Abstract
In this paper, by using the scalarization method and normal subdifferential for set-valued maps, we consider an extension of Minty variational-like inequalities and obtain some relations between their solutions and set-valued optimization problems. An existence result for generalized Minty variational-like inequalities and set-valued optimization problems is also given. Moreover, the concept of approximate efficient solutions due to Kutateladze is investigated and by the Tammer–Weidner nonlinear functional, we characterize them for cone constrained set-valued optimization problems.
Highlights
Giannessi (1980) was the first author who obtained the equivalence between solutions of a Minty variational inequality and efficient solution of differentiable, convex optimization problem
By introducing a scalarized Minty variational-like inequalities (MVLI), we show that any solution of a scalarized set-valued optimization problems (SOP) is a solution of Minty variational-like inequalities under standard assumptions and that the inverse implications hold under the additional generalized K-convexity assumption, where K is an ordering cone of the considered image space
Approximation of weakly efficient solutions we present the concept of approximate efficiency for set-valued maps that is a generalization of the same notion due to Kutateladze (1979)
Summary
Variational inequalities are identified either in the form presented by the Minty (1967) or in the form by Stampacchia (1960). Giannessi (1980) was the first author who obtained the equivalence between solutions of a Minty variational inequality and efficient solution of differentiable, convex optimization problem. Variational inequalities are identified either in the form presented by the Minty (1967) or in the form by Stampacchia (1960). Giannessi (1980) was the first author who obtained the equivalence between solutions of a Minty variational inequality and efficient solution of differentiable, convex optimization problem. Some authors focused their works to nonsmooth functions Al-Homidan & Ansari, 2010; Alshahrani, Ansari, & Al-Homidan, 2014; Chen & Huang, 2012; Yang & Yang, 2006). Al-Homidan and Ansari (2010) obtained these results for invex functions with Clarke’s generalized directional derivative. By using the scalarization method, Santos, Rojas-Medar, M.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
