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The purpose of this paper is to improve the GM(1, 1) model based on concave sequences.

First, the restored sequence of the GM(1, 1) model is proved to be convex, and the residual characters of the GM(1, 1) model for concave sequences are analyzed. Second, two symmetry transformations are introduced to transform an original concave sequence into a convex sequence, and then the GM(1, 1) model is established based on the convex sequence.

Compared with the traditional modeling method, the new method has high accuracy and is applicable for all concave sequence modeling.

Two cases are used to illustrate the superiority of this modeling method. Case A is to predict China’s per capita natural gas consumption, and case B is to predict the annual output of an oilfield.

The application scope of GM (1, 1) model is greatly extended.

In this paper, an online convex optimization method for the exoatmospheric ascent trajectory of space interceptors is proposed. The purpose of this paper is to transform the original trajectory optimization problem into a sequence of convex optimization subproblems.

For convenience in calculating accuracy and efficiency, the complex nonlinear terminal orbital elements constraints are converted into several quadratic equality constraints, which can be better computed by a two-step correction method during the iteration. First, the nonconvex thrust magnitude constraint is convexified by the lossless convexification technique. Then, discretization and successive linearization are introduced to transform the original problem into a sequence of one convex optimization subproblem, considering different flight phases. Parameters of trust-region and penalty are also applied to improve the computation performance. To correct the deviation in real time, the iterative guidance method is applied before orbit injection.

Numerical experiments show that the algorithm proposed in this paper has good convergence and accuracy. The successive progress can converge in a few steps and 3–4 s of CPU time. Even under engine failure or mission change, the algorithm can yield satisfactory results.

The convex optimization method presented in this paper is expected to generate a reliable optimal trajectory rapidly in different situations and has great potential for onboard applications of space interceptors.

The originality of this paper lies in the proposed online trajectory optimization method and guidance algorithm of the space inceptors, especially for onboard applications in emergency situations.

First, the conditions for GM(1,1) modeling are analyzed, and it revealed that the whole trend of the class ratio sequence and the whole concave‐convex of data sequence both affect the precision of the model. Secondly, the action of translation transformation on the features of data sequence is studied theoretically, and the satisfied interval of translation values is derived. Finally, an effective numerical method is presented which is illustrated by a medical example.

Bounds on the rate of convergence of learning processes based on random samples and probability are one of the essential components of statistical learning theory (SLT). The constructive distribution‐independent bounds on generalization are the cornerstone of constructing support vector machines. Random sets and set‐valued probability are important extensions of random variables and probability, respectively. The paper aims to address these issues.

In this study, the bounds on the rate of convergence of learning processes based on random sets and set‐valued probability are discussed. First, the Hoeffding inequality is enhanced based on random sets, and then making use of the key theorem the non‐constructive distribution‐dependent bounds of learning machines based on random sets in set‐valued probability space are revisited. Second, some properties of random sets and set‐valued probability are discussed.

In the sequel, the concepts of the annealed entropy, the growth function, and VC dimension of a set of random sets are presented. Finally, the paper establishes the VC dimension theory of SLT based on random sets and set‐valued probability, and then develops the constructive distribution‐independent bounds on the rate of uniform convergence of learning processes. It shows that such bounds are important to the analysis of the generalization abilities of learning machines.

SLT is considered at present as one of the fundamental theories about small statistical learning.

The purpose of this paper is to introduce some basic knowledge of statistical learning theory (SLT) based on random set samples in set‐valued probability space for the first time and generalize the key theorem and bounds on the rate of uniform convergence of learning theory in Vapnik, to the key theorem and bounds on the rate of uniform convergence for random sets in set‐valued probability space. SLT based on random samples formed in probability space is considered, at present, as one of the fundamental theories about small samples statistical learning. It has become a novel and important field of machine learning, along with other concepts and architectures such as neural networks. However, the theory hardly handles statistical learning problems for samples that involve random set samples.

Being motivated by some applications, in this paper a SLT is developed based on random set samples. First, a certain law of large numbers for random sets is proved. Second, the definitions of the distribution function and the expectation of random sets are introduced, and the concepts of the expected risk functional and the empirical risk functional are discussed. A notion of the strict consistency of the principle of empirical risk minimization is presented.

