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2006 Recent Developments in Nonlinear Optimization Theory (slide)
NUS Graduate University of Chinese Academy of Sciences 1
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Recent Developments
in Nonlinear Optimization Theory
Defeng Sun
Department of Mathematics
National University of Singapore
Republic of Singapore
July 9-11, 2006
NUS Graduate University of Chinese Academy of Sciences 2
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2 Variational Analysis on Metric
Projectors Over Closed Convex Sets
Let Z be a finite-dimensional Hilbert vector space equipped with a
scalar product 〈·, ·〉 and its induced norm ‖ · ‖ and D be a
nonempty closed convex set in Z. For any z ∈ Z, let ΠD(z) denote
the metric projection of z onto D:
min
1
2
〈y ? z, y ? z〉
s.t. y ∈ D.
(1)
The operator ΠD : Z → Z is called the metric projection operator
or metric projector over D.
NUS Graduate University of Chinese Academy of Sciences 3
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Proposition 2.1 Let D be a nonempty closed convex set in Z.
Then the point y ∈ D is an optimal solution to (1) if and only if it
satisfies
〈z ? y, d? y〉 ≤ 0 ? d ∈ D . (2)
Proof. “?” Suppose that y ∈ D is an optimal solution to (1). Let
d be an arbitrary point in D. Then yt := (1? t)y + td ∈ D for any
t ∈ [0, 1]. This, together with the fact that y is an optimal solution,
implies that
‖z ? yt‖2 ≥ ‖z ? y‖2 ? t ∈ [0, 1],
which further implies
‖(1? t)(z ? y) + t(z ? d)‖2 ≥ ‖z ? y‖2 ? t ∈ [0, 1].
NUS Graduate University of Chinese Academy of Sciences 4
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Thus,
(t2? 2t)‖z? y‖2 + 2t(1? t)〈z? y, z? d〉+ t2‖z? d‖2 ≥ 0 ? t ∈ [0, 1].
By taking t ↓ 0 and dividing t on both sides of the above equation,
we obtain
?2‖z ? y‖2 + 2〈z ? y, z ? d〉 ≥ 0 ,
which turns into (2).
“?” Suppose that y ∈ D satisfies (2). Assume on the contrary that
y does not solve (1). Then we have by the assumption,
〈z ? y, ΠD(z)? y〉 ≤ 0
and by the sufficiency part,
〈z ?ΠD(z), y ?ΠD(z)〉 ≤ 0 .
NUS Graduate University of Chinese Academy of Sciences 5
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Summing up the above two inequalities leads to
〈ΠD(z)? y, ΠD(z)? y〉 ≤ 0 .
This implies that y = ΠD(z). The contradiction shows that y solves
(1). ¤
Note that Proposition 2.1 holds even if Z is infin
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