Convexalter

Convexalter is a term used in mathematical optimization and convex analysis to describe a family of operations or algorithms that aim to increase, enforce, or exploit convexity in a problem. It is not a standard term with a single universally accepted definition, but it is commonly understood as any procedure that transforms a given function or set into a form with stronger convexity properties, thereby improving the tractability of optimization tasks.

Definition and interpretations can vary by context. In one view, a convexalter is an operator that maps a function f: X -> R, defined on a convex set X, to a convex function C(f) on X. Two principal interpretations frequently appear: convexification and convex regularization. Convexification typically refers to forming the convex envelope conv(f), the greatest convex function that lies below f. Convex regularization adds a term such as f_r(x) = f(x) + (mu/2)||x||^2 to induce or enhance strong convexity. Some frameworks combine both approaches, producing a function that is easier to optimize while preserving key properties of the original problem.

Properties and limitations are context-dependent. Commonly cited features include preservation of minima under certain conditions, monotonicity with respect to pointwise order, and compatibility with subdifferential calculus. However, the extent to which a convexalter preserves the original problem’s solutions varies with the method used.

Applications span reformulating non-convex programs into convex relaxations in operations research and computer vision, and providing theoretical tools in machine learning to analyze training dynamics or derive convergence bounds. The term is often used descriptively rather than as a formal axiom, serving as a umbrella for convexity-enhancing techniques.