Vertex Solution Research Articles

Proximity and Flatness Bounds for Linear Integer Optimization

This paper deals with linear integer optimization. We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists; proximity bounds)? If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, nonzero, normal vector that intersect the polyhedron (flatness bounds)? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane; if a polygon [Formula: see text] satisfies [Formula: see text], where τ denotes [Formula: see text] counterclockwise rotation and [Formula: see text] denotes the polar of K, then the area of [Formula: see text] is at least three. Funding: J. Paat was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-02475]. R. Weismantel was supported by the Einstein Stiftung Berlin.

Non-Probabilistic Uncertainty Quantification of Fiber-Reinforced Composite Laminate Based on Micro- and Macro-Mechanical Analysis

In this paper, the main aim is to study and predict macro elastic mechanical parameters of fiber-reinforced composite laminates by combining micro-mechanical analysis models and the non-probabilistic set theory. It deals with uncertain input parameters existing in quantification models as interval variables. Here, several kinds of micro-mechanical mathematical models are introduced, and the parameter vertex solution theorem and the Monte Carlo simulation method can be used to perform uncertainty quantification of macro elastic properties for composites. In order to take the correlations between macro elastic properties into consideration, the obtained limited sample points or experimental data are utilized on the basis of the grey mathematical modeling theory, where correlated uncertain macro parameters can be treated as a closed and bounded convex polyhedral model. It can give out a clear analytical description for feasible domains of correlated macro elastic properties in the process of uncertainty quantification. Finally, two numerical examples are carried out to account for the validity and feasibility of the proposed quantification method. The results show that the proposed method can become a powerful and meaningful supplement for uncertainty quantification of composite laminates and provide data support for further uncertainty propagation analysis.

Spectral solutions of PDEs on networks

To solve linear PDEs on metric graphs with standard coupling conditions (continuity and Kirchhoff's law), we develop and compare a spectral, a second-order finite difference, and a discontinuous Galerkin method. The spectral method yields eigenvalues and eigenvectors of arbitrary order with machine precision and converges exponentially. These eigenvectors provide a Fourier-like basis on which to expand the solution; however, more complex coupling conditions require additional research. The discontinuous Galerkin method provides approximations of arbitrary polynomial order; however computing high-order eigenvalues accurately requires the respective eigenvector to be well-resolved. The method allows arbitrary non-Kirchhoff flux conditions and requires special penalty terms at the vertices to enforce continuity of the solutions. For the finite difference method, the standard one-sided second-order finite difference stencil reduces the accuracy of the vertex solution to O(h3/2). To preserve overall second-order accuracy, we used ghost cells for each edge. For all three methods we provide the implementation details, their validation, and examples illustrating their performance for the eigenproblem, Poisson equation, and the wave equation.

Data-based polyhedron model for optimization of engineering structures involving uncertainties

Abstract This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.

New algorithm for the flexibility index problem of quadratic systems

A new flexibility index algorithm for systems under uncertainty and represented by quadratic inequalities is presented. Inspired by the outer‐approximation algorithm for convex mixed‐integer nonlinear programming, a similar iterative strategy is developed. The subproblem, which is a nonlinear program, is constructed by fixing the vertex directions since this class of systems is proved to have a vertex solution if the entries on the diagonal of the Hessian matrix are non‐negative. By overestimating the nonlinear constraints, a linear min–max problem is formulated. By dualizing the inner maximization problem, and introducing new variables and constraints, the master problem is reformulated as a mixed‐integer linear program. By iteratively solving the subproblem and master problem, the algorithm can be guaranteed to converge to the flexibility index. Numerical examples including a heat exchanger network, a process network, and a unit commitment problem are presented to illustrate the computational efficiency of the algorithm. © 2018 American Institute of Chemical Engineers AIChE J, 64: 2486–2499, 2018

Addition-min fuzzy relation inequalities with application in BitTorrent-like Peer-to-Peer file sharing system

The data transmission mechanism in BitTorrent-like Peer-to-Peer (P2P) file sharing systems may be reduced to some addition-min fuzzy relation inequalities. The solution set of addition-min fuzzy relation inequalities plays an important role in the corresponding optimization problem. In this paper, we study some properties of the solutions to such system. Convexity of the solution set and number of minimal solutions are discussed, with comparison to those of the classical max-T fuzzy relation equations or inequalities. Besides, vertex solution and variable-ordering minimal solution are also investigated, with application in BitTorrent-like P2P file sharing system. Two numerical examples are given to illustrate the feasibility and efficiency of the algorithm for solving the variable-ordering solution.

