解线性约束优化问题的新锥模型信赖域法

本文提出了一个解线性等式约束优化问题的新锥模型信赖域方法.论文采用零空间技术消除了新锥模型子问题中的线性等式约束,用折线法求解转换后的子问题,并给出了解线性等式约束优化问题的信赖域方法.论文提出并证明了该方法的全局收敛性,并给出了该方法解线性等式约束优化问题的数值实验.理论和数值实验结果表明新锥模型信赖域方法是有效的,这给出了用新锥模型进一步研究非线性优化的基础.     

A NEW DERIVATIVE FREE OPTIMIZATION METHOD BASED ON CONIC INTERPOLATION MODEL

In this paper, a new derivative free trust region method is developed basedon the conic interpolation model for the unconstrained optimization. The conic inter-polation model is built by means of the quadratic model function, the collinear scalingformula, quadratic approximation and interpolation. All the parameters in this model axedetermined by objective function interpolation condition. A new derivative free method isdeveloped based upon this model and the global convergence of this new method is provedwithout any information on gradient.     

A very simple proof of Pascal’s hexagon theorem and some applications

In this article we present a simple and elegant algebraic proof of Pascal’s hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic.     

Intersection probabilities and kinematic formulas for polyhedral cones

For polyhedral convex cones in \({\mathbb{R}^d}\), we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic random central hyperplane arrangement, we find probabilities for non-trivial intersection, either with a fixed cone, or for two independent random cones of this type.     

Proving Strong Duality for Geometric Optimization Using a Conic Formulation

Geometric optimization¹ is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the well-known associated duality theory, using conic optimization. After introducing suitable convex cones and studying their properties, we model geometric optimization problems with a conic formulation, which allows us to apply the powerful duality theory of conic optimization and derive the duality results valid for geometric optimization.     

Strichartz Estimates for Schrödinger Equations on Scattering Manifolds

The present article is concerned with Schrödinger equations on non-compact Riemannian manifolds with asymptotically conic ends. It is shown that, for any admissible pair (including the endpoint), local in time Strichartz estimates outside a large compact set are centered at origin hold. Moreover, we prove global in space Strichartz estimates under the nontrapping condition on the metric.     

Rational cubic/quartic Said-Ball conics

In CAGD, the Said-Ball representation for a polynomial curve has two advantages over the Bézier representation, since the degrees of Said-Ball basis are distributed in a step type. One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomial curve runs twice as fast as the de Casteljau algorithm of Bézier curve. Another is that the operations of degree elevation and reduction for a polynomial curve in Said-Ball form are simpler and faster than in Bézier form. However, Said-Ball curve can not exactly represent conics which are usually used in aircraft and machine element design. To further extend the utilization of Said-Ball curve, this paper deduces the representation theory of rational cubic and quartic Said-Ball conics, according to the necessary and sufficient conditions for conic representation in rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball basis. The results include the judging method for whether a rational quartic Said-Ball curve is a conic section and design method for presenting a given conic section in rational quartic Said-Ball form. Many experimental curves are given for confirming that our approaches are correct and effective.     

Stability Estimates for Helical Computer Tomography

In this article we analyze an inversion formula for helical computer tomography proposed earlier by the author. Our first result is a global stability estimate. The formula is continuous of order 1 in the Sobolev norms. Then the formula is extended to distributions. Originally it was derived only for C₀ functions. It turns out that there exist distributions, to which the formula does not apply. These exceptional distributions are characterized in terms of their wave fronts. Finally, we show that microlocally away from a critical set the continuity estimate can be mproved: The order goes down from 1 to 1/2.     

Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly feasible points in both the primal and dual spaces can be defined. A second benefit of the modification is an improvement in the complexity analysis of conic cutting surface algorithms. Complexity results for conic cutting surface algorithms proved to date have depended on a condition number of the added constraints. The proposed modification of the constraints leads to a stronger result, with the convergence of the resulting algorithm not dependent on the condition number. Research supported in part by NSF grant number DMS-0317323. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.     

