where the right-hand side is the Lagrangian upper bound associated with the dual vector \(y\) (the middle inequality holds because the multipliers are non-negative, and \(x^{\ast}\ge0\)).
More generally, when the inequality constraint matrix is a general matrix \(G\) instead of the identity (and the equality constraints are \(Ex = 0\)), we have (Peña and Soheili's)