Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Karush-Kuhn-Tucker and duality<br />
Theorem (Karush-Kuhn-Tucker)<br />
If (g, h) satisfies the constra<strong>in</strong>t qualification, then we have strong duality:<br />
Dopt = Popt.<br />
There exist yopt ≥ 0 and zopt, such that Dopt = ℓ(yopt, zopt).<br />
Moreover, xopt is an optimal solution of the primal optimization problem<br />
and (yopt, zopt) is an optimal solution of the dual optimization problem, if<br />
and only if<br />
1 g(xopt) ≤ 0, h(xopt) = 0,<br />
2 yopt ≥ 0 and xopt m<strong>in</strong>imizes L(x, yopt, zopt) over all x ∈ X and<br />
3 〈yopt, g(xopt)〉 = 0.<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 38 / 59
Change language