Conformal geometry of surfaces in Lorentzian space forms

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.Dedicated to Professor T.J. WillmoreSupported by an FPPI Postdoctoral Grant from DGICYT Ministerio de Educación y Ciencia, Spain 1994 and by a DGICYT Grant No. PB94-0750-C02-02     

A Simplified Novel Technique for Solving Fully Fuzzy Linear Programming Problems

This study proposes a novel technique for solving Linear Programming Problems in a fully fuzzy environment. A modified version of the well-known simplex method is used for solving fuzzy linear programming problems. The use of a ranking function together with the Gaussian elimination process helps in solving linear programming problems in a fully uncertain environment. The proposed algorithm is flexible, easy and reasonable.     

A combinatorial abstraction of linear programming

In the context of oriented matroids we establish and elaborate upon an abstraction of linear programming duality foreseen by Rockafellar in his work on elementary vectors. We describe a pivoting operation for oriented matroids and a finite pivoting method, which elucidate the combinatorial nature of Dantzig's simplex method. The pivoting method specializes, when the oriented matroids arise from real vector spaces, to the simplex method with a new pivot selection rule. A very simple pivot selection rule for which finiteness has been established in the linear programming context, but not in the broader setting of oriented matroids, is also described.     

Oriented matroids

In this paper, the basic properties of oriented matroids are examined. A topological representation theorem for oriented matroids is proven, utilizing the notion of an “arrangement of pseudo-hemispheres”. The duality theorem of linear programming is extended to oriented matroids.     

Open conformal cobordisms

Conformal field theories were first axiomatized by Segal (2004) as symmetric monoidal functors from a topological category of conformal cobordisms between compact oriented 1-dimensional manifolds to vector spaces. Costello (2007) later expanded the definition of the category to allow for cobordisms between manifolds with boundaries, and was able to use representations of this category to give a mirror partner for Gromov-Witten invariants. The main goal of this paper is to provide a rigorous definition of the category of open conformal cobordisms. To the best of our knowledge, such a definition does not appear in the literature. Although most results here are probably known to the experts, the proofs are, as far as we can tell, new, and require only elementary results about quasiconformal mappings.     

Preference modelling by estimating local utility functions for multiobjective optimization

This paper is intended to design goal programming models for capturing the decision maker's (DM's) preference information and for supporting the search for the best compromise solutions in multiobjective optimization. At first, a linear goal programming model is built to estimate piecewise linear local utility functions based on pairwise comparisons of efficient solutions as well as objectives. The interactive step trade-off method (ISTM) is employed to generate a typical subset of efficient solutions of a multiobjective problem. Another general goal programming model is then constructed to embed the estimated utility functions in the original multiobjective problem for utility optimization using ordinary nonlinear programming algorithms. This technique, consisting of the ISTM method and the newly investigated search process, facilitates the identification and elimination of possible inconsistent information which may exist in the DM's preferences. It also provides various ways to carry out post-optimality analysis to test the robustness of the obtained best solutions. A modified nonlinear multiobjective management problem is taken as example to demonstrate the technique.     

Local elimination algorithms for solving sparse discrete problems

The class of local elimination algorithms is considered that make it possible to obtain global information about solutions of a problem using local information. The general structure of local elimination algorithms is described that use neighborhoods of elements and the structural graph describing the problem structure; an elimination algorithm is also described. This class of algorithms includes local decomposition algorithms for discrete optimization problems, nonserial dynamic programming algorithms, bucket elimination algorithms, and tree decomposition algorithms. It is shown that local elimination algorithms can be used for solving optimization problems.     

Linear and quadratic programming in oriented matroids

We prove constructively duality theorems of linear and quadratic programming in the combinatorial setting of oriented matroids. One version of our algorithm for linear programing has the interesting feature of maintaining feasibility. The development of the quadratic programming duality result suggests the study of properties of square matrices such as symmetry and positive semi-definiteness in the context of oriented matroids.     

A conformal mapping for a rectilinear congruence

In this paper we deal with oriented rectilinear congruences in a three-dimensional Euclidean space E³ establishing a conformal mapping between their middle surface and their middle envelope. We give some properties and determine a special class of them, which have a minimal middle envelope.     

Extending the mixed algebraic-analysis Fourier–Motzkin elimination method for classifying linear semi-infinite programmes

Motivated by a recent Basu–Martin–Ryan paper, we obtain a reduced primal-dual pair of a linear semi-infinite programming problem by applying an amended Fourier–Motzkin elimination method to the linear semi-infinite inequality system. The reduced primal-dual pair is equivalent to the original one in terms of consistency, optimal values and asymptotic consistency. Working with this reduced pair and reformulating a linear semi-infinite programme as a linear programme over a convex cone, we reproduce all the theorems that lead to the full eleven possible duality state classification theory. Establishing classification results with the Fourier–Motzkin method means that the two classification theorems for linear semi-infinite programming, 1969 and 1974, have been proved by new and exciting methods. We also show in this paper that the approach to study linear semi-infinite programming using Fourier–Motzkin elimination is not purely algebraic, it is mixed algebraic-analysis.     

