方差分量的二次不变估计的非负改进

研究一类方差分量模型中的方差分量的估计改进问题,首先在含两个方差分量模型中给出σ21二次型估计类,并且此估计类还具有无偏性和不变性.考虑二次损失(δ-θ)2,在此估计类基础上放弃无偏性进行非负改进,不仅得到优于二次不变无偏估计类的σ21的非负二次不变估计类,而且还说明了它优于方差分析估计和最小均方误差估计,文献[5]中给出s>2时的非负改进,但是非负改进存在是有条件的,本文克服了这个缺陷.最后给出了非负改进存在的充分必要条件.     

On a Quadratic Optimization Problem with Equality Constraints

The constrained optimization problem with a quadratic cost functional and two quadratic equality constraints has been studied by Bar-on and Grasse, with positive-definite matrix in the objective. In this note, we shall relax the matrix in the objective to be positive semidefinite. A necessary and sufficient condition to characterize a local optimal solution to be global is established. Also, a perturbation scheme is proposed to solve this generalized problem.     

A new variable reduction technique for convex integer quadratic programs

Convex integer quadratic programming involves minimization of a convex quadratic objective function with affine constraints and is a well-known NP-hard problem with a wide range of applications. We proposed a new variable reduction technique for convex integer quadratic programs (IQP). Based on the optimal values to the continuous relaxation of IQP and a feasible solution to IQP, the proposed technique can be applied to fix some decision variables of an IQP simultaneously at zero without sacrificing optimality. Using this technique, computational effort needed to solve IQP can be greatly reduced. Since a general convex bounded IQP (BIQP) can be transformed to a convex IQP, the proposed technique is also applicable for the convex BIQP. We report a computational study to demonstrate the efficacy of the proposed technique in solving quadratic knapsack problems.     

On Extensions of the Frank-Wolfe Theorems

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.     

Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N ³. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.     

Global Optimization of a Quadratic Functional with Quadratic Equality Constraints, Part 2

In this paper, we investigate a constrained optimization problem with a quadratic cost functional and two quadratic equality constraints. It is assumed that the cost functional is positive definite and that the constraints are both feasible and regular (but otherwise they are unrestricted quadratic functions). Thus, the existence of a global constrained minimum is assured. We develop a necessary and sufficient condition that completely characterizes the global minimum cost. Such a condition is of essential importance in iterative numerical methods for solving the constrained minimization problem, because it readily distinguishes between local minima and global minima and thus provides a stopping criterion for the computation. The result is similar to one obtained previously by the authors. In the previous result, we gave a characterization of the global minimum of a constrained quadratic minimization problem in which the cost functional was an arbitrary quadratic functional (as opposed to positive-definite here) and the constraints were at least positive-semidefinite quadratic functions (as opposed to essentially unrestricted here).     

Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse

We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2＋y2-1＋y（ax＋by＋c）,y=x（ax＋by＋c）,and the ellipse becomes x2＋y2=1.We prove that（i） this quadratic system is analytic integrable if and only if a=0;（ii） if x2＋y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2＋y2=1 is not a limit cycle;and（iii） if x2＋y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.     

A Constructive theory of triple and quintuple product identities of the second degree

The groundwork for a theory of quadratic identities involving the classical triple and quintuple products is layed. The approach is through the study and use of affine maps that act on indexing lattices associated with the terms (double sums) in the given identity. The terms of the identity are found to be connected by the invariant of a ternary quadratic form.

     

HOMOCLINIC CYCLES OF A QUADRATIC SYSTEM DESCRIBED BY QUARTIC CURVES

This paper is devoted to discussing the topological classification of the quartic invariant algebraic curves for a quadratic system. We obtain sufficient and necessary conditions which ensure that the homoclinic cycle of the system is defined by the quartic invariant algebraic curve. Finally, the corresponding global phase diagrams are drawn.     

Dynamic Programming Method for Constrained Discrete-Time Optimal Control

A dynamic programming method is presented for solving constrained, discrete-time, optimal control problems. The method is based on an efficient algorithm for solving the subproblems of sequential quadratic programming. By using an interior-point method to accommodate inequality constraints, a modification of an existing algorithm for equality constrained problems can be used iteratively to solve the subproblems. Two test problems and two application problems are presented. The application examples include a rest-to-rest maneuver of a flexible structure and a constrained brachistochrone problem.     

