A cosine maximization method for the priority vector derivation in AHP

The derivation of a priority vector from a pair-wise comparison matrix (PCM) is an important issue in the Analytic Hierarchy Process (AHP). The existing methods for the priority vector derivation from PCM include eigenvector method (EV), weighted least squares method (WLS), additive normalization method (AN), logarithmic least squares method (LLS), etc. The derived priority vector should be as similar to each column vector of the PCM as possible if a pair-wise comparison matrix (PCM) is not perfectly consistent. Therefore, a cosine maximization method (CM) based on similarity measure is proposed, which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a PCM. An optimization model for the CM is proposed to derive the reliable priority vector. Using three numerical examples, the CM is compared with the other prioritization methods based on two performance evaluation criteria: Euclidean distance and minimum violation. The results show that the CM is flexible and efficient.     

Rank-One Approximation of Positive Matrices Based on Methods of Tropical Mathematics

Low-rank matrix approximation finds wide application in the analysis of big data, in recommendation systems on the Internet, for the approximate solution of some equations of mechanics, and in other fields. In this paper, a method for approximating positive matrices by rank-one matrices on the basis of minimization of log-Chebyshev distance is proposed. The problem of approximation reduces to an optimization problem having a compact representation in terms of an idempotent semifield in which the operation of taking the maximum plays the role of addition and which is often referred to as max-algebra. The necessary definitions and preliminary results of tropical mathematics are given, on the basis of which the solution of the original problem is constructed. Using the methods and results of tropical optimization, all positive matrices at which the minimum of approximation error is reached are found in explicit form. A numerical example illustrating the application of the rank-one approximation is considered.     

Convergent Outer Approximation Algorithms for Solving Unary Programs

Interesting cutting plane approaches for solving certain difficult multiextremal global optimization problems can fail to converge. Examples include the concavity cut method for concave minimization and Ramana's recent outer approximation method for unary programs which are linear programming problems with an additional constraint requiring that an affine mapping becomes unary. For the latter problem class, new convergent outer approximation algorithms are proposed which are based on sufficiently deep l-norm or quadratic cuts. Implementable versions construct optimal simplicial inner approximations of Euclidean balls and of intersections of Euclidean balls with halfspaces, which are of general interest in computational convexity. Computational behavior of the algorithms depends crucially on the matrices involved in the unary condition. Potential applications to the global minimization of indefinite quadratic functions subject to indefinite quadratic constraints are shown to be practical only for very small problem sizes.     

欧氏环上一次不定方程组的矩阵解法

利用欧几里德算法从理论上对多元一次不定方程组在欧氏环上的可逆线性变换下的解进行深入的研究 ,并提出用矩阵的初等变换求解欧氏环上多元一次不定方程组的算法 ,即矩阵解法 .     

Decomposition method in correction problems for inconsistent systems of linear inequalities with partitioned matrices

Correction problems for inconsistent systems of linear inequalities with partitioned matrices are examined. The criteria used are based on the minimum of the sum of squares and the minimax weighted Euclidean norm of a block in the correction matrix.     

The block conjugate gradient algorithm and related methods

The development of the Lanczos algorithm for finding eigenvalues of large sparse symmetric matrices was followed by that of block forms of the algorithm. In this paper, similar extensions are carried out for a relative of the Lanczos method, the conjugate gradient algorithm. The resulting block algorithms are useful for simultaneously solving multiple linear systems or for solving a single linear system in which the matrix has several separated eigenvalues or is not easily accessed on a computer. We develop a block biconjugate gradient algorithm for general matrices, and develop block conjugate gradient, minimum residual, and minimum error algorithms for symmetric semidefinite matrices. Bounds on the rate of convergence of the block conjugate gradient algorithm are presented, and issues related to computational implementation are discussed. Variants of the block conjugate gradient algorithm applicable to symmetric indefinite matrices are also developed.     

Axiomatizations of the Euclidean compromise solution

A multicriteria optimization problem is one of choosing an alternative that optimizes several—possibly conflicting—objective functions simultaneously. The utopia point of a multicriteria optimization problem is the vector that specifies for each objective function the most favorable feasible value. The Euclidean compromise solution in multicriteria optimization is a solution that selects from a feasible set the alternative such that its vector of criteria values has minimal Euclidean distance to the utopia point. This paper provides several axiomatic characterizations of the Euclidean compromise solution that are based on consistency properties.     

