Approximate KKT Points for Iterates of an Optimizer

In this technical note, we suggest a new definition for an approximate KKT point. The concept of approximate KKT point can then be used on iterates (points found by an optimization algorithm) to check whether the iterates lead to a KKT point. We will concentrate on the following simple optimization problem (P): Min f(x) subject to gi(x) ≤ 0, i = 1, 2, . . . ,m. (1) We will assume that f and each gi (i = 1, 2, . . . ,m) are smooth functions (see, for example, [1] for more details on problem (P)). Our main aim in this section is to define certain notions of approximate KKT points and show that if a sequence of such points converge to a point where the constraints satisfy some constraint qualification then such a point is a KKT point. We shall introduce two such notions of approximate KKT points and our first one is given below. Definition 1 -KKT points A point x̄ which is feasible to (P) is said to be an -KKT point if given > 0, there exists scalars λi ≥ 0, i = 1, 2, . . . ,m such that 1. ‖∇f(x̄) + mi=1 λi∇gi(x̄)‖ ≤ , 2. λigi(x̄) = 0, for i = 1, 2, . . . ,m In order to state our main result we need the notion of Mangasarian-Fromovitz constraint qualification (MFCQ) [4] which is given as follows: The constraints of the problem (P) is said to satisfy MFCQ at x̄ which is feasible if there exists a vector d ∈ R such that 〈∇gi(x̄), d〉 < 0, ∀i ∈ I(x̄), where I(x̄) = {i : gi(x̄) = 0} is the set of active constraints. The MFCQ can be alternatively stated in the following equivalent form which can be deduced using separation theorem for convex sets. The constraints of (P) satisfies MFCQ at a feasible x̄ if there exists no vector λ = 0, λ ∈ R+ , with λi ≥ 0, i ∈ I(x̄) and λi = 0 for i = I(x̄) with