An exact procedure and LP formulations for the leader—follower location problem

The leader—follower location problem consists of determining an optimal strategy for two competing firms which make decisions sequentially. The leader optimisation problem is to minimise the maximum market share of the follower. The objective of the follower problem is to maximise its market share. We describe linear programming formulations for both problems and analyse the use of these formulations to solve the problems. We also propose an exact procedure based on an elimination process in a candidate list.     

The regularity of conformal target harmonic maps

We establish that any weakly conformal \(W^{1,2}\) map from a Riemann surface S into a closed oriented sub-manifold \(N^n\) of an euclidian space \({\mathbb {R}}^m\) realizes, for almost every sub-domain, a stationary varifold if and only if it is a smooth conformal harmonic map form S into \(N^n\).     

An Algorithm of Global Optimization for Rational Functions with Rational Constraints

To solve a system of nonlinear equations, Wu wen-tsun introduced a new formative elimination method. Based on Wu's method and the theory of nonlinear programming, we here propose a global optimization algorithm for nonlinear programming with rational objective function and rational constraints. The algorithm is already programmed and the test results are satisfactory with respect to precision and reliability.     

A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive internal spanning trees in a graph resp. bounded regions and uniactive internal bases in a hyperplane arrangement or an oriented matroid (in the sense of Tutte activities). This bijection is the basic case of a general activity preserving bijection between reorientations and subsets of an oriented matroid, called the active bijection, providing bijective versions of various classical enumerative results.Fully optimal bases can be considered as a strenghtening of optimal bases from linear programming, with a simple combinatorial definition. Our first construction used this purely combinatorial characterization, providing directly an algorithm to compute in fact the reverse bijection. A new definition uses a direct construction in terms of a linear programming. The fully optimal basis optimizes a sequence of nested faces with respect to a sequence of objective functions (whereas an optimal basis in the usual sense optimizes one vertex with respect to one objective function). This note presents this construction in terms of graphs and linear algebra.     

Fully Optimal Bases and the Active Bijection in Graphs,Hyperplane Arrangements,and Oriented Matroids

In this note, we present the main results of a series of forthcoming papers, dealing with bi-jective generalizations of some counting formulas. New intrinsic constructions in oriented matroids on a linearly ordered set of elements establish notably structural links between counting regions and linear programming. We introduce fully optimal bases, which have a simple combinatorial characterization, and strengthen the well-known optimal bases of linear programming. Our main result is that every bounded region of an ordered hyperplane arrangement, or ordered oriented matroid, has a unique fully optimal basis, providing the active bijection between bounded regions and uniactive internal bases. The active bijec-tion is extended to an activity preserving mapping between all reorientations and all bases of an ordered oriented matroid. It gives a bijective interpretation of the equality of two expressions for the Tutte polynomial, as well as a new expression of this polynomial in terms of beta invariants of minors. There are several refinements, such as an activity preserving bijection between regions (acyclic reorientations) and no-broken-circuit subsets, and others in terms of hyperplane arrangements, graphs, and permutations.     

关于二次规划问题分段线性同伦算法的改进

本文利用Ｃｈｏｌｅｓｋｙ分解，Ｇａｕｓｓ消去等技术和定义适当的同伦映射，将关于二次规划问题的分段线性同伦算法加以改进，改进后的算法，对于严格凸二次规划来说，计算效率与Ｇｏｌｄｆａｒｂ－Ｉｄｎａｎｉ的对偶法相当。     

Bihermitian surfaces with odd first Betti number

Compact bihermitian surfaces are considered, that is, compact, oriented, conformal four-manifolds admitting two distinct compatible complex structures. It is shown that if the first Betti number is odd then, with respect to either complex structure, such a manifold belongs to Class VII of the Enriques-Kodaira classification. Moreover, it must be either a special Hopf or an Inoue surface (in the strongly bihermitian case), or obtained by blowing-up a minimal class VII surface with curves (in the non-strongly bihermitian case). Received: 10 December 1999; in final form: 29 January 2000 / Published online: 25 June 2001     

Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids

Combinatorica - Quadratic programming, symmetry, positive (semi) definiteness and the linear complementary problem were generalized by Morris and Todd to oriented matroids. Todd gave a constructive...     

A branch and bound algorithm with constraint partitioning for integer goal programming problems

Using constraint partitioning and variable elimination, the authors have recently developed an efficient algorithm for solving linear goal programming problems. However, many goal programs require some or all of the decision variables to be integer valued. This paper shows how the new partitioning algorithm can be extended with a modified branch and bound strategy to solve both pure and mixed type integer goal programming problems. A potential problem in combining the partitioning algorithm and the branch and bound search scheme is presented and resolved.     

COBRA: A New Formulation of the Classic p-Median Location Problem

The p-median problem was first formulated as an integer-linear programming problem by ReVelle and Swain (1970) and further revised by Rosing, ReVelle and Rosing-Vogelaar (1979). These two forms have withstood the test of time, as they have been used by virtually everyone since then. We prove that a property associated with geographical proximity makes it possible to eliminate many of the model variables through a substitution process. This new substitution technique has resulted in the elimination of up to 60% of the variables needed in either of these classic model formulations.