a fast numerical algorithm for the composition of power series with complex coefficients abstract the composition problem for univariate complex power series p q i e the computation of the first n 1 coefficients of the composition q p is numerically solved by interpolation methods using multitape turing machines as a model of computation the composition problem of power series with integer coefficients of modulus n is possible in time o n 2 log n log where m bounds the amount of time for the multiplication of m bit numbers e g m cm log m 1 log log m 2 for multitape turing machines this algorithm is asymptotically faster than an implementation of the brent kung algorithm on a multitape turing machine the improvement is of order n 1 2 up to logarithmic terms

root refinement for real polynomials using quadratic interval refinement we consider the problem of approximating all real roots of a square free polynomial f with real coefficients given isolating intervals for the real roots and an arbitrary positive integer l the task is to approximate each root to l bits after the binary point abbott has proposed the quadratic interval refinement method qir for short which is a variant of regula falsi combining the secant method and bisection we formulate a variant of qir denoted aqir approximate qir that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively it returns a certified result for any given real polynomial whose roots are all simple in addition we propose several techniques to improve the asymptotic complexity of qir we prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial the size and the separation of the roots that is parameters exclusively related to the geometric location of the roots for integer coefficients our variant improves in theory and practice the variant with exact integer arithmetic combining our approach with multipoint evaluation we obtain near optimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

continuous piecewise affine dynamical systems do not exhibit zeno behavior in the context of continuous piecewise affine dynamical systems we study the zeno behavior i e infinite number of mode transitions in finite time interval in this note the main result reveals that piecewise affine dynamical systems do not exhibit zeno behavior a direct benefit of the main result is that one can apply smooth ordinary differential equations theory in a local manner for the analysis of piecewise affine systems

generating alternative mixed integer programming models using variable redefinition dropping the complicating constraints in a mixed integer linear program often yields a special structure subproblem that can be reformulated using a different set of decision variables once the new variables have been identified the entire problem can be reformulated in terms of the new variables we develop a theory of variable redefinition based on relating the two sets of decision variables by a linear transformation and describe methods for reformulating the special structure problem the reformulated models have a more accurate linear relaxation than the problems from which they were derived an important property within the context of linear programming based branch and bound modeling approaches

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

lower bounds for special cases of syntactic multilinear abps abstract algebraic branching programs abps are standard models for computing polynomials syntactic multilinear abps smabps are restrictions of abps where every variable is allowed to occur at most once in every path from the source to the sink node proving super polynomial lower bounds for syntactic multilinear abps remains a challenging open question in algebraic complexity theory the current best lower bound for smabps is only quadratic in the number of variables 1 in this article we develop a new approach for proving syntactic multilinear branching program lower bounds convert the smabp into an equivalent multilinear formula with a super polynomial blow up in size and then exploit the structural limitations of the resulting formula to obtain an upper bound on the rank of partial derivative matrix of the polynomial computed by the smabp using this approach we prove exponential lower bounds for special cases of smabps namely sum of read once oblivious smabps multilinear r pass abps and set multilinear abps en route we also prove an exponential lower bound for a special class of syntactic multilinear arithmetic circuits using a similar approach

an approximation scheme for stochastic integer programs arising in capacity expansion planning for capacity expansion forms a crucial part of the strategic level decision making in many applications consequently quantitative models for economic capacity expansion planning have been the subject of intense research however much of the work in this area has been restricted to linear cost models and or limited degree of uncertainty to make the problems analytically tractable this paper addresses a stochastic capacity expansion problem where the economies of scale in expansion costs are handled via fixed charge cost functions and forecast uncertainties in the problem parameters are explicitly considered by specifying a set of scenarios the resulting formulation is a multistage stochastic integer program we develop a fast linear programming based approximation scheme that exploits the decomposable structure and is guaranteed to produce feasible solutions for this problem through a probabilistic analysis we prove that the optimality gap of the heuristic solution almost surely vanishes asymptotically as the problem size increases

a fast algorithm for reversion of power series we give an algorithm for reversion of formal power series based on an efficient way to evaluate the lagrange inversion formula our algorithm requires o n 1 2 m n mm n 1 2 operations where m n and mm n are the costs of polynomial and matrix multiplication respectively this matches an algorithm of brent and kung but we achieve a constant factor speedup whose magnitude depends on the polynomial and matrix multiplication algo rithms used benchmarks confirm that the algorithm performs well in practice

