v2010.10.26 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 73H + = {y | a T (y − y p )≥0}ay pcyd∂H = {y | a T (y − y p )=0} = N(a T ) + y p∆H − = {y | a T (y − y p )≤0}N(a T )={y | a T y=0}Figure 25: Hyperplane illustrated ∂H is a line partially bounding halfspacesH − and H + in R 2 . Shaded is a rectangular piece of semiinfinite H − withrespect to which vector a is outward-normal to bounding hyperplane; vectora is inward-normal with respect to H + . Halfspace H − contains nullspaceN(a T ) (dashed line through origin) because a T y p > 0. Hyperplane,halfspace, and nullspace are each drawn truncated. Points c and d areequidistant from hyperplane, and vector c − d is normal to it. ∆ is distancefrom origin to hyperplane.