Existence of maximal points with respect to ordered bipreference relations

The maximality of a point with respect to an ordered pair of arbitrary preference relations is introduced. Necessary and sufficient conditions for the existence of maximal points are given.The authors are grateful for the referee's comments which led to an improved presentation of the paper.     

On solving a D.C. programming problem by a sequence of linear programs

We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed convex set in ⁿ . This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved.Parts of this research were accomplished while the third author was visiting the University of Trier, Germany, as a fellow of the Alexander von Humboldt foundation.     

Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison

We consider the following global optimization problems for a Lipschitz functionf implicitly defined on an interval [a, b]. Problem P: find a globally-optimal value off and a corresponding point; Problem Q: find a set of disjoint subintervals of [a, b] containing only points with a globally-optimal value and the union of which contains all globally optimal points. A two-phase algorithm is proposed for Problem P. In phase I, this algorithm obtains rapidly a solution which is often globally-optimal. Moreover, a sufficient condition onf for this to be the case is given. In phase II, the algorithm proves the-optimality of the solution obtained in phase I or finds a sequence of points of increasing value containing one with a globally-optimal value. The new algorithm is empirically compared (on twenty problems from the literature) with a best possible algorithm (for which the optimal value is assumed to be known), with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen. For small, the new algorithm requires only a few percent more function evaluations than the best possible one. An extended version of Piyavskii's algorithm is proposed for problem Q. A sufficient condition onf is given for the globally optimal points to be in one-to-one correspondance with the obtained intervals. This result is achieved for all twenty test problems.The research of the authors has been supported by AFOSR grants 0271 and 0066 to Rutgers University. Research of the second author has been also supported by NSERC grant GP0036426, FCAR grant 89EQ4144 and partially by AFOSR grant 0066. We thank Nicole Paradis for her help in drawing the figures.     

Computation of efficient solutions of discretely distributed stochastic optimization problems

In engineering and economics often a certain vectorx of inputs or decisions must be chosen, subject to some constraints, such that the expected costs (or loss) arising from the deviation between the outputA() x of a stochastic linear systemxA()x and a desired stochastic target vectorb() are minimal. Hence, one has the following stochastic linear optimization problem minimizeF(x)=Eu(A()x b()) s.t.xD, (1) whereu is a convex loss function on ^m , (A(), b()) is a random (m,n + 1)-matrix, E denotes the expectation operator andD is a convex subset of ⁿ . Concrete problems of this type are e.g. stochastic linear programs with recourse, error minimization and optimal design problems, acid rain abatement methods, problems in scenario analysis and non-least square regression analysis.Solving (1), the loss functionu should be exactly known. However, in practice mostly there is some uncertainty in assigning appropriate penalty costs to the deviation between the outputA ()x and the targetb(). For finding in this situation solutions hedging against uncertainty a set of so-called efficient points of (1) is defined and a numerical procedure for determining these compromise solutions is derived. Several applications are discussed.     

Higher-order necessary conditions for infinite and semi-infinite optimization

In this paper, we present a unified theory of first-order and higher-order necessary optimality conditions for abstract vector optimization problems in normed linear spaces. We prove general multiplier rules, from which nearly all known first-order, second-order, and higher-order necessary conditions can be derived. In the last section, we prove higher-order necessary conditions for semi-infinite programming problems.This work was developed within the Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, Bremen, West Germany.The author wishes to thank Prof. Dr. D. Hinrichsen for his helpful remarks and discussions during the preparation of this work.     

Facility location in the presence of forbidden regions,I: Formulation and the case of Euclidean distance with one forbidden circle

A constrained form of the Weber problem is formulated in which no path is permitted to enter a prespecified forbidden region $\text{R}$ of the plane. Using the calculus of variations the shortest path between two points $\text{x}\text{,}\text{y}\text{?}\text{R}$ which does not intersect $\text{R}$ is determined. If $\text{d(}\text{x}\text{,}\text{y}\text{)}$ is unconstrained distance, we denote the shortes distance along a feasible path by $\text{d}\text{(}\text{x}\text{y}\text{)}$. The constrained Weber problem is, then: given points $\text{x}{}_{\mathrm{j}}\text{?}\text{R}$ and positive weights w_j, j = 1,2,…,n, find a point $\text{x}\text{?}\text{R}$ such that

$\text{f(}\text{x}\text{)=}\text{Σ}\text{n}\text{j=1}\text{d(}\text{x}\text{,}\text{x}{}_{\mathrm{j}}\text{)}$

is a minimum.An algorithm is formulated for the solution of this problem when $\text{d(}\text{x}\text{,}\text{y}\text{)}$ is Euclidean distance and $\text{R}$ is a single circular region. Numerical results are presented.     

