Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems

In this paper, we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of an infinite family of nonexpansive mappings and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters.     

A genetic algorithm for the generalised assignment problem

A new algorithm for the generalised assignment problem is described in this paper. The algorithm is adapted from a genetic algorithm which has been successfully used on set covering problems, but instead of genetically improving a set of feasible solutions it tries to genetically restore feasibility to a set of near-optimal ones. Thus it may be regarded as operating in a dual sense to the more familiar genetic approach. The algorithm has been tested on generalised assignment problems of substantial size and compared to an exact integer programming approach and a well-established heuristic approach.     

Sampling from the complement of a polyhedron: An MCMC algorithm for data augmentation

We present an MCMC algorithm for sampling from the complement of a polyhedron. Our approach is based on the Shake-and-bake algorithm for sampling from the boundary of a set and provably covers the complement. We use this algorithm for data augmentation in a machine learning task of classifying a hidden feasible set in a data-driven optimization pipeline. Numerical results on simulated and MIPLIB instances demonstrate that our algorithm, along with a supervised learning technique, outperforms conventional unsupervised baselines.     

A simple randomised algorithm for convex optimisation

We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron.     

A Special Algorithm for an Additive Model in Data Envelopment Analysis

A special algorithm is presented for the additive model in data envelopment analysis (DEA). The special algorithm first classifies a data set into several subsets. Then the subset is solved by a different algorithmic framework. In simulation studies, the algorithm outperformed available DEA codes. The proposed algorithm can efficiently deal with a large data set.     

The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented.     

Online Point Location in Planar Arrangements and Its Applications

Recently, Har-Peled [HP2] presented a new randomized technique for online construction of the zone of a curve in a planar arrangement of arcs. In this paper we present several applications of this technique, which yield improved solutions to a variety of problems. These applications include: (i) an efficient mechanism for performing online point-location queries in an arrangement of arcs; (ii) an efficient algorithm for computing an approximation to the minimum weight Steiner tree of a set of points, where the weight is the number of intersections between the tree edges and a given collection of arcs; (iii) a subquadratic algorithm for cutting a set of pseudo-parabolas into pseudo-segments; (iv) an algorithm for cutting a set of line segments (``rods') in 3-space to eliminate all cycles in the vertical depth order; and (v) a near-optimal algorithm for reporting all bichromatic intersections between a set R of red arcs and a set B of blue arcs, where the unions of the arcs in each set are both connected. Received December 22, 1999, and in revised form August 25, 2000. Online publication May 11, 2001.     

Construction of the Solvability Set in Differential Games with Simple Motion and Nonconvex Terminal Set

We consider planar zero-sum differential games with simple motion, fixed terminal time, and polygonal terminal set. The geometric constraint on the control of each player is a convex polygonal set or a line segment. In the case of a convex terminal set, an explicit formula is known for the solvability set (a level set of the value function, maximal u-stable bridge, viability set). The algorithm corresponding to this formula is based on the set operations of algebraic sum and geometric difference (the Minkowski difference). We propose an algorithm for the exact construction of the solvability set in the case of a nonconvex polygonal terminal set. The algorithm does not involve the additional partition of the time interval and the recovery of intermediate solvability sets at additional instants. A list of half-spaces in the three-dimensional space of time and state coordinates is formed and processed by a finite recursion. The list is based on the polygonal terminal set with the use of normals to the polygonal constraints on the controls of the players.     

Iterative algorithm for mathematical programming problems with preconvex constraints

An iterative algorithm is proposed for minimizing a convex function on a set defined as the set-theoretic difference between a convex set and the union of several convex sets. The convergence of the algorithm is proved in terms of necessary conditions for a local minimum.     

A relaxed cutting plane method for semi-infinite semi-definite programming

In this paper, we develop two discretization algorithms with a cutting plane scheme for solving combined semi-infinite and semi-definite programming problems, i.e., a general algorithm when the parameter set is a compact set and a typical algorithm when the parameter set is a box set in the m-dimensional space. We prove that the accumulation point of the sequence points generated by the two algorithms is an optimal solution of the combined semi-infinite and semi-definite programming problem under suitable assumption conditions. Two examples are given to illustrate the effectiveness of the typical algorithm.     

Computing the minimal covering set

We present the first polynomial-time algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists for two variants of the minimal covering set–the minimal upward covering set and the minimal downward covering set–unless P equals NP. Finally, we observe a strong relationship between von Neumann–Morgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other.     

Algorithms for bound constrained quadratic programming problems

Summary We present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds. We show, in particular, that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations. Moreover, if all stationary points are nondegenerate, termination occurs at a local minimizer. A numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy. On nondegenerate problems the number of iterations typically decreases by at least a factor of 10. For strongly degenerate problems, the performance of the gradient projection algorithm deteriorates, but it still performs better than the active set method.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

Further Analysis of an Outcome Set-Based Algorithm for Multiple-Objective Linear Programming

Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, Benson recently developed a finite, outer-approximation algorithm for generating the set of all efficient extreme points in the outcome set, rather than in the decision set, of problem (MOLP). In this article, we show that the Benson algorithm also generates the set of all weakly efficient points in the outcome set of problem (MOLP). As a result, the usefulness of the algorithm as a decision aid in multiple objective linear programming is further enhanced.     

