Operations Preserving Convexity

Optimization · Axiom Academy

LESSON Operations Preserving Convexity Learn how to build new convex functions from existing ones using fundamental composition rules If f and g are convex functions and α, β are nonnegative scalars, then the weighted sum αf + βg is also convex. 2. Composition with Affine Mapping If f is convex, then the composition f(Ax + b) is also convex, where A is a matrix and b is a vector. This means we can apply affine transformations (linear transformations plus translations) to the argument of a convex function without losing convexity. The pointwise maximum of any collection of convex functions is convex. If f₁, f₂, ..., fₙ are convex, then h(x) = max f₁(x), f₂(x), ..., fₙ(x) is convex. This property is powerful because it works for arbitrarily many functions, even infinitely many! For compositions g(f(x)), convexity depends on the properties of both functions: If g is convex and nondecreasing, and f is convex, then g(f(x)) is convex If g is convex and nonincreasing, and f is concave, then g(f(x)) is convex Example: e^f(x) is convex when f is convex (since e^x is convex and nondecreasing). If f(x, y) is convex in (x, y), then the partial minimization g(x) = inf_y f(x, y) is convex in x. This means we can eliminate variables through optimization while preserving convexity, which is crucial for many optimization algorithms and duality theory.

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