Inexact proximal methods for weakly convex functions

This paper proposes and develops inexact proximal methods for finding stationary points of the sum of a smooth function and a nonsmooth weakly convex one, where an error is present in the calculation of the proximal mapping of the nonsmooth term. A general framework for finding zeros of a continuous mapping is derived from our previous paper on this subject to establish convergence properties of the inexact proximal point method when the smooth term is vanished and of the inexact proximal gradient method when the smooth term satisfies a descent condition. The inexact proximal point method achieves global convergence with constructive convergence rates when the Moreau envelope of the objective function satisfies the Kurdyka-Lojasiewicz (KL) property. Meanwhile, when the smooth term is twice continuously differentiable with a Lipschitz continuous gradient and a differentiable approximation of the objective function satisfies the KL property, the inexact proximal gradient method achieves the global convergence of iterates with constructive convergence rates.Comment: 26 pages, 3 table

ĐẶC ĐIỂM ĐỊA HÌNH ĐÁY VÀ TRẦM TÍCH TẦNG MẶT VÙNG BIỂN QUẦN ĐẢO THỔ CHU

Tho Chu archipelago was formed by sedimentary rocks (176 metres in height) with horizontal position. The islands are limited by the abraded slopes and landslide disruption. The shoreline morphology is cliffs, abrasion and boulders with good size, low abrasion, and little change. Vegetation of island is still in the primary condition. The average depth of Tho Chu archipelago is about 20 - 30 m with canyons of -96 m and several underwater hills which are covered by coral reefs. The characteristics of seabed sediment are mostly kinds of coarse - grained sediments such as gravelly sand (gS), slightly gravelly muddy sand ((g)mS), muddy sand (mS), mud (M).Quần đảo Thổ Chu được cấu tạo bởi các đá trầm tích (cao 176 m), có thế nằm ngang. Xung quanh đảo được giới hạn các sườn dốc mài mòn và đỗ vỡ sụp lở. Hình thái đường bờ là các vách dốc, mài mòn và các tảng lăn với kích thước khá lớn, bị mài mòn yếu, ít bị thay đổi, thảm thực vật trên đảo còn nguyên sinh. Độ sâu trung bình quần đảo Thổ Chu khoảng 20 - 30 m, có rãnh sâu có độ sâu -96 m và có rất nhiều đồi ngầm phần lớn được phủ san hô. Đặc điểm trầm tích tầng mặt được phủ bởi các kiểu trầm tích hạt thô: Cát chứa graven, cát chứa bùn sét (và graven), cát chứa bùn sét, bùn sét

DEVELOPMENT AND EVALUATION OF ORAL SUSTAINED-RELEASE RANITIDINE DELIVERY SYSTEM BASED ON BACTERIAL NANOCELLULOSE MATERIAL PRODUCED BY KOMAGATAEIBACTER XYLINUS

Objective: The short biological half-life (2-3 h) and low bioavailability (50 %) of ranitidine (RAN) following oral administration favor the development of a controlled release system. This study was aimed to develop and in vitro evaluate oral sustained-release RAN delivery system based on the bacterial nanocellulose material (BNM) produced by Komagataeibacter xylinus (K. xylinus) from selected culture media. Methods: BNMs are biosynthesized by K. xylinus in the standard medium (SM) and coconut water (CW). RAN was loaded in BNMs by the absorption method. The structural and physicochemical properties of BNMs and BNMs-RAN were evaluated via swelling behavior, FTIR, and FESEM techniques. Moreover, the effect of BNMs on RAN release profile and release kinetics was analyzed and evaluated. Results: The amount of loaded RAN or entrapment efficacy for BNM-CW is higher than for BNM-SM. The BNM-SM-RAN and BNM-CW-RAN exhibited a decreased initial burst release system followed by a prolonged RAN release up to 24 h in relation to the commercial tablets containing RAN. The RAN release from these formulations was found higher in the SGF medium than that of in SIF medium. RAN released from these formulations was found to follow the Korsmeyer-Peppas model and diffusion sustained drug release mechanism. The sustained release of RAN from BNM-SM-RAN was slower than for RAN from BNM-CW-RAN, but the mechanism of sustained RAN release was the same. Conclusion: Oral sustained-release RAN delivery system based on BNMs was successfully prepared and evaluated for various in vitro parameters. The biopolymers like BNM-SM and BNM-CW could be utilized to develop oral sustained RAN release dosage form

Inexact reduced gradient methods in smooth nonconvex optimization

This paper proposes and develops new line search methods with inexact gradient information for finding stationary points of nonconvex continuously differentiable functions on finite-dimensional spaces. Some abstract convergence results for a broad class of line search methods are reviewed and extended. A general scheme for inexact reduced gradient (IRG) methods with different stepsize selections are proposed to construct sequences of iterates with stationary accumulation points. Convergence results with convergence rates for the developed IRG methods are established under the Kurdyka-Lojasiewicz property. The conducted numerical experiments confirm the efficiency of the proposed algorithms

Globally Convergent Coderivative-Based Generalized Newton Methods in Nonsmooth Optimization

This paper proposes and justifies two globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are coderivative-based and employ generalized Hessians (coderivatives of subgradient mappings) associated with objective functions, which are either of class $\mathcal{C}^{1,1}$, or are represented in the form of convex composite optimization, where one of the terms may be extended-real-valued. The proposed globally convergent algorithms are of two types. The first one extends the damped Newton method and requires positive-definiteness of the generalized Hessians for its well-posedness and efficient performance, while the other algorithm is of {the regularized Newton type} being well-defined when the generalized Hessians are merely positive-semidefinite. The obtained convergence rates for both methods are at least linear, but become superlinear under the semismooth$^*$ property of subgradient mappings. Problems of convex composite optimization are investigated with and without the strong convexity assumption {on smooth parts} of objective functions by implementing the machinery of forward-backward envelopes. Numerical experiments are conducted for Lasso problems and for box constrained quadratic programs with providing performance comparisons of the new algorithms and some other first-order and second-order methods that are highly recognized in nonsmooth optimization.Comment: arXiv admin note: text overlap with arXiv:2101.1055