Operations That Preserve Convexity

Certify convexity by decomposition — sums, maxima, affine maps, and perspectives, never a Hessian.

The idea

The convexity-preserving operations are the standard method for proving a function convex: exhibit it as known convex pieces combined by operations that preserve convexity. Checking the definition directly, or forming a Hessian and testing it for positive semidefiniteness, is impractical in more than a few variables, and impossible when the function is built from a maximum or an infimum and need not be differentiable at all. This mirrors how we differentiate $e^{\sin x}$: not from the limit defining the derivative, but by the chain rule applied to known pieces.

The starting pieces are the functions already known to be convex: affine functions $a^{T}x + b$, every norm $\|x\|$, $e^{x}$, $x^{2}$, and $-\log x$ on $x > 0$. The operations include adding with nonnegative weights, composing with an affine map, and taking a pointwise maximum. Each is a theorem in its own right, and once proved it never has to be proved again.

To certify a function, read it from the outside in, so that each layer costs exactly one rule, and check each rule's hypotheses, because several are one-directional: the composition rule depends on whether the outer function is increasing or decreasing, and returns the wrong verdict if that condition is skipped.

Ways to work on it

• Walkthrough. The rule list, applied layer by layer to certify one concrete function.
• Proof. Why a pointwise maximum is convex: the epigraph of a max is the intersection of the epigraphs.
• Practice. Certify functions and sets, and spot the combinations that carry no guarantee.
• Hardest. Certify convexity of a function that the basic rules alone cannot reach.

Not sure where to start? Take the ten-question placement test.