benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

randomness computability and density we study effectively given positive reals more specifically computably enumerable reals under a measure of relative randomness introduced by solovay manuscript ibm thomas j watson research center yorktown heights ny 1975 and studied by calude hertling khoussainov and wang theoret comput sci 255 2001 pp 125 149 calude theoret comput sci 271 2002 pp 3 14 kucera and slaman siam j comput 31 2002 pp 199 211 and downey hirschfeldt and laforte mathematical foundations of computer science 2001 springer verlag berlin 2001 pp 316 327 among others this measure is called domination or solovay reducibility and is defined by saying that alpha dominates beta if there are a constant c and a partial computable function varphi such that for all positive rationals q alpha we have varphi q downarrow beta and beta varphi q leqslant c alpha q the intuition is that an approximating sequence for alpha generates one for beta whose rate of convergence is not much slower than that of the original sequence it is not hard to show that if alpha dominates beta then the initial segment complexity of alpha is at least that of beta in this paper we are concerned with structural properties of the degree structure generated by solovay reducibility we answer a natural question in this area of investigation by proving the density of the solovay degrees we also provide a new characterization of the random computably enumerable reals in terms of splittings in the solovay degrees specifically we show that the solovay degrees of computably enumerable reals are dense that any incomplete solovay degree splits over any lesser degree and that the join of any two incomplete solovay degrees is incomplete so that the complete solovay degree does not split at all the methodology is of some technical interest since it includes a priority argument in which the injuries are themselves controlled by randomness considerations

an improved multi step gradient type method for large scale optimization in this paper we propose an improved multi step diagonal updating method for large scale unconstrained optimization our approach is based on constructing a new gradient type method by means of interpolating curves we measure the distances required to parameterize the interpolating polynomials via a norm defined by a positive definite matrix by developing on implicit updating approach we can obtain an improved version of hessian approximation in diagonal matrix form while avoiding the computational expenses of actually calculating the improved version of the approximation matrix the effectiveness of our proposed method is evaluated by means of computational comparison with the bb method and its variants we show that our method is globally convergent and only requires o n memory allocations

a half product based approximation scheme for agreeably weighted completion time variance a recent informs journal on computing paper presents a fully polynomial time approximation scheme for minimizing the completion time variance with agreeable weights on a single machine this paper presents a much faster scheme for this problem

randomness and recursive enumerability one recursively enumerable real alpha dominates another one beta if there are nondecreasing recursive sequences of rational numbers a n n in omega approximating alpha and b n n in omega approximating beta and a positive constant c such that for all n c alpha a n geq beta b n see r m solovay draft of a paper or series of papers on chaitin s work manuscript ibm thomas j watson research center yorktown heights ny 1974 p 215 and g j chaitin ibm j res develop 21 1977 pp 350 359 we show that every recursively enumerable random real dominates all other recursively enumerable reals we conclude that the recursively enumerable random reals are exactly the omega numbers g j chaitin ibm j res develop 21 1977 pp 350 359 second we show that the sets in a universal martin lof test for randomness have random measure and every recursively enumerable random number is the sum of the measures represented in a universal martin lof test

process and truth table characterisations of randomness this paper uses quick process machines to provide characterisations of computable randomness schnorr randomness and weak randomness the quick process machine is a type of process machine first considered in work of levin and zvonkin a new technique for building process machines and quick process machines is presented this technique is similar to the kc theorem for prefix free machines using this technique a method of translating computable martingales to quick process machines is given this translation forms the basis for these new randomness characterisations quick process machines are also used to provide characterisations of computable randomness schnorr randomness and weak randomness in terms of truth table reducibility

new implicit updates in multi step quasi newton methods for unconstrained optimisation multi step quasi newton methods for optimisation using data from more than one previous step to revise the current approximate hessian were introduced by ford and moghrabi in j comput appl math 50 1994 305 where they showed how to construct such methods by means of interpolating curves these methods also utilise standard quasi newton formulae but with the vectors normally employed in the formulae replaced by others determined from a multi step version of the secant equation some methods the accumulative and fixed point approaches for defining the parameter values which correspond to the iterates on the interpolating curve were presented by ford and moghrabi in optim methods software 2 1993 357 both the accumulative and the fixed point methods measure the distances required to parameterise the interpolating polynomials via a norm defined by a positive definite matrix m the fixed point algorithm which takes m to be the current approximate hessian was found experimentally to be the best of the six multi step methods studied in ford and moghrabi 1993 all of which exhibited improved numerical performance by comparison with the standard single step bfgs method to produce a better parameterisation of the interpolation ford comput math appl 42 2001 1083 developed the idea of implicit update methods the fundamental concept here is to determine an improved version of the hessian approximation to be used in computing the metric while avoiding the computational expense of actually calculating the improved version two implicit methods denoted by i2 and i3 were developed from f2 in ford 2001 the method i2 employed parameter values generated from an implicit single step bfgs update while i3 used values from an implicit two step update in this paper we describe the derivation of new implicit updates which are similar to i3 the experimental results we present show that one of the new implicit methods produces markedly better performance than the existing implicit methods particularly as the dimension of the test problem grows

