Legendre TransformEdit

The Legendre transform is a fundamental construction in mathematics that creates a bridge between a function and a dual description of the same problem. By re-expressing a function in terms of its slopes or rates of change, it helps translate optimization and energy problems into a dual setting where constraints or natural variables become more transparent. It is widely used across physics, engineering, economics, and pure mathematics, and its power comes from clean dual relationships that preserve convexity and connect primal and dual formulations.

In its most common form, the Legendre transform converts a real-valued function f on a vector space to a new function f* defined by a supremum over linear probes. For a function f: R^n → R, the transform is f*(y) = sup_x { ⟨x, y⟩ − f(x) }. When f is differentiable and strictly convex, this transform is an involution on its domain: applying it twice recovers the original function, up to technical conditions. The transform thus establishes a one-to-one correspondence between points in the graph of f and the slopes of f, and it underpins the powerful duality that appears in optimization theory and physics.

Definition

For a function f: R^n → R that is proper, convex, and lower semicontinuous, its Legendre–Fenchel transform (often simply called the Legendre transform) is f*(y) = sup_x { ⟨x, y⟩ − f(x) }.
The domain of f* consists of those y for which there exists at least one x with ⟨x, y⟩ − f(x) finite.
If f is differentiable and strictly convex, then the gradient map ∇f maps x to y = ∇f(x), and the inverse relationship holds: x = ∇f*(y). This ties the geometry of f to that of f*.

In one dimension, the transform has a familiar interpretation: f*(p) measures the maximum difference between a linear function with slope p and the original curve f. The inequality f(x) + f*(p) ≥ x p, known as Young’s inequality, becomes an equality precisely at the x where p = f′(x) when the derivative exists.

Properties

Convexity preservation: If f is convex, then f* is convex; conversely, the biconjugate f** equals the lower semicontinuous convex hull of f.
Involution under regularity: If f is closed and convex, then (f*)* = f.
Dual relationships: If y ∈ ∂f(x) (i.e., y is a subgradient of f at x), then x ∈ ∂f*(y). When f is differentiable, this reduces to y = ∇f(x) and x = ∇f*(y).
Homogeneity and translation: Scaling f by a positive factor or translating f by an affine function induces corresponding transformations of f*.

These properties make the Legendre transform a natural tool for deriving dual problems in optimization, where a primal problem minimizing f(x) becomes a dual problem maximizing −f*(y) subject to consistency with the primal variables.

Examples

Quadratic example: If f(x) = (1/2) x^2, then f*(y) = (1/2) y^2. The transform preserves the form of simple quadratic energy terms, reflecting the self-duality of the standard Euclidean structure.
Linear-quadratic combination: For f(x) = ax + (1/2) b x^2 with b > 0, the Legendre transform yields f*(y) = (1/2b) (y − a)^2, illustrating how affine shifts translate into shifts in the dual coordinates.

Historical context

The transform bears the name of Adrien-Marie Legendre, who developed ideas around transforming problems by exchanging variables and their conjugate quantities in the early 19th century. It later found a central role in the calculus of variations and the development of convex analysis. The broader convex-dual framework that includes the Legendre–Fenchel transform was developed in the 20th century, formalizing how duality can be exploited in optimization problems and in the study of energy functions in physics.

Applications

In physics and thermodynamics

Potentials and natural variables: In thermodynamics, the internal energy U(S, V) can be transformed into potentials with different natural variables by Legendre transforms. For example, the Helmholtz free energy F(T, V) arises from U by transforming with respect to entropy S to switch from the variable S to the temperature T. The Gibbs free energy G(T, P) results from further transforming with respect to pressure P. These transforms explain why certain variables become the natural descriptors of equilibrium states in different thermodynamic ensembles.
Hamiltonian mechanics: In classical mechanics, the Legendre transform connects the Lagrangian L(q, q̇) and the Hamiltonian H(q, p) through the relation p = ∂L/∂q̇ and H(q, p) = sup_{q̇} [ p q̇ − L(q, q̇) ]. The Legendre transform thus provides the bridge between velocity-based and momentum-based formulations.

In optimization and economics

Duality and optimization: Many optimization problems admit a dual formulation in which the Legendre transform appears naturally. This can yield sharper insights, bounds, and efficient algorithms, especially when the primal problem is difficult to solve directly.
Economic interpretation: In convex analysis terms, the Legendre transform relates cost-like and value-like descriptions. The dual perspective can illuminate how changes in marginal prices or marginal costs propagate through a system, which is a central theme in optimization-based economic modeling.

In mathematics

Convex analysis and duality theory: The Legendre transform is a cornerstone of convex analysis, linking a function to its convex conjugate and enabling a rigorous treatment of dual problems, subgradients, and variational inequalities.
Generalizations: The Legendre–Fenchel transform extends the idea to non-differentiable convex functions and higher-dimensional settings. Moreo–Yosida regularization and related constructs use similar duality principles to smooth or approximate nonsmooth energies.

Generalizations

Legendre-Fenchel transform: This is the broad, nondifferentiable version of the Legendre transform, defined for a wide class of functions and playing a central role in modern convex analysis and optimization.
Multidimensional and geometric extensions: The same duality ideas extend to more complex geometric settings, including manifolds and convex bodies, where duality relations encode relationships between support functions and gauge functions.