一类随机多目标二次线性规划模型的交互式算法

针对线性约束条件下带有一个二次目标函数和多个线性目标函数的随机多目标决策问题，借助参考方向法和权重法对该决策问题的期望值模型进行标量化，获得了关于期望值模型的（恰当／弱）有效解的充要条件，引入Ａｃｈｉｅｖｅｍｅｎｔ函数建立了一类随机多目标二次线性规划模型的交互式计算方法．     

Constructing composite search directions with parameters in quadratic interpolation models

In this paper, we introduce the weighted composite search directions to develop the quadratic approximation methods. The purpose is to make fully use of the information disclosed by the former steps to construct possibly more promising directions. Firstly, we obtain these composite directions based on the properties of simplex methods and use them to construct trust region subproblems. Then, these subproblems are solved in the algorithm to find solutions of some benchmark optimization problems. The computation results show that for most tested problems, the improved quadratic approximation methods can obviously reduce the number of function evaluations compared with the existing ones. Finally, we conclude that the algorithm will perform better if the composite directions approach the previous steepest descent direction of the sub-simplex so far. We also point out the potential applications of this improved quadratic interpolation method in business intelligence systems.     

Maximally productive input–output units

Technical or Pareto–Koopmans efficiency models can be based on ratios of weighted sums of outputs to weighted sums of inputs. Differing meanings have been considered for such factor weights. In this paper, we use value or cost rate meanings depending on model orientation. These meanings permit considering the simultaneous assignment of input and output factor weights along with optimal intensity values for a virtual composite unit constructed from the observed units. An optimization principle we call the winners-take-all criterion is proposed for determining the maximally productive unit(s). No assumptions are required on the internal transformation processes of the units. The model simultaneously determines the intensities and factor weights and results in indefinite quadratic programming problems that simplify to linear programming in certain cases. For the general case, genetic search is applied. Numerical illustrations are provided for faculty merit scoring and for the 15 hospital dataset of Sherman.     

Forward-looking variables in deterministic control

In this paper, we adapt the Fair and Taylor [4] method for forward-looking variables in simulation models to control theory models. In particular, we develop a procedure for solving quadratic linear control models when there are forward-looking variables in the system equations. The simplest way to do this for deterministic problems would be to stack up the variables for all time periods using Theil's procedure [9], as suggested by Hughes-Hallet and Rees [5] for simulation models and done by Becker and Rustem [7] for perfect foresight problems. However, we plan to continue from the current paper and develop similar procedures for passive and active learning control problems, and the stacking procedure does not seem as natural for those problems. Therefore, we will use the Fair-Taylor approach here and adapt it for deterministic quadratic linear problems.     

Integer Programming Duality in Multiple Objective Programming

The weighted sums approach for linear and convex multiple criteria optimization is well studied. The weights determine a linear function of the criteria approximating a decision makers overall utility. Any efficient solution may be found in this way. This is not the case for multiple criteria integer programming. However, in this case one may apply the more general e-constraint approach, resulting in particular single-criteria integer programming problems to generate efficient solutions. We show how this approach implies a more general, composite utility function of the criteria yielding a unified treatment of multiple criteria optimization with and without integrality constraints. Moreover, any efficient solution can be found using appropriate composite functions. The functions may be generated by the classical solution methods such as cutting plane and branch and bound algorithms.     

An unconstrained dual approach to solving Karmarkar-type linear programs using conventional barrier functions

This paper proposes an unconstrained dual approach and an efficient algorithm for solving Karmarkar-type linear programming problems. Conventional barrier functions are incorporated as a perturbation term in the derivation of the associated duality theory. An optimal solution of the original linear program can be obtained by solving a sequence of unconstrained concave programs, or be approximated by solving one such dual program with a sufficiently small perturbation parameter. A globally convergent curved-search algorithm with a quadratic rate of convergence is designed for this purpose. Based on our testing results, we find that the computational procedure is very efficient and can be a viable approach for solving linear programming problems.     

An outcome space approach for generalized convex multiplicative programs

This paper addresses the problem of minimizing an arbitrary finite sum of products of two convex functions over a convex set. Nonconvex problems in this form constitute a class of generalized convex multiplicative problems. Convex analysis results allow to reformulate the problem as an indefinite quadratic problem with infinitely many linear constraints. Special properties of the quadratic problem combined with an adequate outer approximation procedure for handling its semi-infinite constrained set enable an efficient constraint enumeration global optimization algorithm for generalized convex multiplicative programs. Computational experiences illustrate the proposed approach.     

Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions

The main purpose of this paper is using estimates for character sums and analytic methods to study the second, fourth, and sixth order moments of generalized quadratic Gauss sums weighted by L-functions. Three asymptotic formulae are obtained.     

Analytic,Computational, and Approximate Forms for Ratios of Noncentral and Central Gaussian Quadratic Forms

Many useful statistics equal the ratio of a possibly noncentral chi-square to a quadratic form in Gaussian variables with all positive weights. Expressing the density and distribution function as positively weighted sums of corresponding F functions has many advantages. The mixture forms have analytic value when embedded within a more complex problem. The mixture forms also have computational value. The expansions work well with quadratic forms having few components and small degrees of freedom. A more general algorithm from earlier literature can take longer or fail to converge in the same setting. Many approximations have been suggested for the problem. A positively weighted noncentral quadratic form can always have two moments matched to a noncentral chi-square. For a single quadratic form, the noncentral form performs neither uniformly more or less accurately than older approximations. The approach also gives a noncentral F approximation for any ratio of a positively weighted noncentral form to a positively weighted central quadratic form. The method provides better accuracy for noncentral ratios than approximations based on a single chi-square. The accuracy suffices for many practical applications, such as power analysis, even with few degrees of freedom. Naturally the approximation proves much faster and simpler to compute than any exact method. Embedding the approximation in analytic expressions provides simple forms which correctly guarantee only positive values have nonzero probabilities, and also automatically reduce to partially or fully exact results when either quadratic form has only one term.     

