标签：12 SDP matrix cvx %% problems conic code quad
Navigator
Basic model
LMIs and SDPs with one variable. The generalized eigenvalues of a matrix pair
(
A
,
B
)
(A, B)
(A,B), where
A
,
B
∈
S
n
A, B\in\mathcal{S}^n
A,B∈Sn, are defined as the roots of the polynomial
det
(
λ
B
−
A
)
\det(\lambda B-A)
det(λB−A).
Suppose
B
B
B is nonsingular, and that
A
A
A and
B
B
B can be simltaneously diagonalized by a congruence, i.e., there exists a nonsingular, and that
A
A
A and
B
B
B can simultaneously diagonalized by a congruence, i.e, there exists a nonsingular
R
∈
R
n
×
n
R\in\mathbb{R}^{n\times n}
R∈Rn×n such that
R
T
A
R
=
d
i
a
g
(
a
)
R
T
B
R
=
d
i
a
g
(
b
)
R^TAR=diag(a)\\ R^TBR=diag(b)
RTAR=diag(a)RTBR=diag(b)
where
a
,
b
∈
R
n
a, b\in\mathbb{R}^n
a,b∈Rn.
Moreover, a sufficient condition for this to hold is that there exists
t
1
,
t
2
t_1, t_2
t1,t2 such that
t
1
A
+
t
2
B
≻
0
t_1A+t_2B\succ 0
t1A+t2B≻0.
Try to find the optimal
t
t
t that would maximize
c
∗
t
c*t
c∗t while
A
−
t
∗
B
A-t*B
A−t∗B is still P.S.D, we could solve the following SDP:
min
c
∗
t
s
.
t
.
t
B
≺
A
\min c*t\\ s.t.\quad tB\prec A
minc∗ts.t.tB≺A
cvx code
%% generate input data
rng(1);
n = 4;
A = randn(n); A = 0.5*(A'+A);
B = randn(n); B = B'*B;
c = -1;
%% cvx
cvx_begin sdp
variable t
minimize c*t
t*B<=A;
cvx_end
disp('the optimal t obtained is');
disp(t);
yalmip code
%% yalmip
t = sdpvar;
F = [t*B<=A];
obj = c*t;
optimize(F, obj);
disp('the optimal t obtained is');
disp(value(t));
A simple QP with inequalities constraints
Prove that
x
∗
=
(
1
,
1
/
2
,
−
1
)
x^*=(1, 1/2, -1)
x∗=(1,1/2,−1) is optimal for the optimization problem
min
(
1
/
2
)
x
T
P
x
+
q
T
x
+
r
s
.
t
.
−
1
≤
x
i
≤
1
,
i
=
1
,
2
,
3
\min (1/2)x^TPx+q^Tx+r\\ s.t.\quad -1\leq x_i\leq 1, i=1,2,3
min(1/2)xTPx+qTx+rs.t.−1≤xi≤1,i=1,2,3
where
P
=
[
13
12
−
2
12
17
6
−
2
6
12
]
q
=
[
−
22.0
−
14.5
13.0
]
r
=
1
P=\left[ \begin{matrix} 13 & 12 & -2\\ 12 & 17 & 6\\ -2 & 6 & 12 \end{matrix} \right]\quad q=\left[ \begin{matrix} -22.0\\ -14.5\\ 13.0 \end{matrix} \right]\quad r=1
P=⎣⎡1312−212176−2612⎦⎤q=⎣⎡−22.0−14.513.0⎦⎤r=1
cvx code
%% cvx: QP
cvx_begin
variable x(n)
minimize ((1/2)*quad_form(x, P)+q'*x+r);
x>=-1;
x<=1;
cvx_end
yalmip code
%% yalmip
x = sdpvar(n, 1);
F = [-1<=x<=1];
obj = (1/2)*x'*P*x+q'*x+r
optimize(F, obj);
Reference
Convex Optimiztion S.Boyd Page 203, 215
标签：12,SDP,matrix,cvx,%%,problems,conic,code,quad 来源： https://blog.csdn.net/qq_18822147/article/details/112509339
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