v2010.10.26 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 7511−1−11[ 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 26: (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,Exercise 2.5.1.2.2, Example 3.4.0.0.2, [61, exer.4.8-exer.4.20]. Each graph isalso interpretable as a contour plot of a real affine function of two variablesas in Figure 73. (e) Ratio |β|/‖α‖ from {x | α T x = β} represents radius ofhypersphere about 0 supported by hyperplane whose normal is α .