v2009.01.01 - Convex Optimization

146 CHAPTER 2. CONVEX GEOMETRY
variables in the primal program. Third, the maximum value achieved by the
dual problem is often equal to the minimum of the primal. [259,2.1.3]
Essentially, duality theory concerns representation of a given optimization
problem as half a minimax problem. [266,36] [53,5.4] Given any real
function f(x,z)
minimize
x
always holds. When
minimize
x
maximize
z
maximize
z
f(x,z) ≥ maximize
z
f(x,z) = maximize
z
minimize f(x,z) (273)
x
minimize f(x,z) (274)
x
we have strong duality and then a saddle value [128] exists. (Figure 51)
[263, p.3] Consider primal conic problem (p) (over cone K) and its
corresponding dual problem (d): [254,3.3.1] [206,2.1] [207] given
vectors α , β and matrix constant C
(p)
minimize α T x
x
subject to x ∈ K
Cx = β
maximize β T z
y , z
subject to y ∈ K ∗
C T z + y = α
(d) (275)
Observe the dual problem is also conic, and its objective function value never
exceeds that of the primal;
α T x ≥ β T z
x T (C T z + y) ≥ (Cx) T z
x T y ≥ 0
(276)
which holds by definition (270). Under the sufficient condition: (275p) is
a convex problem and satisfies Slater’s condition, 2.53 then each problem (p)
and (d) attains the same optimal value of its objective and each problem
is called a strong dual to the other because the duality gap (primal−dual
optimal objective difference) is 0. Then (p) and (d) are together equivalent
to the minimax problem
minimize α T x − β T z
x,y,z
subject to x ∈ K ,
y ∈ K ∗
Cx = β , C T z + y = α
(p)−(d) (277)
2.53 A convex problem, essentially, has convex objective function optimized over a convex
set. (4) In this context, (p) is convex if K is a convex cone. Slater’s condition is satisfied
whenever any primal strictly feasible point exists. (p.256)