正态分布下任意秩有限总体中的Minimax预测

对于任意秩有限总体,在二次损失下,有关文献已给出了线性可预测变量在齐次线性预测类中的唯一线性Minimax预测．本文在正态假设下,证明了这个线性Minimax预测也是线性可预测变量在一切预测类中的唯一Minimax预测．     

Optimal solutions for unrelated parallel machines scheduling problems using convex quadratic reformulations

In this work, we take advantage of the powerful quadratic programming theory to obtain optimal solutions of scheduling problems. We apply a methodology that starts, in contrast to more classical approaches, by formulating three unrelated parallel machine scheduling problems as 0–1 quadratic programs under linear constraints. By construction, these quadratic programs are non-convex. Therefore, before submitting them to a branch-and-bound procedure, we reformulate them in such a way that we can ensure convexity and a high-quality continuous lower bound. Experimental results show that this methodology is interesting by obtaining the best results in literature for two of the three studied scheduling problems.     

Optimal smoothing and interpolating splines with constraints

This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approximation of discontinuous functions, and trajectory planning.     

矩阵损失下一般Gauss-Markov模型线性预测的可容许性

本文研究了一般Gauss-Markov模型中线性可预测变量的线性预测的可容许性.在给出线性预测可容许性定义的基础上,通过构造一个特殊的常量矩阵D0,分别得到了齐次和非齐次线性预测类中可容许的充要条件.     

Active Constraint Set Invariancy Sensitivity Analysis in Linear Optimization

Active constraint set invariancy sensitivity analysis is concerned with finding the range of parameter variation so that the perturbed problem has still an optimal solution with the same support set that the given optimal solution of the unperturbed problem has. However, in an optimization problem with inequality constraints, active constraint set invariancy sensitivity analysis aims to find the range of parameter variation, where the active constraints in a given optimal solution remains invariant.For the sake of simplicity, we consider the primal problem in standard form and consequently its dual may have an optimal solution with some active constraints. In this paper, the following question is answered: “what is the range of the parameter, where for each parameter value in this range, a dual optimal solution exists with exactly the same set of positive slack variables as for the current dual optimal solution?”. The differences of the results between the linear and convex quadratic optimization problems are highlighted too.     

A sequential method for a class of box constrained quadratic programming problems

The aim of this paper is to propose an algorithm, based on the optimal level solutions method, which solves a particular class of box constrained quadratic problems. The objective function is given by the sum of a quadratic strictly convex separable function and the square of an affine function multiplied by a real parameter. The convexity and the nonconvexity of the problem can be characterized by means of the value of the real parameter. Within the algorithm, some global optimality conditions are used as stopping criteria, even in the case of a nonconvex objective function. The results of a deep computational test of the algorithm are also provided. This paper has been partially supported by M.I.U.R.     

BDD-based optimization for the quadratic stable set problem

The quadratic stable set problem (QSSP) is a natural extension of the well-known maximum stable set problem. The QSSP is NP-hard and can be formulated as a binary quadratic program, which makes it an interesting case study to be tackled from different optimization paradigms. In this paper, we propose a novel representation for the QSSP through binary decision diagrams (BDDs) and adapt a hybrid optimization approach which integrates BDDs and mixed-integer programming (MIP) for solving the QSSP. The exact framework highlights the modeling flexibility offered through decision diagrams to handle nonlinear problems. In addition, the hybrid approach leverages two different representations by exploring, in a complementary way, the solution space with BDD and MIP technologies. Machine learning then becomes a valuable component within the method to guide the search mechanisms. In the numerical experiments, the hybrid approach shows to be superior, by at least one order of magnitude, than two leading commercial MIP solvers with quadratic programming capabilities and a semidefinite-based branch-and-bound solver.     

Point prediction of future order statistics from an exponential distribution

Two-sample point prediction is considered for a two-parameter exponential distribution. Several point predictors such as the best unbiased predictor, best invariant predictor and maximum likelihood predictor are obtained for future order statistics on the basis of observed record values in two cases: where the location parameter is known and unknown. These predictors are compared in the sense of their mean squared prediction errors. Finally, some numerical results are given to illustrate the proposed procedures.     

Splitting Patterns and Invariants of Quadratic Forms

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices i_W(φ_k) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN , we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ?(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ? ? φ ∈ Iⁿ F and give some bounds on m depending on n which assure that they have the same splitting pattern.     

一类二阶二次变系数微分算子的不变子空间及其应用

研究了一类二阶二次变系数微分算子的不变子空间,讨论了这类微分算子不变子空间的应用,并给出了具体应用的一些例子.在这些例子中,构造了大量变系数非线性演化方程的精确解.