Nondifferentiable Problems Research Articles

Duality for Unified Higher-Order Minimax Fractional Programming with Support Function under Type-I Assumptions

This article is devoted to discussing the nondifferentiable minimax fractional programming problem with type-I functions. We focus our study on a nondifferentiable minimax fractional programming problem and formulate a higher-order dual model. Next, we establish weak, strong, and strict converse duality theorems under generalized higher-order strictly pseudo ( V , α , ρ , d ) -type-I functions. In the final section, we turn our focus to study a nondifferentiable unified minimax fractional programming problem and the results obtained in this paper naturally unify. Further, we extend some previously known results on nondifferentiable minimax fractional programming in the literature.

Optimal Nonlinear Control Using Hamilton–Jacobi–Bellman Viscosity Solutions on Unstructured Grids

The optimal control of nonlinear systems is traditionally obtained by the application of the Pontryagin minimum principle. Despite the success of this methodology in finding the optimal control for complex systems, the resulting open-loop trajectory is guaranteed to be only locally optimal. Furthermore, the computation of open-loop solutions is computationally intensive, which rules out its application in feedback controllers, reducing its robustness against disturbances. In principle, these issues can be addressed by solving the Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE). However, the space complexity of the problem is exponential with respect to the number of dimensions of the system. Moreover, the value function of the HJB equation may be nondifferentiable, which renders traditional PDE solution methods impractical. Therefore, extant methods are suitable only for special problem classes such as those involving affine systems or where the value function is differentiable. To deal with these issues, this work introduces a methodology for the solution of the HJB equation for general nonlinear systems that combines PDE viscosity solutions, quasi–Monte Carlo grids, and kriging regression to implement globally optimal nonlinear feedback controllers for practical applications. The effectiveness of the method is illustrated with smooth and nondifferentiable problems with finite and infinite horizons.

Solving primal plasticity increment problems in the time of a single predictor–corrector iteration

The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a well-established method for the solution of strictly convex block-separably nondifferentiable minimization problems. It achieves multigrid-like performance even for non-smooth nonlinear problems, while at the same time being globally convergent and without employing penalty parameters. We show that the algorithm can be applied to the primal problem of classical small-strain elastoplasticity with hardening. Numerical experiments show that the method is considerably faster than classical predictor–corrector methods. Indeed, solving an entire increment problem with TNNMG can take less time than a single predictor–corrector iteration for the same problem. At the same time, memory consumption is reduced considerably, in particular for three-dimensional problems. Since the algorithm does not rely on differentiability of the objective functional, nonsmooth yield laws can be easily incorporated. The method is closely related to a predictor–corrector scheme with a consistent tangent predictor and line search. We explain the algorithm, prove global convergence, and show its efficiency using standard benchmarks from the literature.

Signal reconstruction by conjugate gradient algorithm based on smoothing $$l_1$$-norm

The $$l_1$$-norm regularized minimization problem is a non-differentiable problem and has a wide range of applications in the field of compressive sensing. Many approaches have been proposed in the literature. Among them, smoothing $$l_1$$-norm is one of the effective approaches. This paper follows this path, in which we adopt six smoothing functions to approximate the $$l_1$$-norm. Then, we recast the signal recovery problem as a smoothing penalized least squares optimization problem, and apply the nonlinear conjugate gradient method to solve the smoothing model. The algorithm is shown globally convergent. In addition, the simulation results not only suggest some nice smoothing functions, but also show that the proposed algorithm is competitive in view of relative error.

Sufficient optimality conditions for nondifferentiable multiobjective programming problem with generalized uniform invexity

In this paper, a new class of generalized invex functions named generalized uniform pseudoinvex (C, q) – type I, generalized uniform pseudoquasi-invex (C, q) – type I and generalized uniform quasipseudo-invex (C, q) – type I are defined by utilizing the Clarke subdifferential, where C is sublinear in the third argument. Then some sufficient optimality conditions are derived and proved for a class of nondifferentiable multiobjective programming problems involving the new generalized uniform invexity.

