Machine learning model based on non-convex penalized huberized-SVM

The support vector machine(SVM)is a classical machine learning method.Both the hinge loss and least absolute shrinkage and selection operator(LASSO)penalty are usually used in traditional SVMs.However,the hinge loss is not differentiable,and the LASSO penalty does not have the Oracle property.In this paper,the huberized loss is combined with non-convex penalties to obtain a model that has the advantages of both the computational simplicity and the Oracle property,contributing to higher accuracy than traditional SVMs.It is experimentally demonstrated that the two non-convex huberized-SVM methods,smoothly clipped absolute deviation huberized-SVM(SCAD-HSVM)and minimax concave penalty huberized-SVM(MCP-HSVM),outperform the traditional SVM method in terms of the prediction accuracy and classifier performance.They are also superior in terms of variable selection,especially when there is a high linear correlation between the variables.When they are applied to the prediction of listed companies,the variables that can affect and predict financial distress are accurately filtered out.Among all the indicators,the indicators per share have the greatest influence while those of solvency have the weakest influence.Listed companies can assess the financial situation with the indicators screened by our algorithm and make an early warning of their possible financial distress in advance with higher precision.

Homotopy Method for Non-convex Programming in Unbonded Set

In the past few years, much and much attention has been paid to the method for solving non-convex programming. Many convergence results are obtained for bounded sets. In this paper, we get global convergence results for non-convex programming in unbounded sets under suitable conditions.

Simulation of bacterial flagellar phase transition by non-convex and non-local continuum modeling

Bacterial flagellar filament can undergo a stress-induced polymorphic phase transition in both vitro and vivo environments.The filament has 12 different helical forms(phases) characterized by different pitch lengths and helix radii.When subjected to the frictional force of flowing fluid,the filament changes between a left-handed normal phase and a right-handed semi-coiled phase via phase nucleation and growth.This paper develops non-local finite element method(FEM) to simulate the phase transition under a displacement-controlled loading condition(controlled helix-twist).The FEM formulation is based on the Ginzburg-Landau theory using a one-dimensional non-convex and non-local continuum model.To describe the processes of the phase nucleation and growth,viscosity-type kinetics is also used.The non-local FEM simulation captures the main features of the phase transition:two-phase coexistence with an interface of finite thickness,phase nucleation and phase growth with interface propagation.The non-local FEM model provides a tool to study the effects of the interfacial energy/thickness and loading conditions on the phase transition.

Drag Coefficient of a Non-Convex Polygonal Plate during Free Fall

Waterside creatures or aquatic organisms use a fin or web to generate a thrust force. These fins or webs have a non-convex section, referred to as a non-convex shape. We investigate the drag force acting on a non-convex plate during unsteady motion. We perform the experiment in a water tank during free fall. We fabricate the non-convex plate by cutting isosceles triangles from the side of a convex hexagonal plate. The base angle of the triangle is between 0° to 45°. The base angle is 0 indicates the convex hexagonal thin plate. We estimate the drag coefficient with the force balance acting on the model based on the image analysis technique. The results indicate that increasing the base angle by more than 30° increased the drag coefficient. The drag coefficient during unsteady motion changed with the growth of the vortex behind the model. The vortex has small vortices in the shear layer, which is related to the Kelvin-Helmholtz instabilities.

THE SCHWARZ LEMMA AT THE BOUNDARY OF THE NON-CONVEX COMPLEX ELLIPSOIDS

Let B2,p:= {z ∈ C2: |z1|2+ |z2|p< 1}（0 < p < 1）. Then, B2,p（0 < p < 1） is a non-convex complex ellipsoid in C2 without smooth boundary. In this article, we establish a boundary Schwarz lemma at z0 ∈ ?B2,p for holomorphic self-mappings of the non-convex complex ellipsoid B2,p, where z0 is any smooth boundary point of B2,p.

A Parametric Approach to Non-Convex Optimal Control Problem

In this paper we have considered a non convex optimal control problem and presented the weak, strong and converse duality theorems. The optimality conditions and duality theorems for fractional generalized minimax programming problem are established. With a parametric approach, the functions are assumed to be pseudo-invex and v-invex.

Non-Convex Optimization of Resource Allocation in Fog Computing Using Successive Approximation

Fog computing can deliver low delay and advanced IT services to end users with substantially reduced energy consumption.Nevertheless,with soaring demands for resource service and the limited capability of fog nodes,how to allocate and manage fog computing resources properly and stably has become the bottleneck.Therefore,the paper investigates the utility optimization-based resource allocation problem between fog nodes and end users in fog computing.The authors first introduce four types of utility functions due to the diverse tasks executed by end users and build the resource allocation model aiming at utility maximization.Then,for only the elastic tasks,the convex optimization method is applied to obtain the optimal results;for the elastic and inelastic tasks,with the assistance of Jensen’s inequality,the primal non-convex model is approximated to a sequence of equivalent convex optimization problems using successive approximation method.Moreover,a two-layer algorithm is proposed that globally converges to an optimal solution of the original problem.Finally,numerical simulation results demonstrate its superior performance and effectiveness.Comparing with other works,the authors emphasize the analysis for non-convex optimization problems and the diversity of tasks in fog computing resource allocation.

