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Unformatted text preview: ve seen several connections between p⋆ and g :
29 • Slater’s condition and strong duality. Slater’s condition is: there exists u ≺ 0 for which
p⋆ (u) < ∞. Strong duality (which follows) is: p⋆ (0) = supλ g (λ). (Note that we include the
condition λ 0 in the domain of g .)
• A global inequality. We have p⋆ (u) ≥ p⋆ (0) − λ⋆T u, for any u, where λ⋆ maximizes g . • Local sensitivity analysis. If p⋆ is diﬀerentiable at 0, then we have ∇p⋆ (0) = −λ⋆ , where λ⋆
maximizes g .
In fact the two functions are closely related by conjugation. Show that
p⋆ ( u ) = ( − g ) ∗ ( − u ) .
Here (−g )∗ is the conjugate of the function −g . You can show this for u ∈ int dom p⋆ .
Hint. Consider the problem minimize f0 (x)
subject to fi (x) = fi (x) − ui ≤ 0, i = 1, . . . , m. Verify that Slater’s condition holds for this problem, for u ∈ int dom p⋆ .
4.8 Exact penalty method for SDP. Consider the pair of primal and dual SD...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
- Fall '13
- The Aeneid