Conic systems and sublinear mappings: equivalent approaches

It is known that linear conic systems are a special case of set-valued sublinear mappings. Hence the latter subsumes the former. In this note we observe that linear conic systems also contain set-valued sublinear mappings as a special case. Consequently, the former also subsumes the latter.     

Conic mixed-integer rounding cuts

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability. This research has been supported, in part, by Grant # DMI0700203 from the National Science Foundation.     

A generalized Green's formula for elliptic problems in domains with edges

The usual Green's formula connected with the operator of a boundary-value problem fails when both of the solutions u and v that occur in it have singularities that are too strong at a conic point or at an edge on the boundary of the domain. We deduce a generalized Green's formula that acquires an additional bilinear form in u and v and is determined by the coefficients in the expansion of solutions near singularities of the boundary. We obtain improved asymptotic representations of solutions in a neighborhood of an edge of positive dimension, which together with the generalized Green's formula makes it possible, for example, to describe the infinite-dimensional kernel of the operator of an elliptic problem in a domain with edge. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 106–147.     

Pascal构图的探索

本文揭示了帕斯卡构图的一个重要的性质，对深入研究二次曲线的射影理论起到了重要的作用。     

Sur la continuité de Hölder du semi-groupe de la chaleur sur les variétés coniques

In this Note, we study initially the heat kernel, p_t, on conic manifolds of dimension 2. Then, we improve the upper bound of p_t obtained in Li (Bull. Sci. Math. 124 (2000) 365–384) on conic manifolds of dimension ?3. Finally, we study the Hölder continuity of the heat semigroup on conic manifolds. Some new phenomenons are found on conic manifolds, in particular, on conic manifolds of dimension 2. To cite this article: H.-Q. Li, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Zeta regularized determinants for conic manifolds

We study (relative) zeta regularized determinants of Laplace type operators on compact conic manifolds. We establish gluing formulae for relative zeta regularized determinants. For arbitrary self-adjoint extensions of the Laplace-Beltrami operator, we express the relative ζ-determinants for these as a ratio of the determinants of certain finite matrices. For the self-adjoint extensions corresponding to Dirichlet and Neumann conditions, the formula is particularly simple and elegant.     

有限元解三维弹性波方程的并行算法

研究弹性固体中波的传播问题由来已久,由于地震学、石油勘探、地震层析成像等方面的重要应用,三维弹性波方程的求解问题在工业应用中显得日益重要.自七十年代末由石油勘探界首先引入有限元方法求解二维弹性波方程以来,这一公认精度较高的方法在求解三维弹性波方程方面研究成果尚不多见,其主要原因是该问题需要较大的计算机存贮量和计算量.对三维问题使用一般的有限元方法求解,常常需要几十万次反复递推求解高达10~7阶以上的大型线性代数方程组.对于这类超大型的计算课题,目前的     

Ovoids of PG(3,q), q Even, with a Conic Section

It is shown that if a plane of PG(3,q), q even, meets an ovoidin a conic, then the ovoid must be an elliptic quadric. Thisis proved by using the generalized quadrangles T₂(C) (C a conic),W(q) and the isomorphism between them to show that every secantplane section of the ovoid must be a conic. The result thenfollows from a well-known theorem of Barlotti.     

Lifting for conic mixed-integer programming

Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lower-dimensional restrictions. In order to simplify the computations, we also discuss sequence-independent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming as one may expect.     

A methodized short‐cut to conics

It is my firm belief that mathematics is methodology. The briefer is the method, the more effective it is in treating problems. The novel methodized treatment in this article features all the concepts of conics relating to transformation between coordinate systems and standard forms with graphic illustrations.

Denote O’ as the centre of an ellipse or a hyperbola or the vertex of a parabola and F as a focus of a conic. Let O'F=cu, u=[cosθ, sinθ], v = [‐sinθ, cosθ] and P(x,y), then the transformation x’ = u.O'P, y‘ = v.O'P which leads to the standard form for each conic relative to x’ — O'—y‘

The theorem on normal projection in this article is very important in analytic geometry and especially useful for problems involving conies when a directrix or an axis is given.     

Conic approximation to quadratic optimization with linear complementarity constraints

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an \(\epsilon \)-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.