Special Class of Second-Order Non-Differentiable Symmetric Duality Problems with (G,αf)-Pseudobonvexity Assumptions

In this paper, we introduce the various types of generalized invexities, i.e., α f -invex/ α f -pseudoinvex and ( G , α f ) -bonvex/ ( G , α f ) -pseudobonvex functions. Furthermore, we construct nontrivial numerical examples of ( G , α f ) -bonvexity/ ( G , α f ) -pseudobonvexity, which is neither α f -bonvex/ α f -pseudobonvex nor α f -invex/ α f -pseudoinvex with the same η . Further, we formulate a pair of second-order non-differentiable symmetric dual models and prove the duality relations under α f -invex/ α f -pseudoinvex and ( G , α f ) -bonvex/ ( G , α f ) -pseudobonvex assumptions. Finally, we construct a nontrivial numerical example justifying the weak duality result presented in the paper.

Stability analysis of a class of non-newtonian fluids based on generalized vector variational-like inequalities on riemannian manifold

Fluid mechanics is a branch of mechanics. It is the science of studying fluid phenomena and related mechanical behaviors. So far, the mutual soaking and fusion between fluid mechanics and other disciplines have formed many branches, many physicists and Mathematicians are working on this aspect of research. The classical Newtonian fluid mechanics believe that in parallel flow, the shear force is proportional to the shear rate, and the proportional coefficient is called the viscosity coefficient. In recent years, with the increasing importance of non-Newtonian fluids, people have found that research on non-Newtonian flow is necessary. The variational inequalities based on Riemannian manifolds were first proposed by S.Z.Németh. S.Z.Németh studied the existence of solutions to variational inequalities on Hadamard manifolds. For the equivalence of non-differentiable vector optimization problems on Riemannian manifolds and generalized weak vector variational-like inequalities, we give some new conclusions.

Learning a virtual codec based on deep convolutional neural network to compress image

In this paper, we propose a standard-compliant image compression framework based on image representation network (IRN) and post-processing neural network (PNN), which are trained by learning a virtual codec network (VCN). Firstly, we use a mixed-resolution image coding considering different types of distortions caused by image compression with different quality factors. Secondly, the VCN is introduced to learn a differentiable soft-projection from the represented image to the post-processed image to resolve the non-differentiable problem of hard quantization. Thirdly, the PNN is used to greatly enhance the quality of decoded images, since standard codecs always result in visually unpleasant blocking artifacts and ringing artifacts. Finally, our framework is trained in an end-to-end manner, whose convolutional kernels of the IRN, PNN and VCN are initialized by pre-training an auto-encoder network. Experimental results verify that our method has higher coding efficiency than the newest image representation-based compression method and many post-processing approaches.

Stokes Problem with Slip Boundary Conditions of Friction Type: Error Analysis of a Four-Field Mixed Variational Formulation

In this work, a finite element approximation of the Stokes problem under a slip boundary condition of friction type, known as the Tresca boundary condition, is considered. We treat the approximate problem of a four field mixed formulation using the $${\mathbb {P}}^{1}$$ -bubble element for the velocity field, $${\mathbb {P}}^{1}$$ element for the pressure field and the $${\mathbb {P}}^{1}$$ element for the Lagrange multipliers $$\lambda _{n}$$ and $$\lambda _{t}$$ defined on the slip boundary. The multiplier $$\lambda _{t}$$ is introduced to regularize the non-differentiable problem, whereas $$\lambda _{n}$$ treats the impermeability condition. Existence and uniqueness results for both continuous and discrete problems are proven and an a priori error estimate is established. Numerical realization of such problem is discussed and some numerical tests are provided.

Non-differentiable Vector Programming Problem and Duality for Higher-Order Cone Convex Functions

In this paper, non-differentiable fractional program containing support function of a compact convex set is studied. A class of higher-order cone convex function is introduced which not necessarily lies in any of the existing classes. A non-trivial example of such type of functions is also introduced. Then Schaible dual is formulated for the above-said program for which weak and strong duality theorems are stated and proven using the newly defined function.