The vertex solution theorem and its coupled framework for static analysis of structures with interval parameters

SummaryThis work gives new statement of the vertex solution theorem for exact bounds of the solution to linear interval equations and its novel proof by virtue of the convex set theory. The core idea of the theorem is to transform linear interval equations into a series of equivalent deterministic linear equations. Then, the important theorem is extended to find the upper and lower bounds of static displacements of structures with interval parameters. Following discussions about the computational efforts, a coupled framework based on vertex method (VM) is established, which allows us to solve many large‐scale engineering problems with uncertainties using deterministic finite element software. Compared with the previous works, the contribution of this work is not only to obtain the exact bounds of static displacements but also lay the foundation for development of an easy‐to‐use interval finite element software. Numerical examples demonstrate the good accuracy of VM. Meanwhile, the implementation of VM and availability of the coupled framework are demonstrated by engineering example. Copyright © 2017 John Wiley & Sons, Ltd.

Fuzzy Finite Element Applications in the Waste Heat Boiler

A practical approach for analyzing the waste heat boiler with fuzzy parameters is developed. The methodology involves integrated finite element modeling and fuzzy analysis. The uncertainties in the material, loading and structural properties are represented using convex normal fuzzy sets. Vertex solution methodology that is based on α-cut representation is used for the fuzzy analysis. The values of the extreme responses at an α-level are obtained by applying the binary combinations of the fuzzy variables that result in extreme responses to the finite element model. The merit of the proposed methodology is computational efficiency without compromise on accuracy and this is demonstrated through the engineering problem.

Interval analysis of dynamic response of structures using Laplace transform

In this paper, an interval based method for dynamic analysis of structures with uncertain parameters using Laplace transform is proposed. The structural physical parameters and the external loads are considered as interval variables. The structural stiffness matrix, mass matrix and loading vectors are thus described as the sum of two parts corresponding to the deterministic matrix and the uncertainty matrix of the interval parameters. The Laplace transform is used to convert the dynamic equations into a linear system of equations. The matrix perturbation technique is then utilized to remove the higher order terms, and the inverse Laplace transform is employed to obtain the structural dynamic responses. In addition, the element-by-element (EBE) idea used previously in static analysis is extended to dynamic analysis. A special matrix treatment is also used for both the EBE and non-EBE cases to reduce the overestimation of interval analysis and to facilitate the inverse Laplace transform. Finally, the effectiveness of the proposed method is demonstrated by numerical examples compared with the vertex solutions and other researchers’ work. An interesting finding is the divergent response of the undamped system, which is verified by the vertex solution. The proposed method is suitable for small uncertainties of system parameters since the formulation is limited to the first order terms, which results in an inner enclosure of the system response.

Structural Reliability Analysis with Uncertainty-but-Bounded Parameters Based on Meshless Method

The structural reliability analysis with uncertainty-but-bounded parameters is considered in this paper. Each uncertain-but-bounded parameter is represented as an interval number or vector, an interval reliability index is defined and discussed. Due to the wide application of the Meshless method, it is used in structural stress and strain analysis. The target variable of requiring reliability analysis is estimated via Taylor expansion. Based on optimization theory and vertex solution theorem, the upper and lower bounds of the target variables are obtained, and also the interval reliability index. A typical elastostatics example is presented to illustrate the computational aspects of interval reliability analysis in comparison with the traditional probability method, it can be seen that the result calculated by the vertex solution theorem is consistent with that calculated by probability method.

Interval model updating with irreducible uncertainty using the Kriging predictor

Interval model updating in the presence of irreducible uncertain measured data is defined and solutions are made available for two cases. In the first case, the parameter vertex solution is used but is found to be valid only for particular parameterisation of the finite element model and particular output data. In the second case, a general solution is considered, based on the use of a meta-model which acts as a surrogate for the full finite element mathematical model. Thus, a region of input data is mapped to a region of output data with parameters obtained by regression analysis. The Kriging predictor is chosen as the meta-model in this paper and is found to be capable of predicting the regions of input and output parameter variations with very good accuracy. The interval model updating approach is formulated based on the Kriging predictor and an iterative procedure is developed. The method is validated numerically using a three degree of freedom mass-spring system with both well-separated and close modes. A significant advantage of Kriging interpolation is that it enables the use of updating parameters that are difficult to use by conventional correction of the finite element model. An example of this is demonstrated in an experimental exercise where the positions of two beams in a frame structure are selected as updating parameters.

