the conversion is easy forsimplices. [61,2.2.4] Nonetheless, we tacitly assume the two descriptions tobe equivalent. [307,19 thm.19.1] We explore conversions in2.13.4,2.13.9,and2.13.11:2.13 Dual cone & generalized inequality& biorthogonal expansionThese three concepts, dual cone, generalized inequality, and biorthogonalexpansion, are inextricably melded; meaning, it is difficult to completelydiscuss one without mentioning the others. The dual cone is critical in testsfor convergence by contemporary primal/dual methods for numerical solution

152 CHAPTER 2. CONVEX GEOMETRYS = {s | s ≽ 0, 1 T s ≤ 1}1Figure 53: Unit simplex S in R 3 is a unique solid tetrahedron but not regular.

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