v2007.09.17 - Convex Optimization

64 CHAPTER 2. CONVEX GEOMETRY11−1−11[ 1a =1](a)[ −1b =−1]−1−11(b){y | a T y=1}{y | b T y=−1}{y | a T y=−1}{y | b T y=1}(c)[ −1c =1]−11−11−11−1[ 1d =−11](d){y | c T y=1}{y | c T y=−1}{y | d T y=−1}{y | d T y=1}[ 1e =0]−11(e){y | e T y=−1} {y | e T y=1}Figure 17: (a)-(d) Hyperplanes in R 2 (truncated). Movement in normaldirection increases vector inner-product. This visual concept is exploitedto attain analytical solution of linear programs; e.g., Example 2.4.2.6.2,Example 3.1.6.0.1, [46, exer.4.8-exer.4.20]. Each graph is also interpretableas a contour plot of a real affine function of two variables as inFigure 55. (e) Ratio |b|/‖a‖ from {x | a T x = b} represents radius ofhypersphere about 0 supported by hyperplane whose normal is a .