Geometric consideration of duality in vector optimization

Recently, duality in vector optimization has been attracting the interest of many researchers. In order to derive duality in vector optimization, it seems natural to introduce some vector-valued Lagrangian functions with matrix (or linear operator, in some cases) multipliers. This paper gives an insight into the geometry of vector-valued Lagrangian functions and duality in vector optimization. It is observed that supporting cones for convex sets play a key role, as well as supporting hyperplanes, traditionally used in single-objective optimization.The author would like to express his sincere gratitude to Prof. T. Tanino of Tohoku University and to some anonymous referees for their valuable comments.     

Geometric branch-and-bound methods for constrained global optimization problems

Geometric branch-and-bound methods are popular solution algorithms in deterministic global optimization to solve problems in small dimensions. The aim of this paper is to formulate a geometric branch-and-bound method for constrained global optimization problems which allows the use of arbitrary bounding operations. In particular, our main goal is to prove the convergence of the suggested method using the concept of the rate of convergence in geometric branch-and-bound methods as introduced in some recent publications. Furthermore, some efficient further discarding tests using necessary conditions for optimality are derived and illustrated numerically on an obnoxious facility location problem.     

Geometric conditions for optimization in a linear space

In this paper we give new properties of convex functions whose domains are subsets of a linear space; we use them in order to get geometric characterizations of a minimant of a convex function.     

D-optimality andD n -optimal designs for mixtures regression models with logarithmic terms

In this paper the authors put forward a new mixtures regression models with Logarithmic terms^[14] to generalize Draper's models^{[1, 2 or 6]} by using Kiefer-Wolfowitz's equivalence theorem^[3], Fedorov's and Wynn's method^[5]. And we also suggest a method for computer-aided design of combinatorial search^[13].In this study, we have proved and constructed the approximateD-optimal (measure) andD _n -optimal (exact) designs by the use of the first and second order mixtures regression models with logarithmic terms in three and four components.Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

Geometric measures of convex sets and bounds on problem sensitivity and robustness for conic linear optimization

The effect of data perturbation and uncertainty has always been an important consideration in Optimization. It is important to know whether a given problem is very sensible to perturbations on the data or, on the contrary, is more “robust”. Problem geometry does have an impact on the sensitivity of the problem and in this paper we analyze this connection by developing bounds to the change in the optimal value of a conic linear problem in terms of some geometric measures related to the radius of inscribed and circumscribed balls to the feasible region of the problem. We also present a parametric analysis for Linear Programming which allows us to construct an estimate of safety limits for perturbations of the data. These results are developed in relation to questions in robust optimization.     

Geometric non-vanishing

We consider L-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these L-functions by characters of order prime to the characteristic of the ground field and more generally by certain representations with solvable image. We also allow local restrictions on the twisting representation at finitely many places. Our methods are geometric, and include the Riemann-Roch theorem, the cohomological interpretation of L-functions, and monodromy calculations of Katz. As an application, we prove a result which allows one to deduce the conjecture of Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function fields whose L-function vanishes to order at most 1 from a suitable Gross-Zagier formula.     

Geometric designs

Summary Many examples may be found in the literature showing that some sets of configurations in finite geometries are block sets of BIB-designs on the points of the geometry they belong to. See for instance the designsP _t(d, q)., A_t(d, q) in Dembowski [3, 1.4.5] and the designs of quadrics ofPG(d, q) in Primrose [7]. We prove that the theoretical reason for this is the 2-transitivity of the above geometries. This allows us to, state the following general result (Theorem 2): letD be a 2-transitive design; then any “class of isomorphic configurations” ofD is the block set of a design on the points ofD. Designs arising in this way in projective and affine geometries are shown in Sections 3, 4.

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Riassunto Molti esempi rivelano che talune varità dello stesso “tipo” in una geometria finita (proiettiva o, affine) costituiscono i blocchi di un BIB-disegno avente come punti i punti della geometria di appartenenza. Si vedano per esempio i disegniP _t(d, q), A_t(d, q) in Dembowski [3, 1.4.5] e i disegni delle quadriche diPG(d, q) in Primrose [7]. In questo lavoro si dimostra che tale proprietà vale in generale ed è intimamente legata alla proprietà strutturale delle geometrie di possedere un gruppo di automorfismi 2-transitivo. Se ne deduce un procedimento generale per la costruzione di disegni geometrici, di cui si danno alcuni esempi.

     

Geometric probability

The application of probabilities to geometric objects has a history of some two hundred years. We give a brief history, highlighting typical problems and techniques. The abstract phase of the last decade is illustrated by some work of the author.     

Geometric Dynamics

A kinematic differential system on a Riemann (or semi-Riemann) manifold induces a Lorentz-Udrite world-force law, i.e., any local group with one parameter (any local flow) on a Riemann (or semi-Riemann) manifold induces the dynamics of the given vector field or of an associated particle, which will be called geometric dynamics.The cases of Riemann-Jacobi or Riemann-Jacobi-Lagrange structures are imposed by the behavior of an external tensor field of type (1,1). The case of the Finsler-Jacobi structure appears if the initial metric is chosen such that the energy of the given vector field is constant (Sec. 1). At the end of Sec. 1 are formulated open problems regarding some extensions of geometric dynamics.Adequate structures on the tangent bundle describe the geometric dynamics in the Hamilton language (Sec. 2).Section 3 proves the existence of a Finsler-Jacobi structure induced by an almost contact metric structure.The theory is applied to electromagnetic dynamical systems (the starting point of our theory), offering new principles of unification of the gravitation and the electromagnetism. Also, here, one enounces open problems regarding the geometric dynamics induced by the electric intensity and magnetizing force (Sec. 4).From the geometrical point of view, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds ensuring that all trajectories of a given vector field are geodesics. Having T₁M²ⁿ⁺¹ in mind, the problem of creating a wider class of Riemannian manifolds, in which there exists a vector field such that (1) all trajectories of the vector field are geodesics; (2) the flow defined by is incompressible; (3) the condition which corresponds to the property that is the associate vector field of the contact structure is satisfied;was studied intensively by S. Sasaki. The results were not satisfactory, but Sasaki discovered (, , )-structures [10].AMS Subject Classification (1991): 70H35, 53C22, 58F25, 83C22     

Geometric Permutations

A difference permutation is a structure which serves as an instrument to recognize important properties of permutations. Since special permutations are models for orthoschemes in generalized hyperbolic spaces (Minkowskian spaces) of dimension d there difference permutations give informations on the type of such orthoschemes. In this note with the help of difference permutations especially geometric permutations are explained which describe the structure of the concerning orthoschemes. This knowledge is necessary to count the numbers of orthoscheme types and special chains of orthoschemes as shown in B?hm (http://www.minet.uni-jena.de/preprints/boehm_08/Napiercycles.pdf).     

基于共形几何代数的几何分解及程序实现

本文主要讨论了利用共形几何代数来进行几何定理中的几何构型进行几何分解的算法以及它的程序实现问题.利用这个算法可以给出几何量之间的定量依赖关系.所实现的程序能够给出一些较为复杂的几何命题的自动分解的结果.