Matching is an important prerequisite for point clouds registration, which is to establish a reliable correspondence between two point clouds. This paper aims to improve recent theoretical and algorithmic results on discrete optimal transport (DOT), since it lacks robustness for the point clouds matching problems with large-scale affine or even nonlinear transformation. We first consider the importance of the used prior probability for accurate matching and give some theoretical analysis. Then, to solve the point clouds matching problems with complex deformation and noise, we propose an improved DOT model, which introduces an orthogonal matrix and a diagonal matrix into the classical DOT model. To enhance its capability of dealing with cases with outliers, we further bring forward a relaxed and regularized DOT model. Meantime, we propose two algorithms to solve the brought forward two models. Finally, extensive experiments on some real datasets are designed in the presence of reflection, large-scale rotation, stretch, noise, and outliers. Some state-of-the-art methods, including CPD, APM, RANSAC, TPS-ICP, TPS-RPM, RPMNet, and classical DOT methods, are to be discussed and compared. For different levels of degradation, the numerical results demonstrate that the proposed methods perform more favorably and robustly than the other methods.

A distributed optimization problem (DOP) with affine equality and convex inequality constraints is studied in this article. First, the consensus constraint of the considered DOP is relaxed and a related approximate DOP (ADOP) is presented. It is proved that the optimal solutions of the ADOP (i.e., the near-optimal solutions of the original DOP) are able to approach the optimal solutions of the original DOP. A continuous-time algorithm is proposed for the ADOP and it is demonstrated that the state solution of the presented algorithm converges to the critical point set of the ADOP with general locally Lipschitz continuous objective functions. This means the presented algorithm is efficient for distributed nonconvex optimization problems. Particularly, when the objective functions are convex ones, the state solution of the presented algorithm is further proved to converge to a near-optimal solution of the original DOP. One illustrative example and an application on load sharing problems are shown to validate the effectiveness of the proposed algorithm.

In this article, we consider a class of nonsmooth, nonconvex, and non-Lipschitz optimization problems, which have wide applications in sparse optimization. We generalize the Clarke stationary point and define a kind of generalized stationary point of the problems with a stronger optimal capability. Based on the smoothing method, we propose a projected neural network for solving this kind of optimization problem. Under the condition that the level set of objective function in the feasible region is bounded, we prove that the solution of the proposed neural network is globally existent and bounded. The uniqueness of the solution of the proposed network is also analyzed. When the feasible region is bounded, any accumulation point of the proposed neural network is a generalized stationary point of the optimization model. Based on some suitable conditions, any solution of the proposed neural network is asymptotic convergent to one stationary point. In particular, we give some deep analysis on the proposed network for solving a special class of the non-Lipschitz optimization problem, which indicates a lower bound property and the unify identification for the nonzero elements of all accumulation points. Finally, some numerical results are presented to show the efficiency of the proposed neural network for solving some kinds of sparse optimization models.

Pseudoconvex optimization problem, as an important nonconvex optimization problem, plays an important role in scientific and engineering applications. In this paper, a recurrent one-layer neural network is proposed for solving the pseudoconvex optimization problem with equality and inequality constraints. It is proved that from any initial state, the state of the proposed neural network reaches the feasible region in finite time and stays there thereafter. It is also proved that the state of the proposed neural network is convergent to an optimal solution of the related problem. Compared with the related existing recurrent neural networks for the pseudoconvex optimization problems, the proposed neural network in this paper does not need the penalty parameters and has a better convergence. Meanwhile, the proposed neural network is used to solve three nonsmooth optimization problems, and we make some detailed comparisons with the known related conclusions. In the end, some numerical examples are provided to illustrate the effectiveness of the performance of the proposed neural network.

In this paper, a two-layer recurrent neural network is proposed to solve the nonsmooth convex optimization problem subject to convex inequality and linear equality constraints. Compared with existing neural network models, the proposed neural network has a low model complexity and avoids penalty parameters. It is proved that from any initial point, the state of the proposed neural network reaches the equality feasible region in finite time and stays there thereafter. Moreover, the state is unique if the initial point lies in the equality feasible region. The equilibrium point set of the proposed neural network is proved to be equivalent to the Karush-Kuhn-Tucker optimality set of the original optimization problem. It is further proved that the equilibrium point of the proposed neural network is stable in the sense of Lyapunov. Moreover, from any initial point, the state is proved to be convergent to an equilibrium point of the proposed neural network. Finally, as applications, the proposed neural network is used to solve nonlinear convex programming with linear constraints and L1 -norm minimization problems.

A new class of kernels has been designed to enhance the usability of prior knowledge. Prior knowledge is shown to improve the generalization ability of kernel algorithms for binary classification problems. The prior knowledge is expressed in natural language via fuzzy rules. First, the concepts of a fuzzy rule base and prior-confidence region are proposed to formulate the prior knowledge. Then, the new kernels, which are referred to as natural classification kernels (NCKs), are represented by fuzzy equivalence relations based on the formulation of prior knowledge. An NCK is interpreted as a measure of similarities between samples. It is proven that NCKs have two desired properties: 1) transitivity with respect to the triangular norms and 2) the ability to provide higher similarities to spatially closer samples from the same class. Using transitivity, a large number of NCKs may be directly obtained by means of triangular norms. Additionally, the theoretical results show that the second property makes it possible for the support vector machine (SVM) and convex hull separation algorithm to generalize from training samples to test samples in the prior-confidence region. Finally, some synthetic datasets and a benchmark dataset are employed to validate the proposed approach.

This paper presents a subgradient-based neural network to solve a nonsmooth nonconvex optimization problem with a nonsmooth nonconvex objective function, a class of affine equality constraints, and a class of nonsmooth convex inequality constraints. The proposed neural network is modeled with a differential inclusion. Under a suitable assumption on the constraint set and a proper assumption on the objective function, it is proved that for a sufficiently large penalty parameter, there exists a unique global solution to the neural network and the trajectory of the network can reach the feasible region in finite time and stay there thereafter. It is proved that the trajectory of the neural network converges to the set which consists of the equilibrium points of the neural network, and coincides with the set which consists of the critical points of the objective function in the feasible region. A condition is given to ensure the convergence to the equilibrium point set in finite time. Moreover, under suitable assumptions, the coincidence between the solution to the differential inclusion and the "slow solution" of it is also proved. Furthermore, three typical examples are given to present the effectiveness of the theoretic results obtained in this paper and the good performance of the proposed neural network.