Non-convexity Pricing and Allocating Costs in Stochastic Electricity Markets

Stochastic electricity markets have drawn attention due to fast increase of renewable penetrations.This results in two issues:one is to reduce uplift payments arising from non-convexity under renewable uncertainties,and the other one is to allocate reserve costs based on renewable uncertainties.To resolve the first issue,a convex hull pricing method for stochastic electricity markets is proposed.The dual variables of system-wide constraints in a chance-constrained unit commitment model are shown to reduce expected uplift payments,together with developing a linear program to efficiently calculate such prices.To resolve the second issue,an allocation method is proposed to allocate reserve costs to each renewable power plant by explicitly investigating how renewable uncertainties of each renewable power plant affect reserve costs.The proposed methods are validated in a 24-period 3-unit test example and a 24-period 48-unit utility example.

Adaptive Linearized Alternating Direction Method of Multipliers for Non-Convex Compositely Regularized Optimization Problems

We consider a wide range of non-convex regularized minimization problems, where the non-convex regularization term is composite with a linear function engaged in sparse learning. Recent theoretical investigations have demonstrated their superiority over their convex counterparts. The computational challenge lies in the fact that the proximal mapping associated with non-convex regularization is not easily obtained due to the imposed linear composition. Fortunately, the problem structure allows one to introduce an auxiliary variable and reformulate it as an optimization problem with linear constraints, which can be solved using the Linearized Alternating Direction Method of Multipliers （LADMM）. Despite the success of LADMM in practice, it remains unknown whether LADMM is convergent in solving such non-convex compositely regularized optimizations. In this research, we first present a detailed convergence analysis of the LADMM algorithm for solving a non-convex compositely regularized optimization problem with a large class of non-convex penalties. Furthermore, we propose an Adaptive LADMM （AdaLADMM） algorithm with a line-search criterion. Experimental results on different genres of datasets validate the efficacy of the proposed algorithm.

A Non-Convex Optimization Approach to Dynamic Coverage Problem of Multi-agent Systems in an Environment with Obstacles

In this paper,a cooperative region reconnaissance problem is investigated where a group of agents are required to fly across and detect events occur in an environment with static obstacles until an effective coverage is achieved.First,the region reconnaissance is formulated as a non-convex optimization problem.A coverage performance index with additional collision and obstacle avoidance constraints is given.Since the optimization index is an implicit function of state variables and cannot be used to compute gradients on state variables directly,an approximate optimization index is selected.Then,a non-convex optimization-based coverage algorithm is proposed to find the optimal reconnaissance location for each agent and guarantee no collisions trajectories among agents and obstacles.Finally,simulation experiments are performed to verify the effectiveness of the proposed approach.

ITERATIVE l1 MINIMIZATION FOR NON-CONVEX COMPRESSED SENSING

An algorithmic framework, based on the difference of convex functions algorithm （D- CA）, is proposed for minimizing a class of concave sparse metrics for compressed sensing problems. The resulting algorithm iterates a sequence ofl1 minimization problems. An exact sparse recovery theory is established to show that the proposed framework always improves on the basis pursuit （l1 minimization） and inherits robustness from it. Numerical examples on success rates of sparse solution recovery illustrate further that, unlike most existing non-convex compressed sensing solvers in the literature, our method always out- performs basis pursuit, no matter how ill-conditioned the measurement matrix is. Moreover, the iterative l1 （ILl） algorithm lead by a wide margin the state-of-the-art algorithms on l1/2 and logarithimic minimizations in the strongly coherent （highly ill-conditioned） regime, despite the same objective functions. Last but not least, in the application of magnetic resonance imaging （MRI）, IL1 algorithm easily recovers the phantom image with just 7 line projections.

A Generalized Two-Level Bregman Method with Dictionary Updating for Non-Convex Magnetic Resonance Imaging Reconstruction

In recent years, it has shown that a generalized thresholding algorithm is useful for inverse problems with sparsity constraints. The generalized thresholding minimizes the non-convex p-norm based function with p < 1, and it penalizes small coefficients over a wider range meanwhile applies less bias to the larger coefficients.In this work, on the basis of two-level Bregman method with dictionary updating(TBMDU), we use the modified thresholding to minimize the non-convex function and propose the generalized TBMDU(GTBMDU) algorithm.The experimental results on magnetic resonance(MR) image simulations and real MR data, under a variety of sampling trajectories and acceleration factors, consistently demonstrate that the proposed algorithm can efficiently reconstruct the MR images and present advantages over the previous soft thresholding approaches.

