A class of primal affine scaling algorithms

We obtain a class of primal affine scaling algorithms which generalize some known algorithms. This class, depending on a r-parameter, is constructed through a family of metrics generated by −r power, r ? 1, of the diagonal iterate vector matrix. We prove the so-called weak convergence of the primal class for nondegenerate linearly constrained convex programming. We observe the computational performance of the class of primal affine scaling algorithms, accomplishing tests with linear programs from the NETLIB library and with some quadratic programming problems described in the Maros and Mészáros repository.     

Comparative analysis of affine scaling algorithms based on simplifying assumptions

We analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions. We show that, under a strong probabilistic assumption regarding the distribution of the data in an iteration, the decrease in the primal potential function will be with high probability, compared to the guaranteed(1). ( 2n is a parameter in the potential function andn is the number of variables.) Under the same assumption, we further show that the objective reduction rate of Dikin's affine scaling algorithm is with high probability, compared to no guaranteed convergence rate.Research supported in part by NSF Grant DDM-8922636.     

Optimal scaling of balls and polyhedra

The concern is with solving as linear or convex quadratic programs special cases of the optimal containment and meet problems. The optimal containment or meet problem is that of finding the smallest scale of a set for which some translation contains a set or meets each element in a collection of sets, respectively. These sets are unions or intersections of cells where a cell is either a closed polyhedral convex set or a closed solid ball.This work was supported in part by the Department of Energy Contract DE-AC03-76-SF00326, PA No. DE-AT-03-76ER72018; National Science Foundation Grants MCS79-03145 and SOC78-16811; and the Army Research Office—Durham, Contract DAAG-29-78-G-0026.     

Optimal preventive maintenance for equipment with two quality states and general failure time distributions

We present an economic model for the optimization of preventive maintenance in a production process with two quality states. The equipment starts its operation in the in-control state but it may shift to the out-of-control state before failure or scheduled preventive maintenance. The time of shift and the time of failure are generally distributed random variables. The two states are characterized by different failure rates and revenues. We first derive the structure of the optimal maintenance policy, which is defined by two critical values of the equipment age that determine when to perform preventive maintenance depending on the actual (observable) state of the process. We then provide properties of the optimal solution and show how to determine the optimal values of the two critical maintenance times accurately and efficiently. The proposed model and, in particular, the behavior of the optimal solution as the model parameters and the shift and failure time distributions change are illustrated through numerical examples.     

Deriving collinear scaling algorithms as extensions of quasi-Newton methods and the local convergence of DFP- and BFGS-related collinear scaling algorithms

This paper is concerned with collinear scaling algorithms for unconstrained minimization where the underlying local approximants are forced to interpolate the objective function value and gradient at only the two most recent iterates. By suitably modifying the procedure of Sorensen (1980) for deriving such algorithms, we show that two members of the algorithm class derived related to the DFP and BFGS methods respectively are locally and q-superlinearly convergent. This local analysis as well as the results they yield exhibit the same sort of duality exhibited by those of Broyden, Dennis and Moré (1973) and Dennis and Moré (1974) for the DFP and BFGS methods. The results in this paper also imply the local and q-superlinear convergence of collinear scaling algorithms of Sorensen (1982, pp. 154–156) related to the DFP and BFGS methods.Research supported in part by funds provided by the Washington State University Research and Arts Committee, by NSF Grant DMS-8414460 and by DOE Grant DE-FG06-85ER25007.     

Optimal convex hull algorithms on enhanced meshes

In this paper we propose time-optimal convex hull algorithms for two classes of enhanced meshes. Our first algorithm computes the convex hull of an arbitrary set ofn points in the plane inO (logn) time on a mesh with multiple broadcasting of sizen×n. The second algorithm shows that the same problem can be solved inO (1) time on a reconfigurable mesh of sizen×n. Both algorithms achieve time lower bounds for their respective model of computation.This work was supported by NASA under grant NCCI-99.Additional support by the National Science Foundation under grant CCR-8909996 is gratefully acknowledged.     

Proof of algorithms by general snapshots

A constructive approach to the question of proofs of algorithms is to consider proofs that an object resulting from the execution of an algorithm possesses certain static characteristics. It is shown by an elementary example how this possibility may be used to prove the correctness of an algorithm written in ALGOL 60. The stepping stone of the approach is what is called General Snapshots, i.e. expressions of static conditions existing whenever the execution of the algorithm reaches particular points. General Snapshots are further shown to be useful for constructing algorithms.     

