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IE598NH-lecture-18-Stochastic Variational Inequalities.pdf
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IE598NH-lecture-18-Stochastic Variational
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Stochastic Variational Inequalities
Presenter: Peijun Xiao
Department of Industrial and Enterprise Systems Engineering (ISE)
University of Illinois at Urbana-Champaign (UIUC)
IE598 Fall 2020 Course Project
April 2, 2020
Start with Optimality Conditions
I
Consider minimizing a smooth convex function f(x) over a compact set C,
min
x∈C
f(x).
I
The feasible set at x
∗
is S
feasible
(x
∗
) = {s ∈ R
n
| s := x
0
− x
∗
, x
0
∈ C}
I
The descent set at x
∗
is S
descent
(x
∗
) = {s ∈ R
n
| hs, ∇f (x
∗
)i < 0}
I
A point x
∗
is optimal if x
∗
∈ C and S
feasible
(x
∗
) ∩ S
descent
(x
∗
) = ∅, i.e.
h∇f(x
∗
), x
0
− x
∗
i ≥ 0, ∀x
0
∈ C
I
This is equivalent to say
∇f(x
∗
) ∈ N
C
(x
∗
) = {h ∈ R
n
| h
T
(x
0
− x
∗
) ≥ 0, ∀x
0
∈ C}. (1)
Introduction to Variational Inequalities 4 / 47
Start with Optimality Conditions
I
Consider minimizing a smooth convex function f(x) over a compact set C,
min
x∈C
f(x).
I
The feasible set at x
∗
is S
feasible
(x
∗
) = {s ∈ R
n
| s := x
0
− x
∗
, x
0
∈ C}
I
The descent set at x
∗
is S
descent
(x
∗
) = {s ∈ R
n
| hs, ∇f (x
∗
)i < 0}
I
A point x
∗
is optimal if x
∗
∈ C and S
feasible
(x
∗
) ∩ S
descent
(x
∗
) = ∅, i.e.
h∇f(x
∗
), x
0
− x
∗
i ≥ 0, ∀x
0
∈ C
I
This is equivalent to say
∇f(x
∗
) ∈ N
C
(x
∗
) = {h ∈ R
n
| h
T
(x
0
− x
∗
) ≥ 0, ∀x
0
∈ C}. (1)
Introduction to Variational Inequalities 4 / 47
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