标签：psi log barrier min sum point ALGO Interior Methods
Navigator
I.P.M.
Reformulate the problem by replacing the constraint with a penalty term in the objective function
min
x
f
(
x
)
+
c
P
(
x
)
\min_x f(x)+cP(x)
xminf(x)+cP(x)
Barrier Method
If we define a barrier function
B
(
x
)
B(x)
B(x) that is continuous, and
B
(
x
)
→
∞
B(x)\to\infty
B(x)→∞ as
x
x
x approaches the boundary of
X
X
X, then the minimization problem can be expressed as:
min
x
f
(
x
)
+
1
t
B
(
x
)
s
.
t
.
x
∈
i
n
t
X
\min_x f(x)+\frac{1}{t}B(x)\\ s.t.\quad x\in int X
xminf(x)+t1B(x)s.t.x∈intX
To solve constrained problems by looking at convex optimization problems in standard form:
min
x
f
(
x
)
s
.
t
.
{
g
i
(
x
)
≤
0
i
=
1
,
2
,
…
,
m
A
x
=
b
\min_x f(x)\\ s.t. \begin{cases} g_i(x)\leq 0\quad i=1,2,\dots, m\\ Ax=b \end{cases}
xminf(x)s.t.{gi(x)≤0i=1,2,…,mAx=b
One of the most utilized approaches is the logarithmic barrier function
ϕ
(
x
)
=
−
∑
i
=
1
m
log
(
−
g
i
(
x
)
)
\phi(x)=-\sum_{i=1}^m\log(-g_i(x))
ϕ(x)=−i=1∑mlog(−gi(x))
And the original problem can be approximated by
min
x
f
(
x
)
+
1
t
(
−
∑
i
=
1
m
log
(
−
g
i
(
x
)
)
)
s
.
t
.
A
x
=
b
\min_x f(x)+\frac{1}{t}(-\sum_{i=1}^m\log(-g_i(x)))\\ s.t.\quad Ax=b
xminf(x)+t1(−i=1∑mlog(−gi(x)))s.t.Ax=b
A function
ψ
(
x
)
\psi(x)
ψ(x) is a generalized logarithm for a proper cone
K
K
K if
ψ
(
x
)
\psi(x)
ψ(x) is concave, closed, twice continuously differentiable, and
∇
2
ψ
(
x
)
≺
0
\nabla^2\psi(x)\prec0
∇2ψ(x)≺0 for
x
∈
i
n
t
K
x\in int K
x∈intK with its domain as the interior of
K
K
K; and there exists
θ
>
0
\theta>0
θ>0 such that for all
x
∈
K
x\in K
x∈K and
s
>
0
s>0
s>0
ψ
(
x
)
=
∑
i
=
1
n
log
(
x
i
)
\psi(x)=\sum_{i=1}^n\log(x_i)
ψ(x)=i=1∑nlog(xi)
Furthermore, a generalized logarithm for the second-order cone expressed as
{
x
∈
R
n
∣
∑
i
=
1
n
−
1
x
i
2
≤
x
n
}
\bigg\{ x\in\mathbb{R}^n\bigg|\sqrt{\sum_{i=1}^{n-1}x_i^2\leq x_n} \bigg\}
{x∈Rn∣∣∣∣i=1∑n−1xi2≤xn
}
becomes
ψ
(
x
)
=
log
(
x
n
2
−
∑
i
=
1
n
−
1
x
i
2
)
\psi(x)=\log(x_n^2-\sum_{i=1}^{n-1}x_i^2)
ψ(x)=log(xn2−i=1∑n−1xi2)
and the logarithmic barrier function for the SOCP is
ϕ
(
x
)
=
−
∑
i
=
1
m
log
(
(
c
i
′
x
+
d
i
)
2
−
∥
A
i
x
+
b
i
∥
2
2
)
\phi(x)=-\sum_{i=1}^m\log((c_i'x+d_i)^2-\lVert A_ix+b_i\rVert_2^2)
ϕ(x)=−i=1∑mlog((ci′x+di)2−∥Aix+bi∥22)
A generalized algorithm for a positive semidefinite cone
X
X
X can be defined as
Ψ
(
X
)
=
−
log
(
det
(
F
0
+
x
1
F
1
+
⋯
+
x
n
F
n
)
)
\Psi(X)=-\log(\det(F_0+x_1F_1+\dots+x_nF_n))
Ψ(X)=−log(det(F0+x1F1+⋯+xnFn))
We can find the optimal solution of robust problems formulated as SOCP or SDP by iteratively solving the following problem for tracing the central path of
x
x
x:
min
x
f
(
x
)
+
1
t
ϕ
(
x
)
s
.
t
.
A
x
=
b
\min_x f(x)+\frac{1}{t}\phi(x)\\ s.t.\quad Ax=b
xminf(x)+t1ϕ(x)s.t.Ax=b
where these barrier parameter value of
t
t
t is increased each iteration, and the optimal point for a given
t
t
t is computed using the optimal
x
x
x from the previous iteration as the starting point.
Primal-Dual Interior point method
Primal-Dual is actually known to have better performance for problems in SOCP and SDP compared to barrier methods. Barrier methods trace the central path for the primal variable by optimizing for x t x_t xt while increasing the value of t t t. Primal-dual interior point methods, on the other habd, find the primal and dual central paths together.
标签：psi,log,barrier,min,sum,point,ALGO,Interior,Methods 来源： https://blog.csdn.net/qq_18822147/article/details/112070075
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