指数跟踪问题的广义双线性规划模型

本文对指数跟踪问题建立了一种广义的双线性规划模型,其中考虑了交易费用、持仓限制与重平衡问题.根据该模型的特殊结构,本文给出了近似规划算法,通过逐次逼近的线性规划求解最优指数跟踪问题.     

Bilinear programming and structured stochastic games

One-step algorithms are presented for two classes of structured stochastic games, namely, those with additive rewards and transitions and those which have switching controllers. Solutions to such classes of games under the average reward criterion can be derived from optimal solutions to appropriate bilinear programs. The validity of using bilinear programming as a solution method follows from two preliminary theorems, the first of which is a complete classification of undiscounted stochastic games with optimal stationary strategies. The second of these preliminary theorems relaxes the conditions of the classification theorem for certain classes of stochastic games and provides the basis for the bilinear programming results. Analogous results hold for the discounted stochastic games with the above special structures.This research was supported in part by the Air Force Office of Scientific Research and by the National Science Foundation under Grant No. ECS-850-3440.     

On the Construction of Convex and Concave Envelope Formulas for Bilinear and Fractional Functions on Quadrilaterals

Convex and concave envelopes play important roles in various types of optimization problems. In this article, we present a result that gives general guidelines for constructing convex and concave envelopes of functions of two variables on bounded quadrilaterals. We show how one can use this result to construct convex and concave envelopes of bilinear and fractional functions on rectangles, parallelograms and trapezoids. Applications of these results to global optimization are indicated.     

Bilinear multipliers on Lorentz spaces

We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform. The author has been partially supported by grants DGESIC PB98-1246 and BMF 2002-04013.     

Branch-and-Cut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem

The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of system and control theory in the last few years. This inequality permits to reduce in an elegant way various problems of robust control into its form. However, in contrast to the Linear Matrix Inequality (LMI), which can be solved by interior-point-methods, the BMI is a computationally difficult object in theory and in practice. This article improves the branch-and-bound algorithm of Goh, Safonov and Papavassilopoulos (Journal of Global Optimization, vol. 7, pp. 365–380, 1995) by applying a better convex relaxation of the BMI Eigenvalue Problem (BMIEP), and proposes new Branch-and-Bound and Branch-and-Cut Algorithms. Numerical experiments were conducted in a systematic way over randomly generated problems, and they show the robustness and the efficiency of the proposed algorithms.     

Reductivity of the Lie Algebra of a Bilinear Form

Let f: V × V → F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a field F, and let L(f) be the subalgebra of 𝔤𝔩(V) of all skew-adjoint endomorphisms relative to f. Provided F is algebraically closed of characteristic not 2, we determine all f, up to equivalence, such that L(f) is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for L(f) to be simple, semisimple or isomorphic to 𝔰𝔩(n) for some n.     

A Bilinear Algorithm for Optimizing a Linear Function over the Efficient Set of a Multiple Objective Linear Programming Problem

The problem Q of optimizing a linear function over the efficient set of a multiple objective linear program serves several useful purposes in multiple criteria decision making. However, Q is in itself a difficult global optimization problem, whose local optima, frequently large in number, need not be globally optimal. Indeed, this is due to the fact that the feasible region of Q is, in general, a nonconvex set. In this paper we present a monotonically increasing algorithm that finds an exact, globally-optimal solution for Q. Our approach does not require any hypothesis on the boundedness of neither the efficient set EP nor the optimal objective value. The proposed algorithm relies on a simplified disjoint bilinear program that can be solved through the use of well-known specifically designed methods within nonconvex optimization. The algorithm has been implemented in C and preliminary numerical results are reported.     

马尔可夫跳跃双线性离散随机系统的稳定性

本研究了一类带有马尔可夫姚跃参数的双线性离散随机系统的稳定性，获得了该系统的稳定性和能稳定性的充分条件。     

Bilinear forms on homogeneous Sobolev spaces

In this paper we characterize the boundedness of the bilinear form defined by

$(f,g)\in {\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})\to \underset{\mathbb{R}}{\int }{(?\mathrm{\Delta })}^{s/2}(fg)(x){(?\mathrm{\Delta })}^{s/2}(b)(x)dx,$

in the product of homogeneous Sobolev spaces ${\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})$, $0<s<1/2$. We deduce a characterization of the space of pointwise multipliers from ${\dot{H}}^s(\mathbb{R})$ to its dual ${\dot{H}}^{?s}(\mathbb{R})$ in terms of trace measures.     

