Convex Sets: Dual Description

Abstract

• Why. The heart of the matter of convex analysis is the following phenomenon: there is a perfect symmetry (called ‘duality’) between the two ways in which a closed proper convex set can be described: from the inside, by its points (‘primal description’), and from the outside, by the halfspaces that contain it (‘dual description’). Applications of duality include the theorems of the alternative: non-existence of a solution for a system of linear inequalities is equivalent to existence of a solution for a certain other such system. The best known of these results is Farkas’ lemma. Another application is the celebrated result from finance that European call options have to be priced by a precise rule, the formula of Black-Scholes. Last but not least, duality is very important for practitioners of optimization methods, as we will see in Chap. 8.

• What. Duality for convex sets is stated and a novel proof is given: this amounts to just throwing a small ball against a convex set. Many equivalent versions of the duality result are given: the supporting hyperplane theorem, the separation theorems, the theorem of Hahn–Banach, the fact that a duality operator on convex sets containing the origin, the polar set operator B↦B ^∘, is an involution. For each one of the four binary operations $\square$ on convex sets a rule is given of the type $(B_1\square B_2)^\circ = B_1^\circ \odot B_2^\circ$ where ⊙ is another one of the four binary operations on convex sets, and where B ₁, B ₂ are convex sets containing the origin. Each one of these four rules requires usually a separate proof, but homogenization generates a unified proof. This involves a construction of the polar set operator by means of a duality operator for convex cones, the polar cone operator. The following technical problem requires attention: all convex sets that occur in applications have two nice properties, closedness and properness, but they might lose these if you work with them. For example $B_1\square B_2$ need not have these properties if B ₁ and B ₂ have them and $\square$ is one of the four binary operations on convex sets; then the rule above might not hold. This rule only holds under suitable assumptions.

Road Map

1. Definition 4.2.1, Theorem 4.2.5, Figs. 4.5, 4.6 (the duality theorem for convex sets, remarks on the two standard proofs, the novel ball throwing proof).

2. Figure 4.7, Definition 4.3.1, Theorem 4.3.3, hint (reformulation duality theorem: supporting hyperplane theorem).

3. Figure 4.8, Definition 4.3.4, Theorem 4.3.6, hint (reformulations duality theorem: separation theorems).

4. Definitions 4.3.7, 4.3.9, Theorem 4.3.11, hint (reformulation duality theorem: theorem of Hahn–Banach).

5. Definition 4.3.12, Theorem 4.3.15, hint, remark (reformulation duality theorem: polar set operation is involution).

6. Figure 4.9, Definition 4.3.16, Theorem 4.3.19, hint, Theorem 4.3.20, hint (reformulations of duality theorem: (1) polar cone operator is involution, (2) polar cone of $C\neq {\mathbb {R}}^n$ is nontrivial; self-duality of the three golden cones).

7. The main idea of the Sects. 4.5 and 4.6 (construction polar set operator by homogenization; pay attention to the minus sign in the bilinear mapping on $X\times {\mathbb {R}}$ and the proof that the ‘type’ of a homogenization of a convex set containing the origin remains unchanged under the polar cone operator).

8. Figure 4.12 (geometric construction of polar set).

9. Propositions 4.6.1, 4.6.2 (calculus rules for computing polar cones).

10. Propositions 4.6.3, 4.6.4 (calculus rules for computing polar sets, take note of the structure of the proof by homogenization).

11. Section 4.8 (Minkowski-Weyl representation, many properties of convex sets simplify for polyhedral sets).

12. Theorem 4.8.2, structure proofs (theorems of the alternative, short proofs thanks to calculus rules).