Solving the uncapacitated multi-facility Weber problem by vector quantization and self-organizing maps

The uncapacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane and allocating the demands of n customers to these facilities with the minimum total transportation cost. This is a non-convex optimization problem and difficult to solve exactly. As a consequence, efficient and accurate heuristic solution procedures are needed. The problem has different types based on the distance function used to model the distance between the facilities and customers. We concentrate on the rectilinear and Euclidean problems and propose new vector quantization and self-organizing map algorithms. They incorporate the properties of the distance function to their update rules, which makes them different from the existing two neural network methods that use rather ad hoc squared Euclidean metric in their updates even though the problem is originally stated in terms of the rectilinear and Euclidean distances. Computational results on benchmark instances indicate that the new methods are better than the existing ones, both in terms of the solution quality and computation time.     

Entropy of orthogonal matrices and minimum distance orthostochastic matrices from the uniform van der Waerden matrices

In this article we formulate an optimization problem of minimizing the distance from the uniform van der Waerden matrices to orthostochastic matrices of different orders. We find a lower bound for the number of stationary points of the minimization problem, which is connected to the number of possible partitions of a natural number. The existence of Hadamard matrices ensures the existence of global minimum orthostochastic matrices for such problems. The local minimum orthostochastic matrices have been obtained for all other orders except for 11 and 19. We explore the properties of Hadamard, conference and weighing matrices to obtain such minimizing orthostochastic matrices.     

Nonlinear mapping and distance geometry

Distance Geometry Problem (DGP) and Nonlinear Mapping (NLM) are two well established questions: DGP is about finding a Euclidean realization of an incomplete set of distances in a Euclidean space, whereas Nonlinear Mapping is a weighted Least Square Scaling (LSS) method. We show how all these methods (LSS, NLM, DGP) can be assembled in a common framework, being each identified as an instance of an optimization problem with a choice of a weight matrix. We study the continuity between the solutions (which are point clouds) when the weight matrix varies, and the compactness of the set of solutions (after centering). We finally study a numerical example, showing that solving the optimization problem is far from being simple and that the numerical solution for a given procedure may be trapped in a local minimum.

     

Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction

Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP) to reconstruct such faithful representations. While those SDP models are capable of producing high quality configuration numerically, they suffer two major drawbacks. One is that there exist no theoretically guaranteed bounds on the quality of the configuration. The other is that they are slow in computation when the data points are beyond moderate size. In this paper, we propose a convex optimization model of Euclidean distance matrices. We establish a non-asymptotic error bound for the random graph model with sub-Gaussian noise, and prove that our model produces a matrix estimator of high accuracy when the order of the uniform sample size is roughly the degree of freedom of a low-rank matrix up to a logarithmic factor. Our results partially explain why MVU and MVE often work well. Moreover, the convex optimization model can be efficiently solved by a recently proposed 3-block alternating direction method of multipliers. Numerical experiments show that the model can produce configurations of high quality on large data points that the SDP approach would struggle to cope with.

     

Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework

This paper is concerned with a continuous-time mean-variance portfolio selection model that is formulated as a bicriteria optimization problem. The objective is to maximize the expected terminal return and minimize the variance of the terminal wealth. By putting weights on the two criteria one obtains a single objective stochastic control problem which is however not in the standard form due to the variance term involved. It is shown that this nonstandard problem can be ``embedded' into a class of auxiliary stochastic linear-quadratic (LQ) problems. The stochastic LQ control model proves to be an appropriate and effective framework to study the mean-variance problem in light of the recent development on general stochastic LQ problems with indefinite control weighting matrices. This gives rise to the efficient frontier in a closed form for the original portfolio selection problem. Accepted 24 November 1999     

Minimax Location of Two Facilities with Minimum Separation: Interactive Graphical Solutions

We consider the problem of locating two facilities so as to minimize the maximum of combined Euclidean distances to unweighted existing points when the facilities must be separated by at least a specified distance. Interpreting the problem as the minimum covering ellipse problem, we employ an interactive graphical method which easily produces near-optimal solutions. We describe the method, implemented on a microcomputer, because it exemplifies the more general technique of interactive optimization.     

