Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces

We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝ n + 1 {\mathbb{R}^{n+1}} and every p > n {p>n} , the L p {L^{p}} -norm of the trace-free part of the anisotropic second fundamental form controls from above the W 2 , p {W^{2,p}} -closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p ≤ n {p\leq n} , the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.

Experimental Study on The Effect of Arches Setting on Semi-Flexible Monocrystalline Solar Panels

Indonesia has a high potential for renewable energy, especially solar power, due to its location in the equator and blessed with an abundance of sunlight. However, the energy potential from the sun is not maximally utilized. One of the efforts to increase the generated electricity and efficiency is by applied the panels in arches setting. This setting is made possible by the availability of the semi-flexible monocrystalline solar panel. This paper investigates the increment of harvested power and efficiency by arranging the solar panel in concave, convex, and plane settings. The data were taken in August 2019, where Palembang experiences the dry season and January 2020 during the rainy season. The highest power produced (20.27 Watt) and efficiency (13.14%) were achieved in a concave setting during the dry season. The convex setting produced more power and efficiency (13.26 Watt and 9.30%) compared to the plane setting (10.24 Watt and 9.71%). These results show that arches setting are more efficient to harvest solar power and give more extensive applications such as to power a dynamics mobile robot applied in agriculture.

Multivariate risk measures in the non-convex setting

The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g., in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions based on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex portfolios are presented.

Asynchronous Stochastic Frank-Wolfe Algorithms for Non-Convex Optimization

Asynchronous parallel stochastic optimization for non-convex problems becomes more and more important in machine learning especially due to the popularity of deep learning. The Frank-Wolfe (a.k.a. conditional gradient) algorithms has regained much interest because of its projection-free property and the ability of handling structured constraints. However, our understanding of asynchronous stochastic Frank-Wolfe algorithms is extremely limited especially in the non-convex setting. To address this challenging problem, in this paper, we propose our asynchronous stochastic Frank-Wolfe algorithm (AsySFW) and its variance reduction version (AsySVFW) for solving the constrained non-convex optimization problems. More importantly, we prove the fast convergence rates of AsySFW and AsySVFW in the non-convex setting. To the best of our knowledge, AsySFW and AsySVFW are the first asynchronous parallel stochastic algorithms with convergence guarantees for solving the constrained non-convex optimization problems. The experimental results on real high-dimensional gray-scale images not only confirm the fast convergence of our algorithms, but also show a near-linear speedup on a parallel system with shared memory due to the lock-free implementation.

Non-Ergodic Convergence Analysis of Heavy-Ball Algorithms

In this paper, we revisit the convergence of the Heavy-ball method, and present improved convergence complexity results in the convex setting. We provide the first non-ergodic O(1/k) rate result of the Heavy-ball algorithm with constant step size for coercive objective functions. For objective functions satisfying a relaxed strongly convex condition, the linear convergence is established under weaker assumptions on the step size and inertial parameter than made in the existing literature. We extend our results to multi-block version of the algorithm with both the cyclic and stochastic update rules. In addition, our results can also be extended to decentralized optimization, where the ergodic analysis is not applicable.

Contractibility of Fixed Point Sets of Mean-Type Mappings

We establish a convergence theorem and explore fixed point sets of certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces. We not only extend previous works by Matkowski to general normed linear spaces, but also obtain a new result on the structure of fixed point sets of quasi-nonexpansive mappings in a nonstrictly convex setting.