Hyperplane Method Research Articles

Оценка эффективности метода гиперплоскостей в задаче частичной идентификации неизвестного вещества

The problem of partial identification of a homogeneous medium of unknown chemical composition is investigated. The medium is subjected to repeated transillumination by X-ray fluxes collimated in energy and direction. According to the results of measuring the flux density of radiation entering and leaving the medium, it is necessary to establish the difference in the chemical composition of the medium from some in advance given substances. It is assumed that the measurements of radiation fluxes contain errors for which their maximum relative values are known. As a mathematical basis for the study the method of hyperplanes proposed earlier by the author is taken. The results of numerical calculations performed for problems of various dimensions and for a number of groups of specific chemical elements. Numerical estimates have been obtained for several groups of organic substances, under which any two substances will be successfully distinguished from each other by the results one single measurement experiment on transillumination. Using the hyperplane method in the identification problem for groups of organic substances showed enough high efficiency. As a result of computer calculations, it is shown that that any of the 1287 considered pairs of organic substances is well distinguishable if the total relative measurement error of the probing radiation flux does not exceed 9.726 \%. The analysis of the dependence of the mass attenuation coefficients of radiation from the atomic number for all chemical elements of the periodic system has been carried out and conclusions about the behavior of the identification accuracy with an increase in the dimension of the problem were done. A justification is given that the hyperplane method will be effective in solving the problem of identification for other groups of chemical elements and identification accuracy does not get worse as the dimension of the problem increases.

Transient Stability Assessment of Renewable Dominated Power Systems by the Method of Hyperplanes

With continuous increase of penetration of renewable energies and power electronic equipment, power grid stability incidents are emerging frequently worldwide. In this paper, the transient stability assessment of renewable dominated power systems with multiple PLL-based voltage source converters (VSCs) is studied by calculating critical clearing time (CCT). A tangent hyperplane method is developed, which approximates the boundary of basin of attraction near unstable equilibrium point by linear system theory. The detailed algorithm is presented, accompanying with four typical case studies, including a single machine infinite system, two parallel VSCs, modified IEEE 9- bus, and modified 39- bus systems. Broad EMT simulations are performed to verify the accuracy and computational effectiveness of the algorithm. In all these tests, we find that the hyperplane method can give accurate estimations of CCTs. Hence it is expected to be an efficient method. In addition, we find that generally the CCT in renewable dominated power systems is around tens of milliseconds, which is much shorter than that in traditional power systems around hundreds of milliseconds. Therefore, it might be imperative to adopt faster controls and protections for renewable dominated power systems.

A hierarchical wavefront method for LU-SGS

The lower–upper Symmetric-Gauss–Seidel (LU-SGS) method is one of typical implicit methods, especially for an application that requires high convergence and accuracy. However, the data-dependencies among grid points in the LU-SGS method prevent parallelization and vectorization, both of which are important for modern computing systems. As modern computing systems consist of multiple nodes, each of which is equipped with multi-core processors with a vector processing capability, to take advantages of these computing systems, both parallel computing and vector computing are mandatory. This paper proposes a hierarchical wavefront method that combines the wavefront and the hyperplane methods in order to achieve both parallelization and vectorization of the LU-SGS method with block-structured meshes. The evaluation results show that the hierarchical wavefront method is 4.9 times faster than the conventional wavefront method and 24.2 times faster than the hyperplane method.

Hyperplane-based time-aware knowledge graph embedding for temporal knowledge graph completion

Knowledge Graph (KG) embedding approaches have been proved effective to infer new facts for a KG based on the existing ones–a problem known as KG completion. However, most of them have focused on static KGs, in fact, relational facts in KGs often show temporal dynamics, e.g., the fact (US, has president, Barack Obama, [2009–2017]) is only valid from 2009 to 2017. Therefore, utilizing available time information to develop temporal KG embedding models is an increasingly important problem. In this paper, we propose a new hyperplane-based time-aware KG embedding model for temporal KG completion. By employing the method of time-specific hyperplanes, our model could explicitly incorporate time information in the entity-relation space to predict missing elements in the KG more effectively, especially temporal scopes for facts with missing time information. Moreover, in order to model and infer four important relation patterns including symmetry, antisymmetry, inversion and composition, we map facts happened at the same time into a polar coordinate system. During training procedure, a time-enhanced negative sampling strategy is proposed to get more effective negative samples. Experimental results on datasets extracted from real-world temporal KGs show that our model significantly outperforms existing state-of-the-art approaches for the KG completion task.