A goal programming model for the best possible solution to loan allocation problems

In this paper, we propose a multi-criteria decision making approach to address the problem of finding the best possible solution in credit unions. Sensitivity analysis on the priority structure of the goals has been performed to obtain all possible solutions. The study uses the Euclidean distance method to measure distances of all possible solutions from the identified ideal solution. The possible optimum solution is determined from the minimum distance between the ideal solution and other possible solutions of the problem.     

Distance Matrix Completion by Numerical Optimization

Consider the problem of determining whether or not a partial dissimilarity matrix can be completed to a Euclidean distance matrix. The dimension of the distance matrix may be restricted and the known dissimilarities may be permitted to vary subject to bound constraints. This problem can be formulated as an optimization problem for which the global minimum is zero if and only if completion is possible. The optimization problem is derived in a very natural way from an embedding theorem in classical distance geometry and from the classical approach to multidimensional scaling. It belongs to a general family of problems studied by Trosset (Technical Report 97-3, Department of Computational & Applied Mathematics—MS 134, Rice University, Houston, TX 77005-1892, 1997) and can be formulated as a nonlinear programming problem with simple bound constraints. Thus, this approach provides a constructive technique for obtaining approximate solutions to a general class of distance matrix completion problems.     

“MOSS” multiobjective scatter search applied to non-linear multiple criteria optimization

This paper introduces a multiple criteria scatter search to deal with bounded constrained non-linear continuous vector optimization problems of high dimension, applying a MultiStart Tabu Search (TS) as a diversification generation method, each TS works with its own starting point, recency memory, and aspiration threshold. Frequency memory is used to diversify the search and it is shared between the TS. A Pareto relation is applied in order to designate a subset of the best generated solutions to be reference solutions. A choice function called Kramer Choice function is used to divide the reference solutions in two subsets. The Euclidean distance is used as a measure of dissimilarity in order to find diverse solutions to be combined. Linear combinations of the reference solutions are used as a solution combination method. “Balls” in the decision space and the objective space are used to avoid duplications. Different tabu sets with different tabu tenures are employed in the scatter phase to enhance the diversity of the search. The performance of our approach is compared with Pareto-optimal frontiers and three other state-of-the-art MOEAs for a suite test problems taken from the literature.     

Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean.The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.     

A recursive least-squares algorithm for pairwise comparison matrices

Pairwise comparison matrices are commonly used for setting priorities among competing objects. In a leading decision making method called the analytic hierarchy process the principal right eigenvector components represent the weights of the alternatives. The direct least-squares method extracts the weight vector by first finding a rank-one matrix which minimizes the Euclidean distance from the original ratio matrix. We develop a recursive least-squares algorithm and reveal a striking correspondence between these two approaches for these matrices. The recursion applies for merely positive matrices also. We prove that a convergent iteration leads to matrices by which the Perron-eigenvectors and the Perron approximation of the original matrix may be produced. We show that certain useful properties of the recursion advance the development of reliable measures of perturbations of transitive matrices. Numerical analysis is included for a macroeconomic problem taken from the literature.     

A finite algorithm for solving general quadratic problems

Here we propose a global optimization method for general, i.e. indefinite quadratic problems, which consist of maximizing a non-concave quadratic function over a polyhedron inn-dimensional Euclidean space. This algorithm is shown to be finite and exact in non-degenerate situations. The key procedure uses copositivity arguments to ensure escaping from inefficient local solutions. A similar approach is used to generate an improving feasible point, if the starting point is not the global solution, irrespective of whether or not this is a local solution. Also, definiteness properties of the quadratic objective function are irrelevant for this procedure. To increase efficiency of these methods, we employ pseudoconvexity arguments. Pseudoconvexity is related to copositivity in a way which might be helpful to check this property efficiently even beyond the scope of the cases considered here.     

Management decision-making for sugarcane fertilizer mix problems through goal programming

This paper presents a goal-programming (GP) model for management decision-making for sugarcane fertilizer mix problems. Sensitivity analysis on the priority structure of the goals has been performed to obtain all possible solutions. The study uses Euclidean distance function to measure distances of all possible solutions from the ideal solution. The optimum solution is determined from the minimum distance between the ideal solution and other possible solutions of the problem. The optimum solution corresponds to the appropriate priority structure of the problem in the decision-making context. Furthermore, the results obtained from sensitivity analysis on the cost of combination of fertilizers confirm the priority structure.