Past exam of the continuous optimisation course of the University of Oxford / 2011 / 1 / iv / Solution

Convert to minimization with $f=-F$ and retain $g\ge 0$. A continuously differentiable inequality augmented Lagrangian method uses the merit functionFor $g_i\le \mu {\lambda }_i$ its contribution is $-{\lambda }_i{g}_i+g_i^2/(2\mu )$; otherwise it is the constant $-\mu {\lambda }_i^2/2$. This shifted truncation permits multipliers to decrease as well as increase while remaining nonnegative.

A useful initial estimate comes from the preceding pure quadratic penalty method at ${\mu }_{-1}$:Here $g(-1,-1)=(3,18,-2)$, so with ${\mu }_{-1}=2$ chooseThen use this estimate and ${\mu }_0=1$ in the first augmented subproblem. Initial estimates are a choice, rather than uniquely specified data; zero multipliers would also define an initialization.