An analytical approach of filament bundle swinging dynamics, Part II: identifying equivalent dynamic constitutive parameters of filament bundles

A filament bundle is a type of yarn, which is composed of nearly parallel and highly oriented polymer monofilaments. Due to its nonlinearity both in material constitutive properties and structure, the filament bundle possesses nonlinear viscoelastic properties. It is important to study the dynamic behavior of the filament bundle accurately during its high-speed movement. Therefore, an accurate expression of the constitutive relation of the filament bundle is an essential prerequisite for its dynamic simulation and analysis. Continued the previous study in Part I: modeling filament bundle method, in this paper, an approach was proposed to identify the equivalent dynamic constitutive parameters of the filament bundle considering frequency-dependent characteristics. Firstly, the identification formulas of the dynamic elastic modulus and viscoelastic coefficients were derived based on the Kelvin model. Then, a testing method of the cross-sectional parameters of the filament bundle under a certain tension was proposed, and the testing device was developed to obtain the area of the filament bundle; The dynamic loading test of the bundle filament was conducted in a DMA Q800 dynamic mechanical tester. Thirdly, the equivalent dynamic elastic modulus and viscoelastic coefficients were obtained through the experimental test. Finally, an analytical method was proposed to verify the correctness of experimental results through simulation. The results show that the excitation frequency has a significant influence on the dynamic elastic modulus and viscoelastic coefficient, and the curves of the equivalent dynamic elastic modulus and viscoelastic coefficient present nonlinear variation characteristics.

On the Embed and Project Algorithm for the Graph Bandwidth Problem

The graph bandwidth problem, where one looks for a labeling of graph vertices that gives the minimum difference between the labels over all edges, is a classical NP-hard problem that has drawn a lot of attention in recent decades. In this paper, we focus on the so-called Embed and Project Algorithm (EPA) introduced by Blum et al. in 2000,which in the main part has to solve a semidefinite programming relaxation with exponentially many linear constraints. We present several theoretical properties of this special semidefinite programming problem (SDP) and a cutting-plane-like algorithm to solve it, which works very efficiently in combination with interior-point methods or with the bundle method. Extensive numerical results demonstrate that this algorithm, which has only been studied theoretically so far, in practice gives very good labeling for graphs with n≤1000.

Lagrangian Relaxation Based on Improved Proximal Bundle Method for Short-Term Hydrothermal Scheduling

Short-term hydrothermal scheduling (STHS) can improve water use efficiency, reduce carbon emissions, and increase economic benefits by optimizing the commitment and dispatch of hydro and thermal generating units together. However, limited by the large system scale and complex hydraulic and electrical constraints, STHS poses great challenges in modeling for operators. This paper presents an improved proximal bundle method (IPBM) within the framework of Lagrangian relaxation for STHS, which incorporates the expert system (ES) technique into the proximal bundle method (PBM). In IPBM, initial values of Lagrange multipliers are firstly determined using the linear combination of optimal solutions in the ES. Then, each time PBM declares a null step in the iterations, the solution space is inferred from the ES, and an orthogonal design is performed in the solution space to derive new updates of the Lagrange multipliers. A case study in a large-scale hydrothermal system in China is implemented to demonstrate the effectiveness of the proposed method. Results in different cases indicate that IPBM is superior to standard PBM in global search ability and computational efficiency, providing an alternative for STHS.

A Redistributed Bundle Algorithm Based on Local Convexification Models for Nonlinear Nonsmooth DC Programming

For nonlinear nonsmooth DC programming (difference of convex functions), we introduce a new redistributed proximal bundle method. The subgradient information of both the DC components is gathered from some neighbourhood of the current stability center and it is used to build separately an approximation for each component in the DC representation. Especially we employ the nonlinear redistributed technique to model the second component of DC function by constructing a local convexification cutting plane. The corresponding convexification parameter is adjusted dynamically and is taken sufficiently large to make the ”augmented” linearization errors nonnegative. Based on above techniques we obtain a new convex cutting plane model of the original objective function. Based on this new approximation the redistributed proximal bundle method is designed and the convergence of the proposed algorithm to a Clarke stationary point is proved. A simple numerical experiment is given to show the validity of the presented algorithm.

A Proximal Bundle Method with Exact Penalty Technique and Bundle Modification Strategy for Nonconvex Nonsmooth Constrained Optimization

The bundle modification strategy for the convex unconstrained problems was proposed by Alexey et al. [[2007] European Journal of Operation Research, 180(1), 38–47.] whose most interesting feature was the reduction of the calls for the quadratic programming solver. In this paper, we extend the bundle modification strategy to a class of nonconvex nonsmooth constraint problems. Concretely, we adopt the convexification technique to the objective function and constraint function, take the penalty strategy to transfer the modified model into an unconstrained optimization and focus on the unconstrained problem with proximal bundle method and the bundle modification strategies. The global convergence of the corresponding algorithm is proved. The primal numerical results show that the proposed algorithms are promising and effective.

An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization

In this paper, an adaptive proximal bundle method is proposed for a class of nonconvex and nonsmooth composite problems with inexact information. The composite problems are the sum of a finite convex function with inexact information and a nonconvex function. For the nonconvex function, we design the convexification technique and ensure the linearization errors of its augment function to be nonnegative. Then, the sum of the convex function and the augment function is regarded as an approximate function to the primal problem. For the approximate function, we adopt a disaggregate strategy and regard the sum of cutting plane models of the convex function and the augment function as a cutting plane model for the approximate function. Then, we give the adaptive nonconvex proximal bundle method. Meanwhile, for the convex function with inexact information, we utilize the noise management strategy and update the proximal parameter to reduce the influence of inexact information. The method can obtain an approximate solution. Two polynomial functions and six DC problems are referred to in the numerical experiment. The preliminary numerical results show that our algorithm is effective and reliable.

An Infeasible Incremental Bundle Method for Nonsmooth Optimization Problem Based on CVaR Portfolio

For CVaR (conditional value-at-risk) portfolio nonsmooth optimization problem, we propose an infeasible incremental bundle method on the basis of the improvement function and the main idea of incremental method for solving convex finite min-max problems. The presented algorithm only employs the information of the objective function and one component function of constraint functions to form the approximate model for improvement function. By introducing the aggregate technique, we keep the information of previous iterate points that may be deleted from bundle to overcome the difficulty of numerical computation and storage. Our algorithm does not enforce the feasibility of iterate points and the monotonicity of objective function, and the global convergence of the algorithm is established under mild conditions. Compared with the available results, our method loosens the requirements of computing the whole constraint function, which makes the algorithm easier to implement.

An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems

We present an efficient descent method for unconstrained, locally Lipschitz multiobjective optimization problems. The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth multiobjective optimization problems with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method with the multiobjective proximal bundle method by Mäkelä. The results indicate that our method is competitive while being easier to implement. Although the number of objective function evaluations is larger, the overall number of subgradient evaluations is smaller. Our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth problems. Finally, we demonstrate how our method can be used for solving sparse optimization problems, which are present in many real-life applications.