A level set algorithm for a class of reverse convex programs

A new algorithm is presented for minimizing a linear function subject to a set of linear inequalities and one additional reverse convex constraint. The algorithm utilizes a conical partition of the convex polytope in conjuction with its facets in order to remain on the level surface of the reverse convex constraint. The algorithm does not need to solve linear programs on a set of cones which converges to a line segment.     

A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program

Various computational difficulties arise in using decision set-based vector maximization methods to solve multiple objective linear programming problems. As a result, several researchers have begun to explore the possibility of solving these problems by examining subsets of their outcome sets, rather than of their decision sets. In this article, we present and validate a basic weight set decomposition approach for generating the set of all efficient extreme points in the outcome set of a multiple objective linear program. Based upon this approach, we then develop an algorithm, called the Weight Set Decomposition Algorithm, for generating this set. A sample problem is solved using this algorithm, and the main potential computational and practical advantages of the algorithm are indicated.     

An efficient rough feature selection algorithm with a multi-granulation view

Feature selection is a challenging problem in many areas such as pattern recognition, machine learning and data mining. Rough set theory, as a valid soft computing tool to analyze various types of data, has been widely applied to select helpful features (also called attribute reduction). In rough set theory, many feature selection algorithms have been developed in the literatures, however, they are very time-consuming when data sets are in a large scale. To overcome this limitation, we propose in this paper an efficient rough feature selection algorithm for large-scale data sets, which is stimulated from multi-granulation. A sub-table of a data set can be considered as a small granularity. Given a large-scale data set, the algorithm first selects different small granularities and then estimate on each small granularity the reduct of the original data set. Fusing all of the estimates on small granularities together, the algorithm can get an approximate reduct. Because of that the total time spent on computing reducts for sub-tables is much less than that for the original large-scale one, the algorithm yields in a much less amount of time a feature subset (the approximate reduct). According to several decision performance measures, experimental results show that the proposed algorithm is feasible and efficient for large-scale data sets.     

Rough set based maximum relevance-maximum significance criterion and Gene selection from microarray data

Among the large amount of genes presented in microarray gene expression data, only a small fraction of them is effective for performing a certain diagnostic test. In this regard, a new feature selection algorithm is presented based on rough set theory. It selects a set of genes from microarray data by maximizing the relevance and significance of the selected genes. A theoretical analysis is presented to justify the use of both relevance and significance criteria for selecting a reduced gene set with high predictive accuracy. The importance of rough set theory for computing both relevance and significance of the genes is also established. The performance of the proposed algorithm, along with a comparison with other related methods, is studied using the predictive accuracy of K-nearest neighbor rule and support vector machine on five cancer and two arthritis microarray data sets. Among seven data sets, the proposed algorithm attains 100% predictive accuracy for three cancer and two arthritis data sets, while the rough set based two existing algorithms attain this accuracy only for one cancer data set.     

An all-linear programming relaxation algorithm for optimizing over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple objective linear program has many important applications in multiple criteria decision making. Since the efficient set is in general a nonconvex set, problem (P) can be classified as a global optimization problem. Perhaps due to its inherent difficulty, it appears that no precisely-delineated implementable algorithm exists for solving problem (P) globally. In this paper a relaxation algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact optimal solution to the problem after a finite number of iterations. A detailed discussion is included of how to implement the algorithm using only linear programming methods. Convergence of the algorithm is proven, and a sample problem is solved.Research supported by a grant from the College of Business Administration, University of Florida, Gainesville, Florida, U.S.A.     

Quasi-concave functions on meet-semilattices

This paper deals with maximization of set functions defined as minimum values of monotone linkage functions. In previous research, it has been shown that such a set function can be maximized by a greedy type algorithm over a family of all subsets of a finite set. In this paper, we extend this finding to meet-semilattices.We show that the class of functions defined as minimum values of monotone linkage functions coincides with the class of quasi-concave set functions. Quasi-concave functions determine a chain of upper level sets each of which is a meet-semilattice. This structure allows development of a polynomial algorithm that finds a minimal set on which the value of a quasi-concave function is maximum. One of the critical steps of this algorithm is a set closure. Some examples of closure computation, in particular, a closure operator for convex geometries, are considered.     

An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple Objective Linear Programming Problem

Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have suggested that outcome set-based approaches should instead be developed and used to solve problem (MOLP). In this article, we present a finite algorithm, called the Outer Approximation Algorithm, for generating the set of all efficient extreme points in the outcome set of problem (MOLP). To our knowledge, the Outer Approximation Algorithm is the first algorithm capable of generating this set. As a by-product, the algorithm also generates the weakly efficient outcome set of problem (MOLP). Because it works in the outcome set rather than in the decision set of problem (MOLP), the Outer Approximation Algorithm has several advantages over decision set-based algorithms. It is also relatively easy to implement. Preliminary computational results for a set of randomly-generated problems are reported. These results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach. This revised version was published online in July 2006 with corrections to the Cover Date.