Optimality conditions for multiobjective optimization problem constrained by parameterized variational inequalities
A multiobjective problem in finite-dimensional spaces with a parameterized variational inequality is considered, where at the end the feasible set \(B\) is convex by the given assumptions. First by the validity of Theorem 3.1, the problem is equivalent to a family of usual problems with linear scalarizations of the multiobjective. Theorem 3.2 derives a Karush-Kuhn-Tucker statement for the family of scalarized problems. In Chapter 4, the used coderivative for the Jacobian of the constraint function \(g\) in connection with the normal cone is estimated. It gives more detailed optimality conditions for the parametric family. Examples illustrate the theory. Reviewer's remark: However, at least the proofs of both statements of the basic Theorem 3.1 are incorrect. In the proof, the definition of \(A_i\) is unclear w.r.t. its further use. The equation (11) is under the assumptions made not valid. Four lines later, it is stated and used that the image of a closed convex set \(B\) of a vector function with convex coordinate functions is convex using the argumentation that it is the Cartesian product of the images of the coordinate functions. A simple example like \((x,y)\rightarrow (x^2+y^2,x+y): [0,1]^2\rightarrow \mathbb R^2\) shows that both are wrong. Thus the use of the separation theorem for the constructed convex sets remains open. For the second statement the authors use for a nonconvex vector function that the sum of the image of a closed convex set and that the usual open order cone is the Cartesian product of the sum of the images of the coordinate function and the set of positive reals, which is as wrong as the former statement. By the way they write that the sum of a closed convex set with open order cone is closed, but everybody knows that the union of open sets is open. Since the other theorems mainly use Theorem 3.1 it remains open whether they are true.
scientific article; zbMATH DE number 6345537
scientific article