Filled function method for nonlinear equations

AbstractSystems of nonlinear equations are ubiquitous in engineering, physics and mechanics, and have myriad applications. Generally, they are very difficult to solve. In this paper, we will present a filled function method to solve nonlinear systems. We will first convert the nonlinear systems into equivalent global optimization problems with the property: x∗ is a global minimizer if and only if its function value is zero. A filled function method is proposed to solve the converted global optimization problem. Numerical examples are presented to illustrate our new techniques

Multi-block Min-max Bilevel Optimization with Applications in Multi-task Deep AUC Maximization

In this paper, we study multi-block min-max bilevel optimization problems, where the upper level is non-convex strongly-concave minimax objective and the lower level is a strongly convex objective, and there are multiple blocks of dual variables and lower level problems. Due to the intertwined multi-block min-max bilevel structure, the computational cost at each iteration could be prohibitively high, especially with a large number of blocks. To tackle this challenge, we present a single-loop randomized stochastic algorithm, which requires updates for only a constant number of blocks at each iteration. Under some mild assumptions on the problem, we establish its sample complexity of $O(1/\epsilon^4)$ for finding an $\epsilon$-stationary point. This matches the optimal complexity for solving stochastic nonconvex optimization under a general unbiased stochastic oracle model. Moreover, we provide two applications of the proposed method in multi-task deep AUC (area under ROC curve) maximization and multi-task deep partial AUC maximization. Experimental results validate our theory and demonstrate the effectiveness of our method on problems with hundreds of tasks

Oleic acid induces smooth muscle foam cell formation and enhances atherosclerotic lesion development via CD36

<p>Abstract</p> <p>Background</p> <p>Elevated plasma free fatty acid (FFA) levels have been linked to the development of atherosclerosis. However, how FFA causes atherosclerosis has not been determined. Because fatty acid translocase (FAT/CD36) is responsible for the uptake of FFA, we hypothesized that the atherogenic effects of FFA may be mediated via CD36.</p> <p>Results</p> <p>We tested this hypothesis using cultured rat aortic smooth muscle cells (SMCs) treated with oleic acid (OA). We found that OA induces lipid accumulation in SMCs in a dose dependent manner. Rat aortic SMCs treated for 48 hours with OA (250 μmol/L) became foam cells based on morphological (Oil Red O staining) and biochemical (5 times increase in cellular triglyceride) criteria. Moreover, specific inhibition of CD36 by sulfo-N-succinimidyl oleate significantly attenuated OA induced lipid accumulation and foam cell formation. To confirm these results <it>in vivo</it>, we used ApoE-deficient mice fed with normal chow (NC), OA diet, NC plus lipolysis inhibitor acipimox or OA plus acipimox. OA-fed mice showed increased plasma FFA levels and enhanced atherosclerotic lesions in the aortic sinus compared to the NC group (both <it>p </it>< 0.01). This effect was partially reversed by acipimox (lesion area: OA: 3.09 ± 0.10 ×10⁵μm²vs. OA plus acipimox: 2.60 ± 0.10 ×10⁵μm², <it>p </it>< 0.05; FFA: OA: 0.91 ± 0.03 mmol/L vs. OA plus acipimox: 0.78 ± 0.03 mmol/L, <it>p </it>< 0.05).</p> <p>Conclusions</p> <p>These findings suggest that OA induces smooth muscle foam cell formation and enhances atherosclerotic lesions in part though CD36. Furthermore, these findings provide a novel model for the investigation of atherosclerosis.</p