Linear Matrix Inequalities in Control

Upper and lower bounds for convex programs
Lower bound on optimal value
Let x ∈ S. Then for arbitrary y ≥ 0 and z we have
L(x, y, z) := f(x) + 〈y, g(x)〉 + 〈z, h(x)〉 ≤ f(x)
and, in particular,
so that
ℓ(y, z) := inf L(x, y, z) ≤ inf L(x, y, z) ≤ inf f(x) = Popt.
x∈X x∈S x∈S
Dopt := sup ℓ(y, z) = sup
y≥0, z
y≥0, z
defines a lower bound for Popt.
inf
x∈X
L(x, y, z) ≤ Popt
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 34 / 59