On the stationarity for nonlinear optimization problems with polyhedral constraints

For polyhedral constrained optimization problems and a feasible point x, it is shown that the projection of the negative gradient on the tangent cone, denoted del(Omega) f(x), has an orthogonal decomposition of the form beta(x) + phi(x). At a stationary point, del(Omega) f(x) = 0 so parallel to del(Omega) f(x)parallel to reflects the distance to a stationary point. Away from a stationary point, parallel to beta(x)parallel to and parallel to phi(x)parallel to measure different aspects of optimality since beta(x) only vanishes when the KKT multipliers at x have the correct sign, while phi(x) only vanishes when x is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between parallel to beta(x)parallel to and parallel to phi(x)parallel to.