Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Upper and lower bounds for convex programs<br />
Lower bound on optimal value<br />
Let x ∈ S. Then for arbitrary y ≥ 0 and z we have<br />
L(x, y, z) := f(x) + 〈y, g(x)〉 + 〈z, h(x)〉 ≤ f(x)<br />
and, <strong>in</strong> particular,<br />
so that<br />
ℓ(y, z) := <strong>in</strong>f L(x, y, z) ≤ <strong>in</strong>f L(x, y, z) ≤ <strong>in</strong>f f(x) = Popt.<br />
x∈X x∈S x∈S<br />
Dopt := sup ℓ(y, z) = sup<br />
y≥0, z<br />
y≥0, z<br />
def<strong>in</strong>es a lower bound for Popt.<br />
<strong>in</strong>f<br />
x∈X<br />
L(x, y, z) ≤ Popt<br />
Siep Weiland and Carsten Scherer (DISC) <strong>L<strong>in</strong>ear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> <strong>in</strong> <strong>Control</strong> Class 1 34 / 59