Convex Analysis & Optimization Structures

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Abstractions that formalize convexity for use in optimization, including convex sets and their generating operations such as the convex hull, absorbing sets and supporting hyperplanes, dual and conjugate constructions like the convex conjugate, subderivative and proximal operator, and hard-constraint representations such as indicator functions and linear matrix inequalities.

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Absorbing set — A subset of a vector space whose scalar dilations eventually contain every vector, forming a basic neighborhood and boundedness concept in topological vector spaces.
• Convex conjugate — The supremum transform mapping an extended-real function on a vector space to the greatest affine lower-bound gap over its dual space.
• Convex hull — The smallest convex set containing a given set, equivalently all finite convex combinations of its points.
• Indicator function (convex analysis) — An extended-real function that is zero on a selected set and positive infinity outside it, encoding membership as a hard optimization constraint.
• Linear matrix inequality — A convex constraint requiring an affine combination of symmetric or Hermitian matrices to be positive semidefinite.
• Proximal operator — The operator mapping a point to the unique minimizer of a function plus one-half the squared distance to that point, under standard proper lower-semicontinuous convex assumptions.
• Subderivative — A supporting slope or covector that generalizes the derivative of a convex function at a point where an ordinary derivative may not exist.
• Supporting hyperplane — A hyperplane meeting a set while the entire set lies in one of the two closed half-spaces it bounds.