Optimality Conditions for Approximate Pareto Minimality

Exact Penalization and Optimality Conditions for Approximate Directional Minima

“…Exact penalization of Clarke's type is a useful technique that leads us to a precise conclusion about necessary optimality conditions for scalar or vector optimization problems. Notice that the next results are relatively close to some results in [2,5], but it is important to make a careful tracking of the constants involved in the definitions of approximate minima (see Definition 2.1). Inspired by a penalization technique applied for vector optimization problems in [14], we obtain a first penalization result presented below.…”

“…Proof. The first two assumption were discussed in [5,Remark 3.5] and is nothing but a straightforward computation that leads us to the desired conclusion. For (iii), firstly observe that ϕ(0) = ψ(0) = 0 and ϕ(3) = ψ(3) = 4 −1 .…”

“…Besides considering the directional setting, the minimum concept studied in this paper contains two constants that, in some sense, measure the approximate character of efficiency. By this paper, we had aimed to prove that some results from the papers [2,3,5] still hold for approximate directional Pareto minima. The novelty here is the fact that it's required a careful tracking (see Lemma 3.2) of the aforementioned constants that are involved in the definition of the minimality concepts.…”

“…More precisely, we discuss about approximate directional efficiency in constrained and unconstrained problems from the point of view of penalization techniques and optimality conditions. In this sense, the present work is linked with important papers from the literature dedicated to the concept of approximate solution in vector optimization (see [1,5,6,11,13,16] and the references therein).…”

“…The second section starts with the presentation of an abstract optimization problem that we will study together with the classical concept of Pareto minimality. In [5], it was studied the concept of approximate minimizers for vector optimization problems with set-valued maps and some interesting results were obtained only by using a penalization technique and some tools from variational analysis. So, motivated by the aforementioned paper, we start this second section with the adapted definition of approximate minima to the directional framework.…”