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SECTION 04 / 8

Why the biconjugate representation matters

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Figure 37. Biconjugate identity.
Figure 37. Biconjugate identity.
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Illustration: Why the biconjugate representation matters

For a proper, convex, lower-semicontinuous function f, the biconjugate identity is

f(u)=sup⁡t{tu−f∗(t)}f(u)=\sup_t\{tu-f^*(t)\}

This converts an expression involving f into an optimization over a scalar t. The conditions matter: the identity is not valid without qualification for every arbitrary function.

Worked example

Recover the quadratic

  1. Take

    f∗(t)=t22f^*(t)=\frac{t^2}{2}

  2. For a fixed u, maximize tu−t22tu-\frac{t^2}{2} over t.

  3. Derivative with respect to t is u−tu-t, so t=ut=u.

  4. The maximum is

    u2−u22=u22=f(u)u^2-\frac{u^2}{2}=\frac{u^2}{2}=f(u)
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