The paper formulates and proves the key theorem and presents the bounds on the rate of uniform convergence of learning theory based on random sets in set‐valued probability space, which become cornerstones of the theoretical fundamentals of the SLT for random set samples.

The paper provides a studied analysis of some theoretical results of learning theory.

Shaft orbit is an important characteristic for vibration monitoring and diagnosing system of hydroelectric generating set. Because of the low accuracy and poor reliability of traditional methods in identifying the shaft orbit moving direction (MD), the purpose of this paper is to present a novel automatic identification method based on trigonometric function and polygon vector (TFPV).

First, some points on shaft orbit were selected with inter‐period acquisition method and joined together orderly to form a complex plane polygon. Second, by using the coordinate transformation and rotation theory, TFPV were applied comprehensively to judge the concavity or convexity of the polygon vertices. Finally, the shaft orbit MD is identified.

The simulation and experiment demonstrate that the method proposed can effectively identify the common shaft orbit MD.

In order to identity the shaft orbit MD effectively, a novel automatic identification method based on TFPV is proposed in this paper. The problem of identifying the shaft orbit MD is transformed into the problem about orientation of complex polygons, which are formed orderly by points on orbit shaft, and TFPV are applied comprehensively to judge the concavity or convexity of the polygon vertices.

Structural optimisation and reliability theory are considered, and described. A general reliability‐based structural optimisation problem is formulated, and consideration given to procedures for solving it. Two different examples suggest the efficacy of these procedures. The amount of calculations depends to a great degree on the definition of failure of the structure. In order to reduce this by improving optimisation procedures, more research is needed, and the convergence of the optimisation is very dependent on accurate evaluation of the gradients of the reliability constraints.

Topology optimization is a method used for developing optimized geometric designs by distributing material pixels in a given design space that maximizes a chosen quantity of interest (QoI) subject to constraints. The purpose of this study is to develop a problem-agnostic automatic differentiation (AD) framework to compute sensitivities of the QoI required for density distribution-based topology optimization in an unstructured co-located cell-centered finite volume framework. Using this AD framework, the authors develop and demonstrate the topology optimization procedure for multi-dimensional steady-state heat conduction problems.

Topology optimization is performed using the well-established solid isotropic material with penalization approach. The method of moving asymptotes, a gradient-based optimization algorithm, is used to perform the optimization. The sensitivities of the QoI with respect to design variables, required for optimization algorithm, are computed using a discrete adjoint method with a novel AD library named residual automatic partial differentiator (Rapid).

Topologies that maximize or minimize relevant quantities of interest in heat conduction applications are presented. The efficacy of the technique is demonstrated using a variety of realistic heat transfer applications in both two and three dimensions, in conjugate heat transfer problems with finite conductivity ratios and in non-rectangular/non-cuboidal domains.

In contrast to most published work which has either used finite element methods or Cartesian finite volume methods for transport applications, the topology optimization procedure is developed in a general unstructured finite volume framework. This permits topology optimization for flow and heat transfer applications in complex design domains such as those encountered in industry. In addition, the Rapid library is designed to provide a problem-agnostic pathway to automatically compute all required derivatives to machine accuracy. This obviates the necessity to write new code for finding sensitivities when new physics are added or new cost functions are considered and permits general-purpose implementations of topology optimization for complex industrial applications.

The purpose of this paper is to introduce the implicit midpoint rule (IMR) of nonexpansive mappings in 2- uniformly convex hyperbolic spaces and study its convergence. Strong and △-convergence theorems based on this algorithm are proved in this new setting. The results obtained hold concurrently in uniformly convex Banach spaces, CAT(0) spaces and Hilbert spaces as special cases.

In this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.

The concepts of relative ρ-stability and weak ρ-stability are introduced, and conditions in which these multistep iterations are relatively ρ-stable, weakly ρ-stable and ρ-stable are established for the newly introduced strong ρ-quasi-contractive-like class of maps.

Noor type, Ishikawa type and Mann type iterative sequences are deduced as corollaries in this study.

The results obtained in this work are complementary to those proved in normed and metric spaces in the literature.