Method of hyperplanes in the problem of identification of an unknown substance

Method of hyperplanes in the problem of identification of an unknown substance

Method of Hyperplanes in the Problem of Identification of an Unknown Substance

Method of Hyperplanes in the Problem of Identification of an Unknown Substance

Mechanical System Evolution and Reasonable Structural Design Parameters of Long-Span Deck-Type Beam-Arch Composite Rigid Frame Bridge

Long-span deck-type beam-arch composite rigid frame (BACRF) bridge fully integrates the merits of arch bridges and beam bridges, and overcomes the cracking and deflection problems of continuous rigid frame bridges. As a perfect combination of beam bridges and arch bridges, the long-span deck-type BACRF bridge boasts a light structure, a strong bearing capacity, and a powerful spanning capability. From the perspective of mechanical system evolution, this paper theoretically analyzes the structural mechanics of the beam-arch composite system, establishes a half-bridge model for BACRF bridge, and derives the expressions of the internal force and displacement of the beam-arch composite system. Next, finite-element analysis was conducted to analyze how the variation of a single parameter, e.g., rise-span ratio, open web ratio, and side-to-middle span ratio, affects midspan displacement, arch-beam junction displacement, main beam bending moment, and main arch axial force. Finally, the calculation formula for deflection-span ratio of BACRF bridge was proposed based on the maximum hyperplane method. The research results provide a reference for the structural design of similar bridges.

An Improved Sampling-Based Approach for Spacecraft Proximity Operation Path Planning in Near-Circular Orbit

This paper proposes an improved sampling-based approach for spacecraft proximity operation path planning under Clohessy-Wiltshire-Hill dynamics. The proposed approach is based on a modified version of the FMT* (Fast Marching Tree) algorithm with safety strategy which is divided into three parts: (1) incorporating relative ellipse to simplify the sampling state space and avoid collision with target; (2) combining internal/external ellipse-based collision detection algorithms for hovering obstacle and non-coplanar ellipse obstacle; (3) using rotating hyperplane method to handle the coplanar ellipse obstacle and uncertainty obstacle. By referring the safety strategy to simplify the state space before FMT* algorithm, the approach can reduce the complexity of path planning, especially the resampling and collision avoidance detection cyclic process in FMT*, thereby improve the planning efficiency. Two simulated scenarios, a coplanar path planning with/without coplanar ellipse obstacle, are developed to illustrate the approach. As a result, the proposed approach appears to be potential for spacecraft proximity real-time path planning as well as other complex space mission path planning generalized to different dynamics and environments, such as On-Orbit Service, and enabling a real-time computation of low-cost trajectory.

Relaxed convex model for optimal location and sizing of DGs in DC grids using sequential quadratic programming and random hyperplane approaches

This report addresses the problem of optimal location and sizing of constant power sources (distributed generators (DGs)) in direct current (DC) networks for improving network performance in terms of voltage profiles and energy efficiency. An exact mixed-integer nonlinear programming (MINLP) method is proposed to represent this problem, considering the minimization of total power losses as the objective function. Furthermore, the power balance per node, voltage regulation limits, DG capabilities, and maximum penetration of the DG are considered as constraints. To solve the MINLP model, a convex relaxation is proposed using a Taylor series expansion, in conjunction with the transformation of the binary variables into continuous variables. The solution of the relaxed convex model is constructed using a sequential quadratic programming approach to minimize the linearization error using the Taylor series method. The solution of the relaxed convex model is used as the input for a heuristic random hyperplane method that facilitates the recovery of binary variables that solve the original MINLP model. Two DC distribution feeders, one having 21 and the other having 69 nodes, were used as test systems. Simulation results were obtained using the MATLAB/quadprog package and contrasted with the large-scale nonlinear solvers available for General algebraic modeling system (GAMS) software metaheuristic optimization approaches to demonstrate the robustness and effectiveness of our proposed methodology.