a new gradient method via quasi cauchy relation which guarantees descent we propose a new monotone algorithm for unconstrained optimization in the frame of barzilai and borwein bb method and analyze the convergence properties of this new descent method motivated by the fact that bb method does not guarantee descent in the objective function at each iteration but performs better than the steepest descent method we therefore attempt to find stepsize formula which enables us to approximate the hessian based on the quasi cauchy equation and possess monotone property in each iteration practical insights on the effectiveness of the proposed techniques are given by a numerical comparison with the bb method

on kurtz randomness kurtz randomness is a notion of algorithmic randomness for real numbers in particular a real x is called kurtz random or weakly random iff it is contained in every computably enumerable set u of lebesgue measure 1 we prove a number of characterizations of this notion relating it to other notions of randomness such as the well known notions of computable randomness martin lof randomness and schnorr randomness for the first time we give machine characterizations of kurtz randomness whereas the turing degree of every martin lof random c e real is the complete degree and the degrees of schnorr random c e reals are all high we show that kurtz random c e reals occur in every non zero c e degree additionally we show that the sets that are low for kurtz randomness are all hyperimmune and include those that are low for schnorr randomness characterized previously by terwijn and email protected 2004 elsevier b v all rights reserved

the single machine weighted mean squared deviation problem abstract this paper studies a single machine problem related to the just in time jit production objective in which the goal is to minimize the sum of weighted mean squared deviation of the completion times with respect to a common due date in order to solve the problem several structural and dominance properties of the optimal solution are investigated these properties are then integrated within a branch and cut approach to solve a time indexed formulation of the problem the results of a computational experiment with the proposed algorithm show that the method is able to optimally solve instances with up to 300 jobs within reduced running times improving other integer programming approaches

earliness tardiness scheduling problems ii derivation of completion times about a restrictive common due date a companion paper part i considers the problem of minimizing the weighted earliness and tardiness of jobs scheduled on a single machine around a common due date d which is unrestrictively late this paper part ii considers the problem of minimizing the unweighted earliness and tardiness of jobs allowing the possibility that d is early enough to constrain the scheduling decision we describe several optimality conditions the recognition version of the problem is shown to be np complete in the ordinary sense confirming a well known conjecture moreover this complexity definition is precise since we describe a dynamic programming algorithm which runs in pseudopolynomial time this algorithm is also extremely efficient computationally providing an improvement over earlier procedures of almost two orders of magnitude in the size of instance that can be solved finally we describe a special case of the problem which is polynomially solvable

minimization of half products in this paper a special class of quadratic functions the so called half products are considered it is shown that while the minimization over the set of binary n vectors for half products is np complete an e approximating solution can be found in polynomial time for any e 0

approximability issues for unconstrained and constrained maximization of half product related functions we address the boolean programming problem of maximizing a half product function with and without a linear knapsack constraint maximizing the half product can be done in polynomial time adding a knapsack constraint makes the problem non approximable within a constant factor provided that the coefficients in the linear part of the function are negative for maximizing a function with positive coefficients in the linear part we develop a fully polynomial time approximation scheme

single machine scheduling with coefficient of variation minimization abstract the problem of scheduling n jobs on a single machine so as to minimize the coefficient of variation of the job completion times is considered in this paper some results on the characterization of an optimal schedule including its v shape are proved explicit optimal schedules for some small values of n are also given a heuristic method for solving the problem is proposed and its efficiency is investigated

stochastic binary problems with simple penalties for capacity constraints violations this paper studies stochastic programs with first stage binary variables and capacity constraints using simple penalties for capacities violations in particular we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly mathcal n mathcal p hard in general we provide pseudo polynomial algorithms for three special cases of the problem constant weights and capacity uniformly distributed subset sum with gaussian weights and strictly positively distributed random capacity and subset sum with constant weights and arbitrary random capacity we then turn to a branch and cut algorithm based on the outer approximation of the objective function we provide computational results for the stochastic knapsack problem i with gaussian weights and constant capacity and ii with constant weights and capacity uniformly distributed on randomly generated instances inspired by computational results for the knapsack problem