Block coordinate descent energy minimization for dynamic cohesive fracture

A block coordinate descent algorithm within a discontinuous Galerkin finite element formulation is proposed for both explicit and implicit energy minimization in solid bodies in which rigid-cohesive fractures initiate and propagate. The crack opening dependent Barenblatt type surface cohesive energy function is non-differentiable at the origin. In each iteration or each explicit update, the algorithm minimizes the potential with respect to each crack opening displacement unknown and with respect to the block of deformation unknowns, sequentially. This decomposes minimization of the full non-differentiable problem into a number of “small” non-differentiable sub-problems that must be solved locally at the inter-element boundaries of the finite element mesh and an “easy” differentiable sub-problem that characterizes global equilibrium. As a result, non-differentiability is effectively treated locally and the algorithm can be easily incorporated into standard finite element codes. On the basis of a convexity analysis of the proposed non-differentiable energy functional, we obtain a minimum cohesive process zone resolution criterion, known empirically in the previous literature as a requirement for capturing the amount of dissipated fracture energy correctly. The method is free of any regularization parameters and preserves the discrete nature of fracture without introducing a stress discontinuity at initiation of decohesion. Robustness of the method is shown through several numerical examples of fragmentation and branching and through comparisons with existing numerical and experimental results.

Graph- and finite element-based total variation models for the inverse problem in diffuse optical tomography.

Total variation (TV) is a powerful regularization method that has been widely applied in different imaging applications, but is difficult to apply to diffuse optical tomography (DOT) image reconstruction (inverse problem) due to unstructured discretization of complex geometries, non-linearity of the data fitting and regularization terms, and non-differentiability of the regularization term. We develop several approaches to overcome these difficulties by: i) defining discrete differential operators for TV regularization using both finite element and graph representations; ii) developing an optimization algorithm based on the alternating direction method of multipliers (ADMM) for the non-differentiable and non-linear minimization problem; iii) investigating isotropic and anisotropic variants of TV regularization, and comparing their finite element (FEM)- and graph-based implementations. These approaches are evaluated on experiments on simulated data and real data acquired from a tissue phantom. Our results show that both FEM and graph-based TV regularization is able to accurately reconstruct both sparse and non-sparse distributions without the over-smoothing effect of Tikhonov regularization and the over-sparsifying effect of L1 regularization. The graph representation was found to out-perform the FEM method for low-resolution meshes, and the FEM method was found to be more accurate for high-resolution meshes.

Symmetric duality results for second-order nondifferentiable multiobjective programming problem

In this article, we study the existence of Gf-bonvex/Gf -pseudo-bonvex functions and construct various nontrivial numerical examples for the existence of such type of functions. Furthermore, we formulate Mond-Weir type second-order nondifferentiable multiobjective programming problem and give a nontrivial concrete example which justify weak duality theorem present in the paper. Next, we prove appropriate duality relations under aforesaid assumptions.

Local fractional homotopy analysis method for solving non-differentiable problems on Cantor sets

In this paper, we present a semi-analytic method called the local fractional homotopy analysis method (LFHAM) for solving differential equations involving local fractional derivatives based on the local fractional calculus and the homotopy analysis method. The suggested analytical technique always provides a simple way of constructing a series of solutions from the higher-order deformation equation. The LFHAM guarantees the convergence of the series solutions using the nonzero convergence-control parameter. Three examples are provided to illustrate the efficiency and high accuracy of the method.

Motion Compensated Dynamic MRI Reconstruction With Local Affine Optical Flow Estimation.

This paper proposes a novel framework to reconstruct dynamic magnetic resonance imaging (DMRI) with motion compensation (MC). Specifically, by combining the intensity-based optical flow constraint with the traditional compressed sensing scheme, we are able to jointly reconstruct the DMRI sequences and estimate the interframe motion vectors. Then, the DMRI reconstruction can be refined through MC with the estimated motion field. By employing the coarse-to-fine multi-scale resolution strategy, we are able to update the motion field in different spatial scales. The estimated motion vectors need to be interpolated to the finest resolution scale to compensate the DMRI reconstruction. Moreover, the proposed framework is capable of handling a wide class of prior information (regularizations) for DMRI reconstruction, such as sparsity, low rank, and total variation. The formulated optimization problem is solved by a primal-dual algorithm with linesearch due to its efficiency when dealing with non-differentiable problems. Experiments on various DMRI datasets validate the reconstruction quality improvement using the proposed scheme in comparison to several state-of-the-art algorithms.

