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Review of duality so far
- LP/QP duality, cone duality, set duality
- All are halfspace bounds
Review of duality so far LP/QP duality, cone duality, set duality - - PowerPoint PPT Presentation
Review of duality so far LP/QP duality, cone duality, set duality - - PowerPoint PPT Presentation
Review of duality so far LP/QP duality, cone duality, set duality All are halfspace bounds on a cone on a set on objective of LP/QP Set duality Set duality LP/QP objective min z s.t. z x - 1 z 3 - 2x Dual
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Set duality
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LP/QP objective
min z s.t. z ≥ x - 1 z ≥ 3 - 2x
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Dual functions
- Arbitrary function F(x)
- Dual is F*(y) =
- For example: F(x) = xTx/2
- F*(y) =
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Fenchel’s inequality
- F*(y) = supx [xTy - F(x)]
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Duality and subgradients
- Suppose F(x) + F*(y) – xTy = 0
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Duality examples
- 1/2 – ln(-x)
- ex
- x ln(x) – x
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More examples
- F(x) = xTQx/2 + cTx, Q psd:
- F(X) = –ln |X|, X psd:
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Indicator functions
- Recall: for a set S,
IS(x) =
- E.g., I[-1,1](x):
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Duals of indicators
- Ia(x), point a:
- IK(x), cone K:
- IC(x), set C:
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Properties
- F(x) ≥ G(x)
F*(y) G*(y)
- F* is closed, convex
- F** = cl conv F (= F if F closed, convex)
- If F is differentiable:
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Working with dual functions
- G(x) = F(x) + k
- G(x) = k F(x)
k > 0
- G(x) = F(x) + aTx
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Working with dual functions
- G(x1, x2) = F1(x1) + F2(x2)
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An odd-looking operation
- Definition: infimal convolution
- E.g., F1(x) = I[-1,1](x), F2(x) = |x|
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Infimal convolution example
- F1(x) = I≤0(x), F2(x) = x2
F1 F2 F1 F2
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Dual of infimal convolution
- G(x) = F1(x) F2(x)
- G*(y) =
- G(x) = F1(x) + F2(x)
G*(y) =
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Convex program duality
- min f(x) s.t.
Ax = b gi(x) ≤ 0 i ∈ I
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Duality example
- min 3x s.t. x2 ≤ 1
- L(x, y) = 3x + y(x2 – 1)
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Dual function
- L(y) = infx L(x,y) = infx 3x + y(x2 – 1)
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