Statistical tests of heterogeneity for anisotropic multifractional Brownian fields

In this paper, we deal with some anisotropic extensions of the multifractional Brownian fields that account for spatial phenomena whose properties of regularity and directionality may both vary in space. Our aim is to set statistical tests to decide whether an observed field of this kind is heterogeneous or not. The statistical methodology relies upon a field analysis by quadratic variations, which are averages of square field increments. Specific to our approach, these variations are computed locally in several directions. We establish an asymptotic result showing a linear Gaussian relationship between these variations and parameters related to regularity and directional properties of the model. Using this result, we then design a test procedure based on Fisher statistics of linear Gaussian models. Eventually we evaluate this procedure on simulated data.     

A primal–dual regularized interior-point method for convex quadratic programs

Interior-point methods in augmented form for linear and convex quadratic programming require the solution of a sequence of symmetric indefinite linear systems which are used to derive search directions. Safeguards are typically required in order to handle free variables or rank-deficient Jacobians. We propose a consistent framework and accompanying theoretical justification for regularizing these linear systems. Our approach can be interpreted as a simultaneous proximal-point regularization of the primal and dual problems. The regularization is termedexact to emphasize that, although the problems are regularized, the algorithm recovers a solution of the original problem, for appropriate values of the regularization parameters.     

Pontryagin Type Maximum Principle for budget-constrained infinite horizon optimal control problems with linear dynamics

In this paper a class of infinite horizon optimal control problems with an isoperimetrical constraint, also interpreted as a budget constraint, is considered. Herein a linear both in the state and in the control dynamic is allowed. The problem setting includes a weighted Sobolev space as the state space. For this class of problems, we establish the necessary optimality conditions in form of a Pontryagin Type Maximum Principle including a transversality condition. The proved theoretical result is applied to a linear–quadratic regulator problem.     

Inequalities involving partial derivatives

This paper presents a method for obtaining closed form solutions to serial and nonserial dynamic programming problems with quadratic stage returns and linear transitions. Global parametric optimum solutions can be obtained regardless of the convexity of the stage returns. The closed form solutions are developed for linear, convex, and nonconvex quadratic returns, as well as the procedure for recursively solving each stage of the problem. Dynamic programming is a mathematical optimization technique which is especially powerful for certain types of problems. This paper presents a procedure for obtaining analytical solutions to a class of dynamic programming problems. In addition, the procedure has been programmed on the computer to facilitate the solution of large problems.     

Suboptimal control of systems with multiple delays

The application of Pontryagin's maximum principle to the optimization of linear systems with time delays results in a system of coupled two-point boundary-value problems involving both delay and advance terms. The exact solution of this system of TPBV problems is extremely difficult, if not impossible. In this paper, a fast-converging iterative approach is developed for obtaining the suboptimal control for nonstationary linear systems with multiple state and control delays and with quadratic cost. At each step of the proposed method, a linear nondelay system with an extra perturbing input must be optimized. The procedure can be extended for the optimization of nonlinear systems with multiple time-varying delays, provided that some of the nonlinearities satisfy the Lipschitz condition.     

A weighted pseudoinverse,generalized singular values,and constrained least squares problems

The weighted pseudoinverse providing the minimum semi-norm solution of the weighted linear least squares problem is studied. It is shown that it has properties analogous to those of the Moore-Penrose pseudoinverse. The relation between the weighted pseudoinverse and generalized singular values is explained. The weighted pseudoinverse theory is used to analyse least squares problems with linear and quadratic constraints. A numerical algorithm for the computation of the weighted pseudoinverse is briefly described.This work was supported in part by the Swedish Institute for Applied Mathematics.     

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.     

Revisions of constraint approximations in the successive QP method for nonlinear programming problems

In the last few years the successive quadratic programming methods proposed by Han and Powell have been widely recognized as excellent means for solving nonlinea programming problems. However, there remain some questions about their linear approximations to the constraints from both theoretical and empirical points of view. In this paper, we propose two revisions of the linear approximation to the constraints and show that the directions generated by the revisions are also descent directions of exact penalty functions of nonlinear programming problems. The new technique can cope better with bad starting points than the usual one.     

Radially symmetric weighted extended b-spline model

In this paper, the weighted extended basis splines approach in the finite element method is applied to the electrostatic, electromagnetic wave and bioheat problems for inhomogeneous boundary conditions and radially symmetric structures. This new method, which does not need mesh generation, overcomes some of the drawbacks of using meshes and piecewise-uniform or linear trial functions. Two-dimensional radially symmetric electrostatic and electromagnetic wave equations are evaluated. We also attempt to propose a three-dimensional radially symmetric unexposed human eye model for simulating changes in corneal temperature using these new finite elements in conjunction with linear, quadratic and cubic b-splines. Our findings indicate that weighted extended basis spline solutions improve the standard finite element method. The simulation results which are verified using the values reported in the literature, point out to better efficiency in terms of the accuracy level.     

On some hybrid power moments of products of generalized quadratic Gauss sums and Kloosterman sums*

In this paper, we investigate hybrid power moments of generalized quadratic Gauss sums weighted with powers of Kloosterman sums and with powers of values of Dirichlet L-functions at 1. We obtain several exact formulas for prime and prime power modulus and some asymptotic formulas.