Bilinear pseudo-differential operators on local hardy spaces

In this paper, the authors consider a class of bilinear pseudo-differential operators with symbols of order 0 and type (1, 0) in the sense of Ho¨rmander and use the atomic decompositions of local Hardy spaces to establish the boundedness of the bilinear pseudo-differential operators and the bilinear singular integral operators on the product of local Hardy spaces.     

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

We describe the null-cone of the representation of G on M ^p , where either G = SL(W) × SL(V) and M = Hom(V,W) (linear maps), or G = SL(V) and M is one of the representations S ²(V ^*) (symmetric bilinear forms), Λ²(V ^*) (skew bilinear forms), or (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and M ^p is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on M ^p . Results of Kraft and Wallach predict that their number stabilises at a certain value of p, and we determine this value. We also answer the question of when the null-cone in M ^p is defined by the polarisations of the invariants on M; typically, this is only the case if either dim V or p is small. A fundamental tool in our proofs is the Hilbert–Mumford criterion for nilpotency (also known as unstability).     

Construction of Symmetric and Symplectic Bilinear Forms

Let V be a finite-dimensional vector-space. A linear mapping on V is called simple if V( - 1) is 1-dimensional. Let S be a set of simple bijections on V. We discuss conditions entraining that each element of S is orthogonal (respectively symplectic) under an appropriate symmetric (respectively symplectic) bilinear form on V.     

Efficient algorithms for solving rank two and rank three bilinear programming problems

Finitely convergent algorithms for solving rank two and three bilinear programming problems are proposed. A rank k bilinear programming problem is a nonconvex quadratic programming problem with the following structure: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFTbGaa8% xAaiaa-5gacaWFPbGaa8xBaiaa-LgacaWF6bGaa8xzaiaa-bcacaWF% 7bacbiGaa43yamaaDaaaleaacaGFWaaabaGaa4hDaaaakiaa+Hhaca% GFRaGaa4hzamaaDaaaleaacaGFWaaabaGaa4hDaaaakiaa+LhacaGF% RaWaaabuaeaacaGFJbWaa0baaSqaaiaa+PgaaeaacaGF0baaaOGaam% iEaiabl+y6NjaadsgadaqhaaWcbaGaamOAaaqaaiaadshaaaGccaWG% 5bGaaiiFaaWcbaGaa8NAaiaa-1dacaWFXaaabeqdcqGHris5aOGaa4% hEaiabgIGiolaa+HfacaGFSaGaa4xEaiabgIGiolaa+LfacaWF9bGa% a8hlaaaa!5D2E!\[minimize \{ c_0^t x + d_0^t y + \sum\limits_{j = 1} {c_j^t xd_j^t y|} x \in X,y \in Y\} ,\]where X Rⁿ¹ and Y R ⁿ² are non-empty and bounded polytopes. We show that a variant of parametric simplex algorithm can solve large scale rank two bilinear programming problems efficiently. Also, we show that a cutting-cake algorithm, a more elaborate variant of parametric simplex algorithm can solve medium scale rank three problems.This research was supported in part by Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture, Grant No. 63490010.     

Maximal bilinear singular integral operators associated with dilations of planar sets

We obtain square function estimates and bounds for maximal singular integral operators associated with bilinear multipliers given by characteristic functions of dyadic dilations of certain planar sets. As a consequence, we deduce pointwise almost everywhere convergence for lacunary partial sums of bilinear Fourier series with respect to methods of summation determined by the corresponding planar sets.     

Frank separation of two convex sets and Hahn-Banach theorem

First we give some elementary properties of the core of a subset relative to a linear subspace. Then we prove a theorem on the frank separation of two convex sets. This theorem admits as particular cases the known theorems on the frank separation and introduces new cases. Finally, we provide a very general version of the Hahn-Banach theorem in an analytic form.     

辛代数的不变双线性型

考察了辛代数和与它相联系的李三系的双线性型之间的关系,并证明了辛代数的反对称不变双线性型可以唯一扩张到与它相联系的李三系中.作为这种关系的一个应用,得到了二次辛代数是单辛代数的一个充要条件,并证明二次辛代数的唯一分解定理.     

Symmetric Bilinear Forms over Valuation Rings

本文讨论赋值环上的对称线性型、二次型和对称矩阵的合同标准形。     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).