Convex Analysis is the calculus of inequalities while Convex Optimization is its application.

For example, myriad alternative systems of linear inequality can be explained in terms of pointed closed convex cones and their duals.

As another example, the general first-order necessary and sufficient condition for optimality of a solution to a minimization problem, with real differentiable convex objective function, can be expressed as an inequality over a convex feasible set.

For one last example from an infinitude, the Schoenberg criterion for determining existence of a Euclidean distance matrix is accurately interpreted as a discretized membership relation (a generalized inequality) between the EDM cone and its ordinary dual.

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