v2010.10.26 - Convex Optimization

88 CHAPTER 2. CONVEX GEOMETRYIf the two planes are independent (meaning any line in one is linearlyindependent [ ] of any line from the other), they will intersect at a point becauseA1then is invertible;A 2A 1 ∩ A 2 ={ ∣ [ ] [ ]}∣∣∣x∈ R 4 A1 b1x =A 2 b 22.5.1.2.2 Exercise. Linear program.Minimize a hyperplane over affine set A in the nonnegative orthant(155)minimize c T xxsubject to Ax = bx ≽ 0(156)where A = {x | Ax = b}. Two cases of interest are drawn in Figure 30.Graphically illustrate and explain optimal solutions indicated in the caption.Why is α ⋆ negative in both cases? Is there solution on the vertical axis?What causes objective unboundedness in the latter case (b)? Describe allvectors c that would yield finite optimal objective in (b).Graphical solution to linear programmaximize c T xxsubject to x ∈ P(157)is illustrated in Figure 31. Bounded set P is an intersection of manyhalfspaces. Why is optimal solution x ⋆ not aligned with vector c as inCauchy-Schwarz inequality (2055)?2.5.2 Intersection of subspacesThe intersection of nullspaces associated with two matrices A∈ R m×n andB ∈ R k×n can be expressed most simply as([ ]) [ ]A AN(A) ∩ N(B) = N {x∈ R n | x = 0} (158)BBthe nullspace of their rowwise concatenation.