Convex Functions: Dual Description

Abstract

• Why. The phenomenon duality for a proper closed convex function, the possibility to describe it from the outside of its epigraph, by graphs of affine (constant plus linear) functions, has to be investigated. This has to be done for its own sake and as a preparation for the duality theory of convex optimization problems. An illustration of the power of duality is the following task, which is challenging without duality but easy if you use duality: prove the convexity of the main function from geometric programming, $\ln (e^{x_1}+\cdots +e^{x_n})$, a smooth function that approximates the non-smooth function $\max (x_1, \ldots , x_n)$.

• What. The duality for convex functions can be formulated efficiently by the property of a duality operator on convex functions that have the two nice properties (proper and closed), the conjugate function operator f↦f ^∗: this is an involution. For each one of the eight binary operations $\square$ on convex functions, one has a rule of the type $(f\square g)^*= f^*\odot g^*$ where ⊙ is another one of the eight binary operations on convex functions. Again, homogenization generates a unified proof for these eight rules. This requires the construction of the conjugate function operator by means of a duality operator for convex cones (the polar cone operator). There is again a technical complication due to the fact that the rules only hold if all convex functions involved, $f, g, f\square g$ and f ^∗⊙ g ^∗, have the two nice properties.

Even more important, for applications, is that one has, for the two binary operations +  and $\max$ on convex functions, a rule for subdifferentials: the theorem of Moreau-Rockafellar ∂(f ₁ + f ₂)(x) = ∂f ₁(x) + ∂f ₂(x) and the theorem of Dubovitskii-Milyutin $\partial \max (f_1,f_2)(x)=\partial f_1(x) \mathrm {co}\cup \partial f_2(x)$ in the difficult case f ₁(x) = f ₂(x). Homogenization generates a unified proof for these two rules. Again, these rules only hold under suitable assumptions.

Again, there is no need to prove something new: all duality results for convex functions follow from properties of convex sets (their epigraphs) or of convex cones (their homogenizations).

Road Map

1. Figure 6.1, Definition 6.2.3, (conjugate function).

2. Figure 6.2 (dual norm is example of either conjugate function or duality operator by homogenization).

3. Theorem 6.4.2 and structure proof (duality theorem for proper closed convex functions: conjugate operator is an involution, proof by homogenization).

4. Propositions 6.5.1, 6.5.2 (calculus rules for computation conjugate functions).

5. Take note of the structure of Sect. 6.5 (duality between convex sets and sublinear functions; this is a preparation for the proof of calculus rules for subdifferentials).

6. Definitions 6.7.1, 6.7.2, Propositions 6.7.8–6.7.11 (subdifferential convex function at a point, nonemptiness and compactness, at points of differentiability, Fenchel-Moreau, Dubovitskii-Milyutin, subdifferential norm).