The time response of structures with bounded parameters and interval initial conditions

Uncertainty plays an important role in the performance of structures. In this paper, we focus on the dynamic response of structures with bounded parameters and interval initial conditions, and present a new method to determine the supremum and infimum of the time response. The method is based on the vertex solution theorem for the first-order deviation of the dynamic response from its central value and avoids interval extension problems present in current methods, where the length of the interval increases significantly due to the intermediate calculations. The method is more accurate than existing perturbation methods and provides tighter bounds on the response. The approach neglects the second-order terms in the equation of motion, and care should be exercised when the parameter variations are large. The other advantage of this method is its ability to solve problems with uncertainties in the initial conditions.

Vertex solution theorem for the upper and lower bounds on the dynamic response of structures with uncertain-but-bounded parameters

The aim of this paper is to evaluate the effects of uncertain-but-bounded parameters on the dynamic response of structures. By combining the interval mathematics and the finite element analysis, the mass matrix, damping matrix, stiffness matrix and the external loads are represented as interval matrices and vector. With the help of the optimization theory, we present the vertex solution theorem for determining both the exact upper bounds or maximum values and the exact lower bounds or minimum values of the dynamic response of structures, in which these parameters reach their extreme values on the boundary of the interval mass, damping, stiffness matrices and the interval external loads vector. Three examples are used to illustrate the computational aspects of the presented vertex solution theorem.

Parallel computing for static response analysis of structures with uncertain-but-bounded parameters

The vertex solution for estimation on the static displacement bounds of structures with uncertain-but-bounded parameters is studied in this paper. For the linear static problem, when there are uncertain interval parameters in the stiffness matrix and the vector of applied forces, the static response may be an interval. Based on the interval operations, the interval solution obtained by the vertex solution is more accurate and more credible than other methods (such as the perturbation method). However, the vertex solution method by traditional serial computing usually needs large computational efforts, especially for large structures. In order to avoid its disadvantages of large calculation and much runtime, its parallel computing which can be used in large-scale computing is presented in this paper. Two kinds of parallel computing algorithms are proposed based on the vertex solution. The parallel computing will solve many interval problems which cannot be resolved by traditional interval analysis methods.

Some disadvantages of a Mehrotra-type primal-dual corrector interior point algorithm for linear programming

Employing a new primal-dual corrector algorithm, we investigate the impact that corrector directions may have on the convergence behaviour of predictor–corrector methods. The Primal-Dual Corrector ( pdc) algorithm that we propose computes on each iteration a corrector direction in addition to the direction of the standard primal-dual path-following interior point method [M. Kojima, S. Mizuno, A. Yoshise, A primal-dual interior point algorithm for linear programming, in: Progress in Mathematical Programming, Pacific Grove, CA, 1987, Springer, New York, 1989, pp. 29–47] for Linear Programming ( lp), in an attempt to improve performance. The new iterate is chosen by moving along the sum of these directions, from the current iterate. This technique is similar to the construction of Mehrotra's highly popular predictor–corrector algorithm [S. Mehrotra, On finding a vertex solution using interior point methods, Linear Algebra Appl. 152 (1991) 233–253]. We present examples, however, that show that the pdc algorithm may fail to converge to a solution of the lp problem, in both exact and finite arithmetic, regardless of the choice of stepsize that is employed. The cause of this bad behaviour is that the correctors exert too much influence on the direction in which the iterates move.

