Convex Functions: Basic Properties

Abstract

• Why. A second main object of convex analysis is a convex function. Convex functions can help to describe convex sets: these are infinite sets, but they can often be described by a formula for a convex function, so in finite terms. Moreover, in many optimization applications, the function that has to be minimized is convex, and then the convexity is used to solve the problem.

• What. In the previous chapters, we have invested considerable time and effort in convex sets and convex cones, proving all their standard properties. Now there is good news. No more essentially new properties have to be established in the remainder of this book. It remains to reap the rewards. In Chaps. 5 and 6, we consider to begin with convex functions: the dual properties in Chap. 6, and the properties that do not require duality—the primal properties in this chapter. This requires convex sets: convex functions are special functions on convex sets. Even better, they can be expressed entirely in terms of convex sets: convex functions are functions for which the epigraph, that is, the region above the graph, is a convex set. Some properties of convex functions follow immediately from a property of a convex set, applied to the epigraph. An example is the continuity property of convex functions. For other properties, a deeper investigation is required: one has to take the homogenization of the epigraph, and then one should apply a property of convex cones. An example is the unified construction of the eight standard binary operations on convex functions—sum, maximum, convex hull of the minimum, infimal convolution, Kelley’s sum, and three nameless ones—by means of the homogenization method. The defining formulas for them look completely different from each other, but they can all be generated in exactly the same systematic way by a reduction to convex cones (‘homogenization’). One has for convex functions the same technical problem as for convex sets: all convex functions that occur in applications have the nice properties of being closed and proper, but if you work with them and make new functions out of them, then they might lose these properties. Two examples of such work are: (1) the consideration, in a sensitivity analysis for an optimization problem, of the optimal value function, and (2) the application of a binary operation. Finally, twice continuously differentiable functions f(x) are convex iff their Hessian f ⁽²⁾ is positive semidefinite for all x.

Road Map

1. Definitions 5.2.1, 5.2.4, 5.2.6, 5.2.9 and 5.2.10 (convex function, defined either by Jensen’s inequality or by the convexity of a set: its (strict) epigraph).

2. Proposition 5.3.1 (continuity property of a convex function).

3. Propositions 5.3.6, 5.3.9 (first and second order characterizations of convex functions).

4. Figure 5.7 and Definition 5.4.2 (construction convex function by homogenization).

5. Definitions 5.2.12, 5.3.4 (two nice properties for convex functions, closedness and properness).

6. Definitions 5.5.1, 5.5.2 (the two actions by linear functions on convex functions).

7. Figure 5.8, Definition 5.6.1, Proposition 5.6.6 (binary operations on convex functions—pointwise sum, pointwise maximum, convex hull of pointwise minimum, infimal convolution, Kelley’s sum and three nameless ones—defined by formulas and constructed by homogenization).