Polynomiality of primal-dual affine scaling algorithms for nonlinear complementarity problems

This paper provides an analysis of the polynomiality of primal-dual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu’s scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primal-dual affine scaling algorithms generates an approximate solution (given a precision ε) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial ofn, ln(1/ε) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in Jansen et al., SIAM Journal on Optimization 7 (1997) 126–140. Research supported in part by Grant-in-Aids for Encouragement of Young Scientists (06750066) from the Ministry of Education, Science and Culture, Japan. Research supported by Dutch Organization for Scientific Research (NWO), grant 611-304-028     

On the stability and ergodicity of adaptive scaling Metropolis algorithms

The stability and ergodicity properties of two adaptive random walk Metropolis algorithms are considered. Both algorithms adjust the scaling of the proposal distribution continuously based on the observed acceptance probability. Unlike the previously proposed forms of the algorithms, the adapted scaling parameter is not constrained within a predefined compact interval. The first algorithm is based on scale adaptation only, while the second one also incorporates covariance adaptation. A strong law of large numbers is shown to hold assuming that the target density is smooth enough and has either compact support or super-exponentially decaying tails.     

A numerical evaluation of some collinear scaling algorithms for unconstrained

Collinear scaling algorithms for unconstrained minimization were first proposed by Davidon (1977,80) so that they may incorporate more information about the problem than is possible with quasi–Newton algorithms. Sorensen (1980,82), and Ariyawansa (1983,90) have derived collinear scaling algorithms as natural extensions of quasi–Newton algorithms. In this paper we describe the results of a comprehensive numerical evaluation of four members in the classes of collinear scaling algorithms derived by Sorensen (1980,82) and Ariyawansa (1983,90), relative to the quasi–Newton algorithms they extend.     

Optimal conditioning of self-scaling variable Metric algorithms

Variable Metric Methods are Newton—Raphson-like algorithms for unconstrained minimization in which the inverse Hessian is replaced by an approximation, inferred from previous gradients and updated at each iteration. During the past decade various approaches have been used to derive general classes of such algorithms having the common properties of being Conjugate Directions methods and having quadratic termination. Observed differences in actual performance of such methods motivated recent attempts to identify variable metric algorithms having additional properties that may be significant in practical situations (e.g. nonquadratic functions, inaccurate linesearch, etc.). The SSVM algorithms, introduced by this first author, are such methods that among their other properties, they automatically compensate for poor scaling of the objective function. This paper presents some new theoretical results identifying a subclass of SSVM algorithms that have the additional property of minimizing a sharp bound on the condition number of the inverse Hessian approximation at each iteration. Reducing this condition number is important for decreasing the roundoff error. The theoretical properties of this subclass are explored and two of its special cases are tested numerically in comparison with other SSVM algorithms.This work has been done while this author was a visiting fellow at the Engineering Economic System Department, Stanford University.     

Portfolio construction based on stochastic dominance and target return distributions

Mean-risk models have been widely used in portfolio optimization. However, such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk-averse investors. This paper considers the problem of constructing a portfolio which is non-dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties. The problem is multi-objective and is transformed into a single objective problem by using the reference point method, in which target levels, known as aspiration points, are specified for the objective functions. A model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return. This concept is extended by additionally specifying reservation points, which act pre-emptively in the optimization model. The theoretical properties of the models are studied. The performance of the models on real data drawn from the Hang Seng index is also investigated.     

Optimal block diagonal scaling of block 2-cyclic matrices

We describe a class of optimal block diagonal scalings (preconditionings) of a symmetric positive definite block 2-cyclic matrix, generalizing a result of Forsythe and Strauss [1] for (point) 2-cyclic matrices.     

带两个服务等级的三台机最优在线算法

研究了带服务等级约束的三台平行机在线排序问题.每台机器和每个工件的服务等级为1或者2,工件只能在等级不高于它的机器上加工,即等级为1的工件只能在等级为1的机器上加工,等级为2的工件可在所有机器上加工.每个工件的加工时间为一个单位,目标是极小化所有工件的总完工时间.考虑两种情形:当一台机器等级为1,两台机器等级为2时,给出了竞争比为17/14的最优在线算法;当两台机器等级为1,一台机器等级为2时,给出了竞争比为43/36的最优在线算法.     

Optimal algorithms for total exchange without buffering on the hypercube

Two methods are given for constructing total exchange algorithms for hypercubic processor networks. This is done by means of bit sequences with special properties. The algorithms are optimal with respect to a given time model, need no intermediate message buffering and are local in the sense that every processor executes basically the same program.     

SOME PROPERTIES OF ROSEN’S MLE FOR GENERAL DISTRIBUTIONS

1 IntroductionConsider the lnultivariate linear model (MLM) as follows:mX = Z AiBiC E (1)i= 1where X, Ai, Bi and C are p x nfp x qi(qi 5 p), qi x ki and ki x n matrices respectively, Z is ap x p definite positive matrix with p(C1) p 5 n and R(CL) G R(Cfu--,) g' g R(CI), p(.)and R(.) stand for the rank and the colunu spanned linear space Of a matriX respbctively.e = (e1,'2,... f e.), e1le21',f n are iid. p--variate random vectors with D(e1) = Z > 0,E(El) = 0, A: aild C: are …     

Optimal unbiased estimators for finite families of distributions

Necessary and sufficient conditions for the existence of a nontrivial optimal algebra are established on the basis of a statistical structure with finitely many measures. A method is proposed for constructing an optimal unbiased estimator of a given function of a parameter, if it exists.Translated from Statisticheskie Metody, pp. 156–163, 1980.