The conjugate function (Fenchel conjugate)
- Counterpart to Fourier transform in Convex Analysis
The conjugate of f (f not necessarily convex):
f∗(y)=supx∈dom f(yTx−f(x))
The domain of f∗ consists of y∈Rn for which the supermum is finite, i.e. for which the difference yTx−f(x) is bounded above on dom f.
f∗ is convex, since it is the pointwise supremum of a family of affine functions of y.
Examples:
f∗(y)={by=a∞otherwise with dom f∗={a}.
The function xy+logx is unbound above if y≥0 and reaches its max at x=−1/y otherwise.
f∗(y)={1−log(−y)y<0∞otherwise with dom f∗={y|y<0}
Fenchel's inequality
f∗ is convex, since it is the pointwise supremum of a family of affine functions of y.
Examples:
- Affine function. f(x)=ax+b. f∗(y)=yx−ax−b=(y−a)x−b.
f∗(y)={by=a∞otherwise with dom f∗={a}.
- Negative logarithm. f(x)=−logx, with dom f=R++.
The function xy+logx is unbound above if y≥0 and reaches its max at x=−1/y otherwise.
f∗(y)={1−log(−y)y<0∞otherwise with dom f∗={y|y<0}
Fenchel's inequality
Fenchel's inequality can be obtained from the definition of conjugate function
f(x)+f∗(y)≥xTy,∀x,y
since for each y,
f∗(y)≥xTy−f(x)⇔f(x)+f∗(y)≥xTy
since for each y,
f∗(y)≥xTy−f(x)⇔f(x)+f∗(y)≥xTy
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