alternating multi step quasi newton methods for unconstrained optimization we consider multi step quasi newton methods for unconstrained optimization these methods were introduced by the authors ford and moghrabi 5 6 8 who showed how an interpolating curve in the variable space could be used to derive an appropriate generalization of the secant equation normally employed in the construction of quasi newton methods one of the most successful of these multi step methods employs the current approximation to the hessian to determine the parametrization of the interpolating curve and hence the derivatives which are required in the generalized updating formula however certain approximations were found to be necessary in the process in order to reduce the level of computation required which must be repeated at each iteration to acceptable levels in this paper we show how a variant of this algorithm which avoids the need for such approximations may be obtained this is accomplished by alternating on successive iterations a single step and a two step method the results of a series of experiments which show that the new algorithm exhibits a clear improvement in numerical performance are reported

minimizing mean squared deviation of completion times about a common due date this paper addresses a nonpreemptive single machine scheduling problem where all jobs have a common due date and have zero ready time the scheduling objective is to minimize mean squared deviation msd of job completion times about the due date this nonregular measure of performance is appropriate when earliness and tardiness are both penalized and when large deviations of completion time from the due date are undesirable a special case of the msd problem referred to as the unconstrained msd problem is shown to be equivalent to the completion time variance problem ctv studied by merten and muller merten a g m e muller 1972 variance minimization in single machine sequencing problems management sci 18september 518 528 and schrage schrage l 1975 minimizing the time in system variance for a finite jobset management sci 21may 540 543 strong results for this latter problem are combined with several new propositions to develop a reasonably efficient procedure for solving the unconstrained msd problem this enables us to improve the existing procedures for the ctv problem we also propose a branching procedure for the constrained msd problem and present computational results

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

on the minimization of completion time variance with a bicriteria extension we discuss a single machine scheduling problem where the objective is to minimize the variance of job completion times to date the problem has not been solved in polynomial time this paper presents a dynamic programming algorithm that is pseudopolynomial in complexity we also propose a fully polynomial approximation scheme and derive a lower bound that is useful in its implementation furthermore we show that the dynamic programming solution is easy to extend to a bicriteria version of the problem in which it is desired to simultaneously minimize the mean completion time

scheduling about a given common due date to minimize mean squared deviation of completion times abstract in this paper the problem of minimizing the mean squared deviation msdp of job completion times about a given common due date in n job single machine scheduling is considered the release times for all jobs are assumed to be zero no polynomial time algorithms have been found to identify an optimal solution to msdp for the tightly restricted version of msdp no even pseudo polynomial time algorithms have been found to identify an optimal solution we present some dominance properties for msdp based on which a pseudo polynomial dynamic programming algorithm which is an extension of the dynamic programming algorithm proposed by lee danusaputro and lin is presented to optimally solve the tightly restricted version of msdp computational results show that the proposed dynamic programming algorithm can optimally solve tightly restricted instances of msdp with up to 100 jobs in less than two seconds

multi step quasi newton methods for optimization abstract quasi newton methods update at each iteration the existing hessian approximation or its inverse by means of data deriving from the step just completed we show how multi step methods employing in addition data from previous iterations may be constructed by means of interpolating polynomials leading to a generalization of the secant or quasi newton equation the issue of positive definiteness in the hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original single step methods the results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach by comparison with single step methods particularly as the dimension of the problem increases

a lower bound for weighted completion time variance we consider a single machine scheduling problem to minimize the weighted completion time variance this problem is known to be np hard we propose a heuristic and a lower bound based on job splitting and the viswanathkumar and srinivasan procedure the test on more than 2000 instances shows that this lower bound is very tight and the heuristic yields solutions very close to optimal ones since the gap between the solution given by the heuristic and the lower bound is very small

a new two step gradient type method for large scale unconstrained optimization in this paper we propose some improvements on a new gradient type method for solving large scale unconstrained optimization problems in which we use data from two previous steps to revise the current approximate hessian the new method which we considered resembles to that of barzilai and borwein bb method the innovation features of this approach consist in using approximation of the hessian in diagonal matrix form based on the modified weak secant equation rather than the multiple of the identity matrix in the bb method using this approach we can obtain a higher order accuracy of hessian approximation when compares to other existing bb type method by incorporating a simple monotone strategy the global convergence of the new method is achieved practical insights into the effectiveness of the proposed method are given by numerical comparison with the bb method and its variant

