Extended-real convex functions

Here a proper function maps $\mathbb R^d$ into $\mathbb R\cup\{+\infty\}$, is finite somewhere, and never takes negative infinity. Its effective domain is the set where its value is finite. Calling a function proper does not mean that it is bounded below. The affine function f(x)=x is proper but has infimum negative infinity. Likewise, properness alone says nothing about continuity, closedness, or whether an infimum is attained. These are additional properties that must be checked rather than inferred from a familiar-looking formula.

For a nonempty set C, define its extended-value indicator to be zero on C and positive infinity outside. Minimizing h over C becomes minimizing h plus this indicator over the whole space, provided h is finite on C or their common domain is explicitly identified. The sum of two proper functions can fail to be proper when their domains do not meet. With the present convention there is no negative-infinity value to combine with positive infinity, but expressions involving subtraction of two infinite objective values remain undefined.

An extended-real function is convex when its epigraph is convex, equivalently when the convexity inequality holds for interior mixture weights and finite-domain endpoints. Checking finite endpoints is sufficient because a positive multiple of infinity makes the right side infinite otherwise. Endpoint weights zero and one can be handled separately, avoiding any need to interpret zero times infinity. The inequality forces the effective domain to be convex. For an indicator function, that condition is also sufficient: indicators are convex exactly when their allowed sets are convex.

A Hessian test addresses only the smooth interior of a domain. It does not independently prove that the entire extended-value model is convex. For instance, assigning x² to two disjoint intervals and infinity elsewhere produces positive curvature wherever differentiable, but the midpoint of two finite points can be forbidden. Conversely, a convex indicator has no useful ordinary derivative across its boundary and can still be a perfectly valid convex objective. Domain geometry and finite-value inequalities should therefore come before derivative-based recognition rules.

Adding a convex indicator to a convex loss keeps the combined objective convex whenever their common domain is nonempty. Composing a convex extended-real function with an affine map also preserves the inequality, although the new domain can be empty. A resource constraint Ax=b becomes the indicator of an affine inverse image. A nonnegative-variable restriction becomes the indicator of an orthant. Such notation makes later subgradient and conjugate calculations uniform, but the modeler must still verify that at least one candidate has finite total cost.

Convexity implies that every local minimizer relative to the domain is global. It does not imply a minimizer exists: exp(x) has infimum zero on the real line but never reaches it. Strict convexity gives at most one minimizer, still without guaranteeing existence. Finally, an open domain can prevent attainment even for a simple linear loss. The next lesson adds lower semicontinuity and compactness conditions that control these issues. Keeping the concepts separate makes it possible to diagnose a modeling failure instead of blaming an optimization routine for a mathematically nonexistent answer.