v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 147
whose optimal objective always has the saddle value 0 (regardless of the
particular convex cone K and other problem parameters). [312,3.2] Thus
determination of convergence for either primal or dual problem is facilitated.
Were convex cone K polyhedral (2.12.1), then problems (p) and (d)
would be linear programs. 2.54 Were K a positive semidefinite cone, then
problem (p) has the form of prototypical semidefinite program 2.55 (584) with
(d) its dual. It is sometimes possible to solve a primal problem by way of its
dual; advantageous when the dual problem is easier to solve than the primal
problem, for example, because it can be solved analytically, or has some special
structure that can be exploited. [53,5.5.5] (4.2.3.1)
2.13.1.1 Key properties of dual cone
For any cone, (−K) ∗ = −K ∗
For any cones K 1 and K 2 , K 1 ⊆ K 2 ⇒ K ∗ 1 ⊇ K ∗ 2 [286,2.7]
(Cartesian product) For closed convex cones K 1 and K 2 , their
Cartesian product K = K 1 × K 2 is a closed convex cone, and
K ∗ = K ∗ 1 × K ∗ 2 (278)
(conjugation) [266,14] [92,4.5] When K is any convex cone, the dual
of the dual cone is the closure of the original cone; K ∗∗ = K . Because
K ∗∗∗ = K ∗ K ∗ = (K) ∗ (279)
When K is closed and convex, then the dual of the dual cone is the
original cone; K ∗∗ = K .
If any cone K has nonempty interior, then K ∗ is pointed;
K nonempty interior ⇒ K ∗ pointed (280)
2.54 The self-dual nonnegative orthant yields the primal prototypical linear program and
its dual.
2.55 A semidefinite program is any convex program constraining a variable to any subset of
a positive semidefinite cone. The qualifier prototypical conventionally means: a program
having linear objective, affine equality constraints, but no inequality constraints except
for cone membership.