On solving generalized convex MINLP problems using supporting hyperplane techniques

Solution methods for convex mixed integer nonlinear programming (MINLP) problems have, usually, proven convergence properties if the functions involved are differentiable and convex. For other classes of convex MINLP problems fewer results have been given. Classical differential calculus can, though, be generalized to more general classes of functions than differentiable, via subdifferentials and subgradients. In addition, more general than convex functions can be included in a convex problem if the functions involved are defined from convex level sets, instead of being defined as convex functions only. The notion generalized convex, used in the heading of this paper, refers to such additional properties. The generalization for the differentiability is made by using subgradients of Clarke’s subdifferential. Thus, all the functions in the problem are assumed to be locally Lipschitz continuous. The generalization of the functions is done by considering quasiconvex functions. Thus, instead of differentiable convex functions, nondifferentiable \(f^{\circ }\)-quasiconvex functions can be included in the actual problem formulation and a supporting hyperplane approach is given for the solution of the considered MINLP problem. Convergence to a global minimum is proved for the algorithm, when minimizing an \(f^{\circ }\)-pseudoconvex function, subject to \(f^{\circ }\)-pseudoconvex constraints. With some additional conditions, the proof is also valid for \(f^{\circ }\)-quasiconvex functions, which sums up the properties of the method, treated in the paper. The main contribution in this paper is the generalization of the Extended Supporting Hyperplane method in Eronen et al. (J Glob Optim 69(2):443–459, 2017) to also solve problems with \(f^{\circ }\)-pseudoconvex objective function.

Reformulations for utilizing separability when solving convex MINLP problems

Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optimal solution and verify global optimality, these methods often require a quite accurate polyhedral approximation. In case the nonlinear functions are convex and separable to some extent, it is possible to obtain a tighter approximation by using a lifted polyhedral approximation, which can be achieved by reformulating the problem. We prove that under mild assumptions, it is possible to obtain tighter linear approximations for a type of functions referred to as almost additively separable. Here it is also shown that solvers, by a simple reformulation, can benefit from the tighter approximation, and a numerical comparison demonstrates the potential of the reformulation. The reformulation technique can also be combined with other known transformations to make it applicable to some nonseparable convex functions. By using a power transform and a logarithmic transform the reformulation technique can for example be applied to p-norms and some convex signomial functions, and the benefits of combining these transforms with the reformulation technique are illustrated with some numerical examples.

Maximum Principle of Optimal Stochastic Control with Terminal State Constraint and Its Application in Finance

This paper considers the optimal control problem for a general stochastic system with general terminal state constraint. Both the drift and the diffusion coefficients can contain the control variable and the state constraint here is of non-functional type. The author puts forward two ways to understand the target set and the variation set. Then under two kinds of finite-codimensional conditions, the stochastic maximum principles are established, respectively. The main results are proved in two different ways. For the former, separating hyperplane method is used; for the latter, Ekeland’s variational principle is applied. At last, the author takes the mean-variance portfolio selection with the box-constraint on strategies as an example to show the application in finance.

Stochastic dynamic analysis of floating production systems using the First Order Reliability Method and the Secant Hyperplane Method

The dynamic analysis of a deepwater floating production systems has many complexities, such as the dynamic coupling between the vessel and the riser, the coupling between the first-order and second-order wave forces, several sources of nonlinearities. These complexities can be captured by fully coupled time domain analyses. Moreover, the sea state is random; hence the need of stochastic dynamic analysis. In this paper the evaluation of the non-Gaussian distributions of the responses of the systems is developed through the well-known First-Order Reliability Method (FORM) and the recently proposed Secant Hyperplane Method (SHM). They give rise to two stochastic Equivalent Linear Systems (ELS) allowing to determine any quantity of engineering interest: the Tail Equivalent Linear System (TELS) based on FORM and the Tail Probability Equivalent Linear System (TPELS) based on SHM. The TELS is the Equivalent Linear System (ELS) having the same design point of the original nonlinear system. The Tail Probability Equivalent Linear System (TPELS) is the ELS where the difference in terms of tail probability between the TPELS and the original system is minimized. A simplified 2-degrees-of-freedom model is used to demonstrate how these methods can be effective for stochastic dynamic analysis of a marine riser.

Convexity and optimization with copulæ structured probabilistic constraints

Probability constraints play a key role in optimization problems involving uncertainties. These constraints request that an inequality system depending on a random vector has to be satisfied with a high enough probability. In specific settings, copulæ can be used to model the probabilistic constraints with uncertainty on the left-hand side. In this paper, we provide eventual convexity results for the feasible set of decisions under local generalized concavity properties of the constraint mappings and involved copulæ. The results cover all Archimedean copulæ. We consider probabilistic constraints wherein the decision and random vector are separated, i.e. left/right-hand side uncertainty. In order to solve the underlying optimization problem, we propose and analyse convergence of a regularized supporting hyperplane method: a stabilized variant of generalized Benders decomposition. The algorithm is tested on a large set of instances involving several copulæ among which the Gaussian copula. A Numerical comparison with a (pure) supporting hyperplane algorithm and a general purpose solver for non-linear optimization is also presented.

