On a class of quadratic programs

Most of the known methods for finding the efficient set of a multiple objective linear programming (MOLP) problem are bottom-up search methods. Main difficulties of the known bottom-up search methods are to find all efficient extreme points adjacent to and to enumerate all efficient faces incident to an efficient degenerate extreme point. Main drawbacks of these methods are that the computational cost is still large and an implementation of them is still difficult. In this paper we propose a new local bottom-up search method for finding all maximal efficient faces for an MOLP problem. Our method is based on the maximal descriptor index sets for efficient edges and extreme rays for the MOLP problem in which the maximal descriptor index sets for edges and extreme rays incident to an efficient degenerate extreme point are easily found on the basis of solving some special linear programming problems. In addition, all efficient extreme points adjacent to and all efficient faces incident to an efficient extreme point can be easily found without using the simplex tables corresponding to bases of this point. Our method can overcome difficulties caused by the degeneracy of faces and is easy to implement. Some comparisons of our method with the known bottom-up search methods are presented. A numerical example is given to illustrate the method.

This paper looks at the task of computing efficient extreme points in multiple objective linear programming. Vector maximization software is reviewed and the ADBASE solver for computing all efficient extreme points of a multiple objective linear program is described. To create MOLP test problems, models for random problem generation are discussed. In the computational part of the paper, the numbers of efficient extreme points possessed by MOLPs (including multiple objective transportation problems) of different sizes are reported. In addition, the way the utility values of the efficient extreme points might be distributed over the efficient set for different types of utility functions is investigated. Not surprisingly, results show that it should be easier to find good near-optimal solutions with linear utility functions than with, for instance, Tchebycheff types of utility functions.

In this work, we consider numerical methods for solving a class of block three-by-three saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and a class of quadratic programs. We present a variant of Uzawa method with two variable parameters for the saddle point problems. These two parameters can be updated easily in each iteration, similar to the evaluation of the two iteration parameters in the conjugate gradient method. We show that the new iterative method converges to the unique solution of the saddle point problems under a reasonable condition. Numerical experiments highlighting the performance of the proposed method for problems are presented.

A multi-stage methodology for the solution of orientated DEA models

Stability of efficient sets: continuity of mobile polarities

It is a well known property that the efficient frontier of multiple criteria objective functions can be generated by parametrically optimizing linear combinations of the separate objectives when the set of decisions generate convex sets in the criterion space. It is also common knowledge that when the criterion space is not convex, as in the case when the decision criteria are discrete, the parametric linear combinations will generate some but not necessarily all of the efficient points. This paper shows that parametrically optimizing a Tchebycheff Norm applied to the criteria function will generate all efficient points.

This paper proposes a new approach for multicriteria optimization which aggregates the objective functions and uses a line search method in order to locate an approximate efficient point. Once the first Pareto solution is obtained, a simplified version of the former one is used in the context of Pareto dominance to obtain a set of efficient points, which will assure a thorough distribution of solutions on the Pareto frontier. In the current form, the proposed technique is well suitable for problems having multiple objectives (it is not limited to bi-objective problems) and require the functions to be continuous twice differentiable. In order to assess the effectiveness of this approach, some experiments were performed and compared with two well known population-based meta-heuristics. When compared to the population-based meta-heuristic, the proposed approach not only assures a better convergence to the Pareto frontier but also illustrates a good distribution of solutions. We propose a fuzzy logic controller to adapt the parameter required to control the distribution of solutions in the spreading phase. Our goal is to find a good distribution of solutions as quick as possible. From a computational point of view, both stages of the line search converge within a short time (average about 150 milliseconds for the first stage and about 20 milliseconds for the second stage). Apart from this, the proposed technique is very simple, easy to implement to solve multiobjective problems.

We address the two-commodity maximum flow problem on undirected networks. As a result of a change of variables, we introduce a new formulation that solves the problem through classical maximum flow techniques with only one-commodity. Therefore, a general strategy, based on this change of variables, is defined to deal with other undirected multi-commodity problems. Finally, we extend the single objective problem to a bicriteria environment. We show that the set of efficient solutions of the biobjective undirected two-commodity maximum flow problem is the set of alternative optimum solutions of the undirected two-commodity maximum flow problem. In addition, we prove that the set of efficient extreme points in the objective space has, at most, cardinality two.

Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al. 2006), this is essentially the only class of linear programs for which such an algorithm is known. In this work we provide a distributed algorithm for solving a different class of convex programs which we call “distance-bounded network design convex programs”. These can be thought of as relaxations of network design problems in which the connectivity requirement includes a distance constraint (most notably, graph spanners). Our algorithm runs in O((D/e) log n) rounds in the LOCAL model and with high probability finds a (1+e)-approximation to the optimal LP solution for any 0 < e ≤ 1, where D is the largest distance constraint. While solving linear programs in a distributed setting is interesting in its own right, this class of convex programs is particularly important because solving them is often a crucial step when designing approximation algorithms. Hence we almost immediately obtain new and improved distributed approximation algorithms for a variety of network design problems, including Basic 3- and 4-Spanner, Directed k-Spanner, Lowest Degree k-Spanner, and Shallow-Light Steiner Network Design with a spanning demand graph. Our algorithms do not require any “heavy” computation and essentially match the best-known centralized approximation algorithms, while previous approaches which do not use heavy computation give approximations which are worse than the best-known centralized bounds.

In this paper, combine spatial clustering and Gaussian mixture models to propose an efficient and robust point set alignment scheme. This work aims to balance recall, accuracy, robustness and operational efficiency. First, the point set alignment problem is transformed into a spatial clustering problem with outliers. The main idea is to adaptively cluster the assumed matches into multiple motion-consistent clusters and one outlier cluster. The Gaussian mixture model is then fitted to each of the multiple kinematic consistency clusters. In order to pursue a higher alignment, the mixing coefficients of the Gaussian mixture model are assigned values using descriptors. This scheme is suitable for alignment problems with complex point sets. In addition, it offers better robustness and efficiency compared to existing related schemes. Analysis of the combined performance and simulation results demonstrates the effectiveness and robustness of the scheme.

One of the aim of this paper is to present new properties for the efficient (Pareto) point sets in separated locally convex spaces. Thus, for any non-empty and compact subset, the coincidence result between the set of corresponding Pareto points with respect to a convex cone and the Choquet boundary with respect to a convenient cone of real continuous functions established by us in a previous research work gives interesting topological properties of Pareto sets. An important result shows immediate applications of spline optimal interpolation in multiple criteria decision making problems and two numerical examples based on spline functions in H-locally convex spaces illustrate the possibility of calculus for Choquet boundaries through the agency of Pareto sets and conversely.KeywordsEfficient pointPareto setChoquet boundaryH-locally convex spacespline function

We consider a multiobjective linear program and the coefficients of the multiobjective function are supposed to be uncertain. Let x* be an efficient point. We propose a procedure computing a tolerance for each objective function coefficient, such that all these coefficients may simultaneously and independently vary within their tolerances while preserving the efficiency of x*. If x* is a non-degenerate basic solution, then the procedure runs in a polynomial time. Our method is also applicable for the intervals of multiobjective linear programming for checking the necessary efficiency of x*, i.e. whether x* is efficient for all the realizations of interval values.

A two-phase procedure for calculation of multiple linear classification functions has been developed. Phase I approximates the distribution of samples in the measurement space by a series of linear functions and preselects only those samples lying in the critical region. Phase II utilizes the reduced number of samples for calculation by a linear programming algorithm, the multiple linear classification functions that can separate given samples strictly. The procedure was applied to the problem of spoken-word recognition. About 3300 samples of Japanese spoken digits, uttered by 55 male speakers and represented as 56-dimensional measurement vectors by a vocoder-type frequency analyzer are collected. During Phase I, the number was reduced to 16 linear functions on the average. Precise classification functions were then calculated for each digit in Phase II. Most of the digits were separated by a single linear function while others were separated by two functions. An error rate of less than 0.2% was achieved for 550 samples newly uttered by the same 55 speakers.

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015