Dualwert

Dualwert is a term used in theoretical discussions of duality to denote a paired value that combines a primal quantity with its dual counterpart. In this sense, a dualwert summarizes both the original measure and the sensitivity information that accompanies it, often within optimization, economics, and functional analysis.

Etymology: The word combines "dual" with "wert," the German word for value, reflecting the idea of a value and its dual representation.

Formal framework: Let V be a vector space and V* its dual, with a bilinear pairing ⟨v, φ⟩. A dualwert is a pair (v, φ) with v ∈ V and φ ∈ V* that are compatible under this pairing and record both the primal quantity v and a dual measure φ. In optimization, v may be a primal solution and φ a dual variable; together they convey objective value and marginal information.

Properties: Dualwerts emphasize the link between primal and dual descriptions, reflecting optimality through conditions such as complementarity and stability under perturbations. They are useful for sensitivity analysis because φ encodes marginal values or shadow prices associated with constraints.

Examples and applications: In linear programming, the optimal primal solution x* and the optimal dual solution y* form a canonical dualwert for the problem. In convex analysis, the gradient of a convex function or a subgradient offers a dualwert representation of local behavior via the conjugate function.

Related concepts include duality, shadow price, conjugate function, and sensitivity analysis.