Regularity results for a class of non-differentiable obstacle problems

In this paper we prove the higher differentiability in the scale of Besov spaces of the solutions to a class of obstacle problems of the type min∫ΩF(x,z,Dz):z∈Kψ(Ω). Here Ω is an open bounded set of Rn, n≥2, ψ is a fixed function called obstacle and Kψ(Ω) is set of admissible functions z∈W1,p(Ω) such that z≥ψ a.e. in Ω. We assume that the gradient of the obstacle belongs to a suitable Besov space. The main novelty here is that we are not assuming any differentiability on the partial maps x↦F(x,z,Dz) and z↦F(x,z,Dz), but only their Hölder continuity.

Making Augmented Lagrangian Methods Computer Amenable for Equilibrium Problems

ABSTRACTWe develop three algorithms to solve the subproblems generated by the augmented Lagrangian methods introduced by Iusem-Nasri (2010) for the equilibrium problem. The first algorithm that we propose incorporates the Newton method and the other two are instances of the subgradient projection method. One of our algorithms is also capable of solving nondifferentiable equilibrium problems. Using well-known test problems, all algorithms introduced here are implemented and numerical results are reported to compare their performances.

All-in-one multicategory Ramp loss maximum margin of twin spheres support vector machine

Maximum margin of twin spheres support vector machine (MMTSSVM) is effective to deal with imbalanced data classification problems. However, it is sensitive to outliers because of the use of the Hinge loss function. To enhance the stability of MMTSSVM, we propose a Ramp loss maximum margin of twin spheres support vector machine (Ramp-MMTSSVM) in this paper. In terms of the Ramp loss function, the outliers can be given fixed loss values, which reduces the negative effect of outliers on constructing models. Since Ramp-MMTSSVM is a non-differentiable non-convex optimization problem, we adopt Concave-Convex Procedure (CCCP) approach to solve it. We also analyze the properties of parameters and verify them by one artificial experiment. Besides, we use Rest-vs.-One(RVO) strategy to extend Ramp-MMTSSVM to multi-class classification problems. The experimental results on twenty benchmark datasets indicate that no matter in binary or multi-class classification cases, our approaches both can obtain better experimental performance than the compared algorithms.

Generating Semantically Similar and Human-Readable Summaries With Generative Adversarial Networks

The application of neural networks in natural language processing, including abstractive text summarization, is increasingly attractive in recent years. However, teaching a neural network to generate a human-readable summary that reflects the core idea of the original source text (i.e., semantically similar) remains a challenging problem. In this paper, we explore using generative adversarial networks to solve this problem. The proposed model contains three components: a generator that encodes the long input text into a shorter representation; a discriminator to teach the generator to create human-readable summaries and another discriminator to restrict the output of the generator to reflect the core idea of the input text. The main training process can be carried out in an adversarial learning process. To solve the non-differentiable problem caused by the words sampling process, we use the policy gradient algorithm to optimize the generator. We evaluate the proposed model on the CNN/Daily Mail summarization task. The experimental results show that the model outperforms previous state-of-the-art models.

Toward Optimal Multiperiod Network Reconfiguration for Increasing the Hosting Capacity of Distribution Networks

A multiperiod network reconfiguration to meet the requirement of hourly hosting capacity under a minimal required number of switching operations, taking into consideration the uncertainty of renewable generation, is formulated as a nonlinear, nondifferentiable integer optimization problem with nonlinear equality and inequality constraints. A four-stage solution methodology, including an assessment stage, a time-partitioning stage, an optimization stage, and an evaluation stage, is developed to solve the constrained large-scale nonlinear integer optimization problem. A fuzzy C-means clustering algorithm is applied to partition the time period to facilitate the design of a minimum-number switching solution. A hybrid particle swarm optimization algorithm is developed to find an optimal network topology for each partitioned time period. The IEEE 123-bus case and a 1001-node system, three-phase and unbalanced, are used to evaluate the proposed method with promising results.