Secant Hyperplane Method for Structural Reliability Analysis

AbstractThe first-order reliability method (FORM) is a powerful tool for structural reliability analysis, providing a good trade-off between computational accuracy and efficiency. Nevertheless, it has two main drawbacks: (1) its approximation is not necessarily adequate for limit state surfaces which depart significantly from linearity, and (2) it does not give information about the degree of accuracy achieved. In this paper, the secant hyperplane method (SHM) is proposed, which is a new linear approximation of the limit state surface based on the support vector method (SVM). The key idea is to treat structural reliability as a classification problem and to seek a suitable secant hyperplane to the limit state that gives improved approximation over the use of a tangent hyperplane. The two drawbacks of FORM are thereby resolved, while keeping the geometrical simplicity and ease of implementation of FORM. The implication of very high-dimensional geometry is examined, demonstrating that FORM can give good app...

Minimization of classification samples for supercritical and subcritical patterns of supersonic inlet

In order to investigate sample minimization for classification of supercritical and subcritical patterns in supersonic inlet, three optimization methods, namely, opposite one towards nearest method, closest one towards the hyper-plane method and random selection method, are proposed for investigation on minimization of classification samples for supercritical and subcritical patterns of supersonic inlet. The study has been carried out to analyze wind tunnel test data and to compare the classification accuracy based on those three methods with or without priori knowledge. Those three methods are different from each other by different selecting methods for samples. The results show that one of the optimization methods needs the minimization samples to get the highest classification accuracy without priori knowledge. Meanwhile, the number of minimization samples needed to get highest classification accuracy can be further reduced by introducing priori knowledge. Furthermore, it demonstrates that the best optimization method has been found by comparing all cases studied with or without introducing priori knowledge. This method can be applied to reduce the number of wind tunnel tests to obtain the inlet performance and to identify the supercritical/subcritical modes for supersonic inlet.

An interior approximal method for solving pseudomonotone equilibrium problems

In this paper, we present an interior approximal method for solving equilibrium problems for pseudomonotone bifunctions without Lipschitz-type continuity on polyhedra. The method can be viewed as combining a special interior proximal function, which replaces the usual quadratic function, Armijo-type linesearch techniques and the cutting hyperplane methods. Convergence properties of the method are established, among them the global convergences are proved under few assumptions. Finally, we present some preliminary computational results to Cournot-Nash oligopolistic market equilibrium models.

An interior proximal cutting hyperplane method for equilibrium problems

We propose a new method for solving equilibrium problems on polyhedra, where the underlying function is continuous and satisfies a pseudomonone assumption which is called an interior proximal cutting hyperplane method. The method is based on the special interior proximal function which replaces the usual quadratic function. This leads to an interior proximal algorithm. The algorithm can be viewed as combining the cutting hyperplane method and the special interior proximal function. Finally some preliminary computational results are given. AMS Mathematics Subject Classification 2000: 65K10; 90 C25.

A DIAMOND method of inducing classification rules for biological data

Identifying the classification rules for patients, based on a given dataset, is an important role in medical tasks. For example, the rules for estimating the likelihood of survival for patients undergoing breast cancer surgery are critical in treatment planning. Many well-known classification methods (as decision tree methods and hyper-plane methods) assume that classes can be separated by a linear function. However, these methods suffer when the boundaries between the classes are non-linear. This study presents a novel method, called DIAMOND, to induce classification rules from datasets containing non-linear interactions between the input data and the classes to be predicted. Given a set of objects with some classes, DIAMOND separates the objects into different cubes, and assigns each cube to a class. Via the unions of these cubes, DIAMOND uses mixed-integer programs to induce classification rules with better rates of accuracy, support and compact. This study uses three practical datasets (Iris flower, HSV patients, and breast cancer patients) to illustrate the advantages of DIAMOND over some current methods.

New separating hyperplane method with application to the optimisation of direct marketing campaigns

In this article we present a new class of separating hyperplane methods for the binary classification task. Our hyperplanes have a very low Vapnik–Chervonenkis dimension, so they generalise well. Geometrically, our approach is based on searching of a proper pair of observations from different classes of the explained variable. Once this pair is found the discriminant hyperplane becomes orthogonal to the line connecting these observations. This method allows the direct optimisation of any prediction criterion, not necessary the fraction of correctly classified observations. Models generated by this technique have low computational complexity and allow fast classification. We illustrate the performance of our method by applying it to the problem of optimisation of direct marketing campaigns, where the natural measure of the prediction performance is the lift curve.