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On Strong Quasiconvexity of Functions in Infinite Dimensions
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abstract
In this paper, we explore the concept of $\sigma$-quasiconvexity for functions defined on normed vector spaces. This notion encompasses two important and well-established concepts: quasiconvexity and strong quasiconvexity. We start by analyzing certain operations on functions that preserve $\sigma$-quasiconvexity. Next, we present new results concerning the strong quasiconvexity of norm and Minkowski functions in infinite dimensions. Furthermore, we extend a recent result by F. Lara [16] on the supercoercive properties of strongly quasiconvex functions, with applications to the existence and uniqueness of minima, from finite dimensions to infinite dimensions. Finally, we address counterexamples related to strong quasiconvexity.
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Cited by 2 Pith papers
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Asymptotic Convergence Analysis of High-Order Proximal-Point Methods Beyond Sublinear Rates
HiPPA's claimed local linear convergence for order p<2 on strongly convex functions is contradicted by a simple quadratic example.
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Heavy Ball and Nesterov Accelerations with Hessian-driven Damping for Nonconvex Optimization
The authors prove exponential convergence for a Hessian-damped heavy-ball ODE and linear convergence for its Heavy Ball and Nesterov-type discretizations on strongly quasiconvex functions.
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