The static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem

In practical engineering, on account of physical imperfections, model inaccuracies, and system complexities, almost all structures have physical and geometrical uncertainties in some degrees. In this paper, we focus on the static response problem and the stress distribution of structures with bounded uncertainties. In terms of the vertex notation in interval analysis, a new mathematical proof of the vertex solution theorem which is used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties is given. Then, we extend this theorem for calculating the interval stress and the deformation distributions, which was only used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties. The basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations. The resulting deterministic linear equations are then solved by using the familiar techniques like the Gauss elimination method or the Gauss–Seidel iteration method. Another advantage of this method is that the increase in number of the uncertain parameters in a structure element will not result in the increase in number of the uncertain elements in the stiffness matrix. For practical engineering problem with bounded uncertainties, the chief thing is to avoid the disadvantages of large calculation and much runtime of the vertex solution. In this paper, we present the parallel arithmetic of the vertex solution theorem, which can be used in large-scale computations in practical engineering to avoid much runtime. On the other hand, we can use the symmetrical characteristic of the stiffness matrix and the substructure method to reduce the computational effort in practical engineering. The numerical results of the typical example presented by Deif calculated by the proposed vertex solution theorem show that the width between the supremum and the infimum on the static response yielded by the vertex solution theorem is tighter than those produced by the KCN’s method presented by Köylüoğlu, Cakmak, and Nielson. The numerical examples of the two-dimensional shell with an edge fixed and the two-dimensional shell with surrounding fixed are used to illustrate the computational feasibility of the vertex solution theorem.

A universal scaling theory for complexity of analog computation

We discuss the computational complexity of solving linear programming problems by means of an analog computer. The latter is modeled by a dynamical system which converges to the optimal vertex solution. We analyze various probability ensembles of linear programming problems. For each one of these we obtain numerically the probability distribution functions of certain quantities which measure the complexity. Remarkably, in the asymptotic limit of very large problems, each of these probability distribution functions reduces to a universal scaling function, depending on a single scaling variable and independent of the details of its parent probability ensemble. These functions are reminiscent of the scaling functions familiar in the theory of phase transitions. The results reported here extend analytical and numerical results obtained recently for the Gaussian ensemble.

Quark-gluon vertex dressing and meson masses beyond ladder-rainbow truncation

We include a generalized infinite class of quark-gluon vertex dressing diagrams in a study of how dynamics beyond the ladder-rainbow truncation influences the Bethe-Salpeter description of light-quark pseudoscalar and vector mesons. The diagrammatic specification of the vertex is mapped into a corresponding specification of the Bethe-Salpeter kernel, which preserves chiral symmetry. This study adopts the algebraic format afforded by the simple interaction kernel used in previous work on this topic. The new feature of the present work is that in every diagram summed for the vertex and the corresponding Bethe-Salpeter kernel, each quark-gluon vertex is required to be the self-consistent vertex solution. We also adopt from previous work the effective accounting for the role of the explicitly non-Abelian three-gluon coupling in a global manner through one parameter determined from recent lattice-QCD data for the vertex. Within the current model, the more consistent dressed vertex limits the ladder-rainbow truncation error for vector mesons to be never more than 10% as the current quark mass is varied from the $u/d$ region to the $b$ region.

Exact bounds for the static response set of structures with uncertain-but-bounded parameters

The numerical estimation of the static displacement bounds of structures with uncertain-but-bounded parameters is considered in this paper. By representing each uncertain-but-bounded parameter as an interval number or vector, a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations, in which the coefficient matrix and the right-hand side term are, respectively, the interval matrix and the interval vector. In this study, we present two new simple mathematical proofs of the vertex solution theorem using Cramer’s rule for solving linear interval equations, different from the other proof methods, to find the upper and lower bounds on the set of solutions. The first takes advantage of optimization theory, while the second is based on interval extension. By means of a typical example considered first by Hansen, it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the Oettli–Prager criterion. Examples of a three-stepped beam and a 10-bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method.

Several solution methods for the generalized complex eigenvalue problem with bounded uncertainties

The aim of this paper is to evaluate the effects of uncertain-but-bounded parameters on the complex eigenvalues of the non-proportional damping structures. By combining the interval mathematics and the finite element analysis, the mass matrix, the damping matrix and the stiffness matrix were represented as the interval matrices. Firstly, with the help of the optimization theory, we presented an exact solution—the vertex solution theorem, for determining the exact upper bounds or maximum values and exact lower bounds or minimum values of complex eigenvalues of structures, where the extreme values are reached on the boundary of the interval mass, damping and stiffness matrices. Then, an interval perturbation method was proposed, which needs less computational efforts. A numerical example of a seven degree-of-freedom spring-damping-mass system was used to illustrate the computational aspects of the presented vertex solution theorem and the interval perturbation method in comparison with Deif’s method.