v2009.01.01 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 69
1
1
−1
−1
1
[ 1
a =
1
]
(a)
[ −1
b =
−1
]
−1
−1
1
(b)
{y | a T y=1}
{y | b T y=−1}
{y | a T y=−1}
{y | b T y=1}
(c)
[ −1
c =
1
]
−1
1
−1
1
−1
1
−1
[ 1
d =
−1
1
]
(d)
{y | c T y=1}
{y | c T y=−1}
{y | d T y=−1}
{y | d T y=1}
[ 1
e =
0
]
−1
1
(e)
{y | e T y=−1} {y | e T y=1}
Figure 22: (a)-(d) Hyperplanes in R 2 (truncated). Movement in normal
direction increases vector inner-product. This visual concept is exploited
to attain analytical solution of linear programs; e.g., Example 2.4.2.6.2,
Exercise 2.5.1.2.2, Example 3.1.6.0.2, [53, exer.4.8-exer.4.20]. Each graph is
also interpretable as a contour plot of a real affine function of two variables
as in Figure 63. (e) Ratio |β|/‖α‖ from {x | α T x = β} represents radius of
hypersphere about 0 supported by hyperplane whose normal is α .