v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
200 CHAPTER 2. CONVEX GEOMETRY-x 2x 3∂K MK ∗ Mx 2x 3∂K MK ∗ MFigure 66: Two views of monotone cone K M and its dual K ∗ M (drawntruncated) in R 3 . Monotone cone is not pointed. Dual monotone coneis not full-dimensional. Cartesian coordinate axes are drawn for reference.
2.13. DUAL CONE & GENERALIZED INEQUALITY 201where N = n −1, and because K M is closed and convex, we may adapt ConeTable 1 (p.194) as follows:Cone Table 1* K ∗ K ∗∗ = Kvertex-description X ∗ X ∗†T , ±X ∗⊥halfspace-description X ∗† , X ∗⊥T X ∗TThe vertex-description for K M is thereforewhere X ∗⊥ = 1 andK M = {[X ∗†T X ∗⊥ −X ∗⊥ ]a | a ≽ 0} ⊂ R n (439)⎡⎤n − 1 −1 −1 · · · −1 −1 −1n − 2 n − 2 −2... · · · −2 −2X ∗† = 1 . n − 3 n − 3 . .. −(n − 4) . −3n3 . n − 4 . ∈ R.. n−1×n−(n − 3) −(n − 3) .⎢⎥⎣ 2 2 · · ·... 2 −(n − 2) −(n − 2) ⎦while1 1 1 · · · 1 1 −(n − 1)(440)K ∗ M = {y ∈ R n | X ∗† y ≽ 0, X ∗⊥T y = 0} (441)is the dual monotone cone halfspace-description.2.13.9.4.4 Exercise. Inside the monotone cones.Mathematically describe the respective interior of the monotone nonnegativecone and monotone cone. In three dimensions, also describe the relativeinterior of each face.2.13.9.5 More pointed cone descriptions with equality conditionConsider pointed polyhedral cone K having a linearly independent set ofgenerators and whose subspace membership is explicit; id est, we are giventhe ordinary halfspace-descriptionK = {x | Ax ≽ 0, Cx = 0} ⊆ R n(287a)
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200 CHAPTER 2. CONVEX GEOMETRY-x 2x 3∂K MK ∗ Mx 2x 3∂K MK ∗ MFigure 66: Two views of monotone cone K M and its dual K ∗ M (drawntruncated) in R 3 . Monotone cone is not pointed. Dual monotone coneis not full-dimensional. Cartesian coordinate axes are drawn for reference.
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