In [1], some parts of the proofs of Theorems 3.1 and 3.2(ii) are needed to be revised. The conclusion of [1, Theorem 3.1] holds under an additional assumption, thus it is restated and proved. Also, the proof of [1, Theorem 3.2(ii)] is modified. With this method of proof, the USRC of the function (Formula presented.) at (Formula presented.) becomes a smaller set in comparison with its counterpart in [1, Theorem 3.2(ii)]. Thus we can say, the statement of [1, Theorem 3.2(ii)] is improved. Furthermore, [1, Lemmas 3.2 and 3.3] are not required and hence are omitted. Accordingly, the paragraph before these Lemmas is also removed. Further, [1, Theorem 3.3] gives a better form of the optimality condition and is updated. No other changes are required regarding the preliminaries, definitions, main conclusions and examples. First, we update [1, Theorem 3.1] by adding the following additional assumption from [2]: We say that the function F is calm at (Formula presented.) with some modulus (Formula presented.) , if there exists a positive scalar (Formula presented.) satisfying (Formula presented.) for each (Formula presented.) ([1, Theorem 3.1] updated) Assume that the function F is calm at (Formula presented.) with some modulus (Formula presented.) and (Formula presented.) is directionally differentiable at (Formula presented.). If EBCQ holds at (Formula presented.) with a constant (Formula presented.) , then ACQ is satisfied at (Formula presented.) with the same constant. (modified) Let (Formula presented.) (otherwise there is nothing to prove) and EBCQ be satisfied at (Formula presented.) with (Formula presented.). Assume also that (Formula presented.) (if (Formula presented.) , the ACQ obviously holds). Thus, there is a sequence (Formula presented.) such that (Formula presented.) The closedness of (Formula presented.) , gives us a sequence (Formula presented.) such that for each k, (Formula presented.) We assert that the sequence (Formula presented.) is bounded. Fixing (Formula presented.) and observing (1), we obtain the following inequalities for all k sufficiently large: (Formula presented.) which shows the boundedness of (Formula presented.) and the assertion is proved. Thus by passing to a subsequence, without relabelling, (Formula presented.) converges to some vector (Formula presented.). Now, By EBCQ one has (Formula presented.) Next, we claim that (Formula presented.) From [1, Lemma 3.1] and the fact that (Formula presented.) is directionally differentiable at (Formula presented.) we get (Formula presented.) which proves the claim. Now, it follows from (2), (3) and (1) that (Formula presented.) Using again [1, Lemma 3.1], the above especially implies that (Formula presented.) and completes the proof of the theorem. (Formula presented.) In what follows, the proof of [1, Theorem 3.2(ii)] is modified. By this modification, USRC of the function (Formula presented.) at (Formula presented.) becomes a smaller set and gives a better result; hence its statement is also improved. ([1, Theorem 3.2(ii)] updated) Assume that (Formula presented.) is an u.s.c. PJ of (Formula presented.) at (Formula presented.). Suppose also that (Formula presented.) and (Formula presented.) is a bounded USRC of (Formula presented.) at (Formula presented.). Then the closure of the set (Formula presented.) is an USRC of the function (Formula presented.) at (Formula presented.). (revised) Put (Formula presented.) and fix (Formula presented.). First, let us show that (Formula presented.) For given (Formula presented.) , we have (Formula presented.) Thus, using the definition of A, we get (Formula presented.) There are two possible cases: If (Formula presented.) , then trivially we obtain (Formula presented.) Hence, let (Formula presented.) If the following inequality holds: (Formula presented.) due to the cone property of (Formula presented.) , we get (Formula presented.) and the inequality in (4) holds trivially. Finally, the following case remains (Formula presented.) For each fixed (Formula presented.) one has (Formula presented.) Utilizing [1, Lemma 3.1], we have (Formula presented.) which means that (Formula presented.) , for all (Formula presented.) Now, since (Formula presented.) there exits some positive number c such that (Formula presented.). Thus for some sequence (Formula presented.) and for all k sufficiently large, one has (Formula presented.) Applying now the mean value Theorem in [1, Propostion 2.3], we have for each k, (Formula presented.) Using the upper semicontinuity of (Formula presented.) at (Formula presented.) , for given sequence (Formula presented.) , there exits (Formula presented.) satisfying (Formula presented.) Thus, there exists (Formula presented.) such that (Formula presented.) Choosing now subsequences (Formula presented.) and (Formula presented.) , and using the inequality in (6), we deduce that (Formula presented.) Observing that (Formula presented.) and taking limit as (Formula presented.) in the latter inequality, we arrive at the contradiction (Formula presented.) , which shows the case (Formula presented.) and the equality (5) do not occur together and the proof is completed. (Formula presented.) Since the USRC of the function (Formula presented.) is changed, the optimality condition in [1, Theorem 3.3] is improved and updated, accordingly. ([1, Theorem 3.3] updated) Suppose that ACQ is satisfied at the local optimal point (Formula presented.) of GOP. Let (Formula presented.) and (Formula presented.) are USRC and u.s.c. PJ of f and F at (Formula presented.) respectively and (Formula presented.) is a bounded USRC of (Formula presented.) at (Formula presented.). Then (Formula presented.) where (Formula presented.) is the positive constant of ACQ and l is the Lipschitz constant of the function f in a neighborhood of (Formula presented.). The proofs of [1, Theorems 3.1 and 3.2(ii)] are rectified and their statements are updated. Also, [1, Theorem 3.3] gives a better form of the optimality condition which is improved. © 2023, Springer Science+Business Media, LLC, part of Springer Nature.
