A Note on the Paper “Optimality Conditions for Vector Optimization Problems with Difference of Convex Maps”

Abstract

In this work, some counterexamples are given to refute some results reported in the paper by Guo and Li [8] (J Optim Theory Appl 162,(2014), 821-844). We correct the faulty in some of their theorems and we present alternative proofs. Moreover, we extend the definition of approximately pseudo-dissipative in the setting of metrizable topological vector spaces.

<i>ε</i>-weak Pareto minimality in D. C. vector optimization

“…Besides being a correction of [11,Theorem 4.3], Theorem 4.1 concerns also a more general situation than those treated in [11].…”

“…• In [11], Shafie and Bozorgnia proposed necessary optimality conditions of (P ) based on a convexlike assumption. • Looking closely, we realized that both the main result [11,Theorem 4.3] and its proof are erroneous. For example, on page 10, line -6, the authors should have taken L ∈ ∂S (x) instead of L ∈ ∂H (x) .…”

“…In case F is K-convexlike and H is D-convexlike, one deduces the following result. In fact, Corollary 4.3 is a corrected version of [11,Theorem 4.1].…”

“…Nonconvex mappings arising in nonsmooth optimization are frequently of this type. Extensive work on the analysis and optimization of DC mappings has been done in recent years [1,2,3,4,5,6,7,11]. However, there is still a lot of work to be done, such as the investigation of ε-approximate solutions in the Pareto sense for vector optimization problems.…”

“…Due to the fact that some functions can be approximate pseudo dissipative without being Diff-Max (see [6,Example 3.2]), Guo and Li [6] gave sufficient optimality conditions for an ε-weak Pareto minimal point of (P ) using the concept of approximate pseudo-dissipativity [9]. Unfortunately, their main result turned out to be false and was corrected by Shafie and Bozorgnia in [11]. Recently, Gadhi [2] pointed out that Shafie and Bozorgnia's proposed improvement is also incorrect, rectified the errors, and gave an alternate proof for the main result.…”