Spectroscopic and Quantum Mechanical Studies of Substituted Anilines and their Charge-Transfer Complexes with Iodine in Different Solvents

Summary. Equilibrium constants and molar extinction coefficients for 1:1 charge-transfer complexes between 2-hydroxyaniline (HA), 5-chloro-2-hydroxyaniline (CHA), and 4-bromo-2,6-dimethylaniline (BMA) as donors and iodine, as a typical σ-acceptor were determined spectrophotometrically in chloroform, dichloromethane, and carbontetrachloride solutions. Spectral characteristics and formation constants are discussed in terms of donor molecular structure and solvent polarity. The stoichiometry of the complexes was established to be 1:1. For this purpose, optical data were subjected to the form of the Rose-Drago equation for 1:1 equilibria. Electronic absorption spectra of the studied anilines were measured in different solvents. Spectral data were reported and band maxima were assigned to the appropriate molecular orbital transitions (π–π^* and n–π^* electronic transition). Solvent effects on the electronic transitions were discussed. Optimized geometry of the studied anilines was obtained at B3LYP/6-31 + G(d). The effect of the electronic factors of the substituents on the geometrical parameters of the ring has been explored. Geometrical values of the ring deviate from the regular hexagonal ring. Intramolecular H-bonds in HA and CHA have been computed at B3LYP/6-31 + G(d) and MP2/6-31 + G(d) levels. The H-bonding distance was calculated to be 2.105 ? in HA and 2.127 ? in CHA. Abstracted from her M.Sc. thesis     

Properties of the random search in global optimization

From theorems which we prove about the behavior of gaps in a set ofN uniformly random points on the interval [0, 1], we determine properties of the random search procedure in one-dimensional global optimization. In particular, we show that the uniform grid search is better than the random search when the optimum is chosen using the deterministic strategy, that a significant proportion of large gaps are contained in the uniformly random search, and that the error in the determination of the point at which the optimum occurs, assuming that it is unique, will on the average be twice as large using the uniformly random search compared with the uniform grid. In addition, some of the properties of the largest gap are verified numerically, and some extensions to higher dimensions are discussed. The latter show that not all of the conclusions derived concerning the inadequacies of the one-dimensional random search extend to higher dimensions, and thaton average the random search is better than the uniform grid for dimensions greater than 6.This paper is based on work started in the Statistics Department of Princeton University when the first author was visiting as a Research Associate. Part of this research was supported by the Office of Naval Research, Contract No. 0014-67-A-0151-0017, and by the US Army Research Office—Durham, Contract No. DA-31-124-ARO-D-215.2 The authors wish to thank B. Omodei for his careful work in preparing the programs for the results of Figs. 1–2 and Table 1. The computations were performed on the IBM 360/50 of the Australian National University's Computer Centre. Thanks are also due to R. Miles for suggestions regarding the extension of the results to multidimensional regions, and to P. A. P. Moran and R. Brent for suggestions regarding the evaluation of the integral ₀ ¹ ... ^0/1 (x ₁ ² + ... +x _n ^/2 )^1/2 dx ₁ ...dx _n.     

分块镜扇形子镜驱动点位置优化计算

在光学综合孔径共相位检测微调系统中,分块镜子镜三自由度微动调节机构通常采用3个并联的微位移致动器来驱动各子镜,使单个分块镜具有独立的姿态调整能力,从而实现了光学共相位误差的纳米级精确校正。以理论力学原理作为切入点展开了分析,针对微动机构的稳定性、tilt-tip调节灵敏度等,根据所给定几何参数的子镜三驱动点位置对其进行了优化计算。经数学建模和仿真计算,得到了定参数子镜的三驱动点最佳位置坐标,进而归纳出了普遍适用的优化评价方法。     

原子荧光光谱法测定砷过程中还原反应条件的优化

自然界中砷和砷的化合物一般可通过水、大气和食物等途径进入人体,危害人体健康,原子荧光光谱法测定砷含量被广泛应用。为准确测定样品中砷的含量,以原子荧光光谱法(AFS)测定土壤中总砷含量为例,重点研究高价态砷(Ⅴ)还原为低价态砷(Ⅲ)的实验条件,采用正交实验进行条件优化,建立适合的实验方法。通过设计L9(34)正交实验,高价态砷的还原反应与四个因素有关,即与反应温度、反应时间、盐酸加入量和硫脲的用量呈一定的正相关,结果表明,四个因素中硫脲用量的影响最为显著,其次是盐酸加入量、反应时间和反应温度。通过对四个因素的多水平检验,优化后的实验条件为：硫脲用量2.0 mL、盐酸加入量10%、反应时间20 min、温度20～35 ℃(即普通室温)。采用土壤标准物质验证优化后条件的可行性,得出砷的测定值在标准值范围内,RSD在2.2%～4.2%,精密度和准确性均满足质控要求。优化后的实验条件使得操作简便易行,提高了工作效率。