minimizing weighted mean absolute deviation of job completion times from their weighted mean we address a single machine scheduling problem where the objective is to minimize the weighted mean absolute deviation of job completion times from their weighted mean this problem and its precursors aim to achieve the maximum admissible level of service equity it has been shown earlier that the unweighted version of this problem is np hard in the ordinary sense for that version a pseudo polynomial time dynamic program and a 2 approximate algorithm are available however not much except for an important solution property exists for the weighted version in this paper we establish the relationship between the optimal solution to the weighted problem and a related one in which the deviations are measured from the weighted median rather than the mean of the job completion times this generalizes the 2 approximation result mentioned above we proceed to give a pseudo polynomial time dynamic program establishing the ordinary np hardness of the problem in general we then present a fully polynomial time approximation scheme as well finally we report the findings from a limited computational study on the heuristic solution of the general problem our results specialize easily to the unweighted case they also lead to an approximation of the set of schedules that are efficient with respect to both the weighted mean absolute deviation and the weighted mean completion time

deterministic and random single machine sequencing with variance minimization we discuss the problem of scheduling several jobs on a single machine so as to minimize the variance of the jobs completion times by exploiting the well known pooled variance formula we obtain a general formula for the change in variance due to the interchange of two specified jobs whose positions have not been fixed in the sequence this result permits us to obtain explicit optimal sequences for problems with 6 or 7 jobs and to show that schrage s conjecture on positioning the job with the third largest processing time is true for problems with up to 18 jobs we then develop a heuristic procedure based on a variance interchange formula and compare it with other known heuristic methods we also consider a problem never before examined namely the case when processing times are not known deterministically but are random variables we obtain some general results under certain assumptions on the parameters of the distributions of processing times and extend to the random situation many results applicable to the deterministic case finally we give some results for the exponential distribution of processing times

fast approximation schemes for boolean programming and scheduling problems related to positive convex half product we address a version of the half product problem and its restricted variant with a linear knapsack constraint for these minimization problems of boolean programming we focus on the development of fully polynomial time approximation schemes with running times that depend quadratically on the number of variables applications to various single machine scheduling problems are reported minimizing the total weighted flow time with controllable processing times minimizing the makespan with controllable release dates minimizing the total weighted flow time for two models of scheduling with rejection

projected barzilai borwein methods for large scale box constrained quadratic programming this paper studies projected barzilai borwein pbb methods for large scale box constrained quadratic programming recent work on this method has modified the pbb method by incorporating the grippo lampariello lucidi gll nonmonotone line search so as to enable global convergence to be proved we show by many numerical experiments that the performance of the pbb method deteriorates if the gll line search is used we have therefore considered the question of whether the unmodified method is globally convergent which we show not to be the case by exhibiting a counter example in which the method cycles a new projected gradient method pabb is then considered that alternately uses the two barzilai borwein steplengths we also give an example in which this method may cycle although its practical performance is seen to be superior to the pbb method with the aim of both ensuring global convergence and preserving the good numerical performance of the unmodified methods we examine other recent work on nonmonotone line searches and propose a new adaptive variant with some attractive features further numerical experiments show that the pabb method with the adaptive line search is the best bb like method in the positive definite case and it compares reasonably well against the gpcg algorithm of more and toraldo in the indefinite case the pbb method with the adaptive line search is shown on some examples to find local minima with better solution values and hence may be preferred for this reason

note a note on the minimization of mean squared deviation of completion times about a common due date this paper points out a conceptual flaw present in a recent publication it shows how the methodology can be modified to correct this flaw it also presents improved bounds to determine the nature of a given problem

implicit updates in multistep quasi newton methods abstract we consider multistep quasi newton methods for unconstrained optimization these methods were introduced by ford and moghrabi 1 2 who showed how interpolating curves could be used to derive a generalization of the secant equation the relation normally employed in the construction of quasi newton methods one of the most successful of these multistep methods makes use of the current approximation to the hessian to determine the parametrization of the interpolating curve in the variable space and hence the generalized updating formula in this paper we investigate the use of implicit updates to the approximate hessian in an attempt to determine a better parametrization of the interpolation while avoiding the computational burden of actually carrying out the update and thus improve the numerical performance of such algorithms

positive half products and scheduling with controllable processing times we study the single machine scheduling problem with controllable job processing times to minimize a linear combination of the total weighted job completion time and the total weighted processing time compression we show that this scheduling problem is a positive half product minimization problem positive half products make up an interesting subclass of half products and are introduced in this paper to provide a conceptual framework for the problem with controllable job processing times as well as other problems this framework allows to readily derive in one fell swoop a number of results for the problem with controllable processing times from more general results obtained earlier for the half product we also present fast fully polynomial time approximation schemes for the problem with controllable processing times the schemes apply to all positive half products