v2010.10.26 - Convex Optimization

156 CHAPTER 2. CONVEX GEOMETRYof conic problems. [391] [277,4.5] For unique minimum-distance projectionon a closed convex cone K , the negative dual cone −K ∗ plays the role thatorthogonal complement plays for subspace projection. 2.58 (E.9.2.1) Indeed,−K ∗ is the algebraic complement in R n ;K ⊞ −K ∗ = R n (2023)where ⊞ denotes unique orthogonal vector sum.One way to think of a pointed closed convex cone is as a new kind ofcoordinate system whose basis is generally nonorthogonal; a conic system,very much like the familiar Cartesian system whose analogous cone is thefirst quadrant (the nonnegative orthant). Generalized inequality ≽ K is aformalized means to determine membership to any pointed closed convexcone K (2.7.2.2) whereas biorthogonal expansion is, fundamentally, anexpression of coordinates in a pointed conic system whose axes are linearlyindependent but not necessarily orthogonal. When cone K is the nonnegativeorthant, then these three concepts come into alignment with the Cartesianprototype: biorthogonal expansion becomes orthogonal expansion, the dualcone becomes identical to the orthant, and generalized inequality obeys atotal order entrywise.2.13.1 Dual coneFor any set K (convex or not), the dual cone [109,4.2]K ∗ {y ∈ R n | 〈y , x〉 ≥ 0 for all x ∈ K} (297)is a unique cone 2.59 that is always closed and convex because it is anintersection of halfspaces (2.4.1.1.1). Each halfspace has inward-normal x ,belonging to K , and boundary containing the origin; e.g., Figure 55a.When cone K is convex, there is a second and equivalent construction:Dual cone K ∗ is the union of each and every vector y inward-normalto a hyperplane supporting K (2.4.2.6.1); e.g., Figure 55b. When K is2.58 Namely, projection on a subspace is ascertainable from projection on its orthogonalcomplement (Figure 165).2.59 The dual cone is the negative polar cone defined by many authors; K ∗ = −K ◦ .[199,A.3.2] [307,14] [41] [26] [330,2.7]