Basic model

LMIs and SDPs with one variable. The generalized eigenvalues of a matrix pair (A,B)(A, B)(A,B), where A,B∈SnA, B\in\mathcal{S}^nA,B∈Sn, are defined as the roots of the polynomial det⁡(λB−A)\det(\lambda B-A)det(λB−A).
Suppose BBB is nonsingular, and that AAA and BBB can be simltaneously diagonalized by a congruence, i.e., there exists a nonsingular, and that AAA and BBB can simultaneously diagonalized by a congruence, i.e, there exists a nonsingular R∈Rn×nR\in\mathbb{R}^{n\times n}R∈Rn×n such that
RTAR=diag(a)RTBR=diag(b)R^TAR=diag(a)\\ R^TBR=diag(b) RTAR=diag(a)RTBR=diag(b)
where a,b∈Rna, b\in\mathbb{R}^na,b∈Rn.
Moreover, a sufficient condition for this to hold is that there exists t1,t2t_1, t_2t1,t2 such that t1A+t2B≻0t_1A+t_2B\succ 0t1A+t2B≻0.

Try to find the optimal ttt that would maximize c∗tc*tc∗t while A−t∗BA-t*BA−t∗B is still P.S.D, we could solve the following SDP:
min⁡c∗ts.t.tB≺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 sdpvariable tminimize c*tt*B<=A;
cvx_enddisp('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)xTPx+qTx+rs.t.−1≤xi≤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=[1312−212176−2612]q=[−22.0−14.513.0]r=1P=\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_beginvariable 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

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【CVX】SDP and conic form problems

Navigator

Basic model

cvx code

yalmip code

A simple QP with inequalities constraints

cvx code

yalmip code

Reference

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