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First Order Evolution Equations
Published in Behzad Djafari-Rouhani, Hadi Khatibzadeh, Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces, 2019
Behzad Djafari-Rouhani, Hadi Khatibzadeh
Obviously convexity implies pseudo-convexity and pseudo-convexity implies quasiconvexity. We refer the reader to the interesting book by Cambini and Martein [CAM-MAR] for the definitions, properties and illustrative examples of convexity and its extensions. Throughout this section, we assume that ϕ:H→R is a continuously differentiable quasi-convex function with Argminϕ ≠ Ø, and ∇ϕ Lipschitz continuous on bounded subsets of H.
Set-valued optimization in variable preference structures with new variants of generalized convexity
Published in Optimization, 2023
Generalized convexity has been developed from the 50s of the last century (see, a possibly first paper [15]). A main classical condition for a local minimum to be a global one is pseudoconvexity. A close classical concept of quasiconvexity is known as, when combined with pseudoconvexity, ensuring a local solution of a constrained minimization problem to be a global one. Such combinations are also employed as generalized convexity assumptions to establish optimality conditions. Furthermore, they guarantee strong duality for pairs of optimization problems. Since these topics are in the center of mathematical optimization, we investigate them for a general setting of set-valued vector problems with variable general preferences by proposing relaxed concepts of quasiconvexity, which can also replace types of pseudoconvexity, and applying them.
Well-posedness and existence for the weak multicriteria Nash equilibrium of multicriteria games
Published in Optimization, 2023
Wenyan Zhang, Jing Zeng, Ruiting Hu
The quasiconvexity is weaker than the convexity. When and (), Definition 2.3 reduces to the definition of convex and quasiconvex functions in the scalar case.