Two-component design with an active demand constraint

At $x_0$, $\mathrm{\nabla }f(x_0)=0$ and the linearized inequality is $1-p_1-p_2\le 0$. The unconstrained minimizer $p=0$ is infeasible. On the active face $p_1+p_2=1$, stationarity for $q(p)=\frac{1}{2}\mathrm{\parallel }p{\mathrm{\parallel }}_2^2$ is $p-\lambda (1,1{)}^T=0$; combining with the face equation gives $p_1=p_2=\lambda =1\mathrm{/}2$. This point is feasible with nonnegative multiplier.

The positive-definite QP has a unique minimizer; its feasible KKT tuple passes the active-set acceptance filters in [F2]. By [F3] the returned step is exactly $(1\mathrm{/}2,1\mathrm{/}2)$.

The nonlinear point $x_{\ast }=(1\mathrm{/}2,1\mathrm{/}2)$ is feasible and has $\mathrm{\nabla }f(x_{\ast })=(1\mathrm{/}2,1\mathrm{/}2)$, $\mathrm{\nabla }g(x_{\ast })=(-1,-1)$, and multiplier ${\lambda }_{\ast }=1\mathrm{/}2$. Thus stationarity holds, the multiplier is nonnegative, and complementarity holds with $g(x_{\ast })=0$. Since the objective is convex and the constraint affine, [F4] proves global optimality; strict convexity of the objective makes the minimizer unique.