Effect of local non-convexity on the critical shear strength of granular materials determined via the discrete element method

Multi-sphere clumps are commonly used to simulate non-spherical particles in discrete element method simulations.It is of interest whether the degree of local non-convexity λ affects the mechanical behaviour of granular materials with the same non-convexity η.A series of discrete-element-method biaxial shear tests are conducted on rough particle packings with rη=0.075 and different λ values(ranging from 0.134 to 0.770).The microscale results show that the contact type changes with an increase in λ.However,the critical strength is independent of λ.The evaluation of the contributions of different contact types to the critical shear strength and a detailed analysis of the anisotropies help clarify the microscopic mechanisms that result in the independence of the critical shear strength from λ.

Extended finite element methods for optimal control problems governed by Poisson equation in non-convex domains

This paper analyzes two extended finite element methods(XFEMs)for linear quadratic optimal control problems governed by Poisson equation in non-convex domains.We follow the variational discretization concept to discretize the continuous problems,and apply an XFEM with a cut-off function and a classic XFEM with a fixed enrichment area to discretize the state and co-state equations.Optimal error estimates are derived for the state,co-state and control.Numerical results confirm our theoretical results.

ASYMPTOTIC STABILITY OF VISCOUS SHOCK PROFILE FOR NON-CONVEX SYSTEM OF ONE-DIMENSIONAL VISCOELASTIC MATERIALS WITH BOUNDARY EFFECT

This paper is concerned with the asymptotic behavior of solution to the initial-boundary value problem on the half space R+ for a one-dimensional non-convex system of viscoelastic materials. The initial data has constant state at infinity and the velocity is imposed zero at the boundary x = 0. By virture of the boundary effect, the solution is expected to behave as outgoing viscous shock profile. When the initial data is suitably close to the corresponding outgoing viscous shock profile which is suitably away from the boundary, it is proved that the unique global solution exists in time and tends toward the properly shifted shock profile as the time goes to infinity. The result is given by a weighted energy method.

Global Convergence of a Modified Limited Memory BFGS Method for Non-convex Minimization

In this paper, a modified limited memory BFGS method for solving large-scale unconstrained optimization problems is proposed. A remarkable feature of the proposed method is that it possesses global convergence property without convexity assumption on the objective function. Under some suitable conditions, the global convergence of the proposed method is proved. Some numerical results are reported which illustrate that the proposed method is efficient.

Global Optimization Approach to Non-convex Problems

A new approach to find the global optimal solution of the special non-convex problems is proposed in this paper. The non-convex objective problem is first decomposed into two convex sub-problems. Then a generalized gradient is introduced to determine a search direction and the evolution equation is built to obtain a global minimum point. By the approach, we can prevent the search process from some local minima and search a global minimum point. Two numerical examples are given to prove the approach to be effective.

Two-dimensional densely ordered packings of non-convex bending and assembled rods

The investigation of the problem of particle packing has provided basic insights into the structure,symmetry,and physical properties of condensed matter.Dense packings of non-spherical particles have many applications,both in research and industry.We report the two-dimensional dense packing patterns of bending and assembled rods,which are non-convexly deformed from simple objects and modeled as entangled particles.Monte Carlo simulations and further analytical constructions are carried out to explore possible densely packed structures.Two typical densely packed structures of C-bending rods are found,and their packing densities are identified as being functions of the aspect ratio and central angle.Six shapes of assembled rods,representing the combined deformations of rods,are employed in simulations with the packing structures classified into three types.The dense packing density of each packing pattern is derived as a function of different shape parameters.In contrast with the case of disordered packings,both the shape and order are verified to affect the packing density.

Regions of Applicability of Aubry-Mather Theory for Non-convex Hamiltonian

Herman constructed an autonomous system of two degrees of freedom which says that in non-convex situations, oscillations do happen and Aubry-Mather Theory cannot apply (see the results due to W. F. Chen in 1992). In this paper, it is shown that although the orbits could visit a region far away from the initial point in phase space, they can only exist in some fixed regions in I = (I1 , I2 ) plane. Moreover, Aubry-Mather Theory can be applied outside the regions.

基于非凸与不可分离正则化算法的电容层析成像图像重建

搅拌器内两相混合是化工生产中常见的现象,电容层析成像(ECT)技术主要对两相分布进行可视化重构,以达到监测的目的。受稀疏贝叶斯学习的启发,提出了一种非凸与不可分离正则化(NNR)算法重建ECT图像。在稀疏先验的基础上引入矩阵低秩特性,采用最大后验估计在潜在空间中提出一个新的优化问题,利用对偶变量将潜在空间的目标函数映射到原始空间进行迭代求解,用来恢复同时稀疏与低秩的矩阵。与凸近似L1范数相比,NNR算法可获得更准确的重建图像,同时比非凸可分离方法更容易收敛到全局最优解。为验证NNR算法的重建效果,通过数值仿真与静态实验的方法分别与其他5种算法进行重建对比。结果表明:NNR算法可以有效减少重建伪影,提升中心物体的重建质量,为搅拌器内两相分布提供了高质量的重建算法。