lot size models with back logging strong reformulations and cutting planes we examine mixed integer programming reformulations of the uncapacitated lot sizing problem with backlogging first we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables leading to a known plant location reformulation and a shortest path reformulation each of these reformulations is strong in the sense that its linear programming relaxation solves the lot sizing problem secondly we attempt to treat the problem in the space of the original variables we give an implicit description of the convex hull of solutions and show how the problem of finding a violated cutting plane can be solved as a linear program we also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm the efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi item capacitated lot sizing problems with backlogging near optimal solutions have been found to problems with 8 periods and up to 100 times

a new cross decomposition method for stochastic mixed integer linear programming abstract two stage stochastic mixed integer linear programming milp problems can arise naturally from a variety of process design and operation problems these problems with a scenario based formulation lead to large scale milps that are well structured when first stage variables are mixed integer and second stage variables are continuous these milps can be solved efficiently by classical decomposition methods such as dantzig wolfe decomposition dwd lagrangian decomposition and benders decomposition bd or a cross decomposition strategy that combines some of the classical decomposition methods this paper proposes a new cross decomposition method where bd and dwd are combined in a unified framework to improve the solution of scenario based two stage stochastic milps this method alternates between dwd iterations and bd iterations where dwd restricted master problems and bd primal problems yield a sequence of upper bounds and bd relaxed master problems yield a sequence of lower bounds the method terminates finitely to an optimal solution or an indication of the infeasibility of the original problem case study of two different supply chain systems a bioproduct supply chain and an industrial chemical supply chain show that the proposed cross decomposition method has significant computational advantage over bd and the monolith approach when the number of scenarios is large

an extension of strassen s degree bound with every set p 1 cdots p m of multivariate polynomials we associate in a natural way several algebraic varieties i e irreducible zariski closed sets the degree of each of these closed sets can be nicely bounded in terms of the number l ns p 1 cdots p m of nonscalar operations which are necessary to evaluate p 1 cdots p m we establish lower bounds l ns p geqq omega k lg n for single specific polynomials p of degree n depending on o k variables with 0 1 coefficients typical examples are l ns sum i 1 k x i n y i geqq frac 1 2 k lg n l ns sum i 1 k x 1 x 2 cdots x i n y i frac 1 2 k lg n provided k omega k lg n for almost all polynomials p of degree n depending on k variables k ll n these lower bounds hold for any field characteristic

a stochastic programming approach for planning horizons of infinite horizon capacity planning problems planning horizon is a key issue in production planning different from previous approaches based on markov decision processes we study the planning horizon of capacity planning problems within the framework of stochastic programming we first consider an infinite horizon stochastic capacity planning model involving a single resource linear cost structure and discrete distributions for general stochastic cost and demand data non markovian and non stationary we give sufficient conditions for the existence of an optimal solution furthermore we study the monotonicity property of the finite horizon approximation of the original problem we show that the optimal objective value and solution of the finite horizon approximation problem will converge to the optimal objective value and solution of the infinite horizon problem when the time horizon goes to infinity these convergence results together with the integrality of decision variables imply the existence of a planning horizon we also develop a useful formula to calculate an upper bound on the planning horizon then by decomposition we show the existence of a planning horizon for a class of very general stochastic capacity planning problems which have complicated decision structure

solving embedded generalized network problems abstract a complete unified description is given of the design and implementation of a fast and efficient program which solves linear programming problems with network substructures the very efficient generalized network procedures which have recently been developed are extended in this paper to produce an algorithm called e mnet which can solve generalized network problems with additional constraints and additional complicating variables the only requirement on the network substructure is that it contain at most two nonzero entries in each column this requirement is the most general possible for network substructures many large linear programming problems can be placed in this form by a simple rearrangement of its rows and columns preliminary computational experience indicates e mnft is about five times faster than m inos the algorithm presented should provide a very efficient and fast solution for many linear programming problems involving production scheduling distribution facility location and personnel assignment these problems tend to contain large embedded network substructures

algorithmic applications of baur strassen s theorem shortest cycles diameter and matchings consider a directed or an undirected graph with integral edge weights from the set w w that does not contain negative weight cycles in this article we introduce a general framework for solving problems on such graphs using matrix multiplication the framework is based on the usage of baur strassen s theorem and of strojohann s determinant algorithm it allows us to give new and simple solutions to the following problems finding shortest cycles we give a simple o wn time algorithm for finding shortest cycles in undirected and directed graphs for directed graphs and undirected graphs with nonnegative weights this matches the time bounds obtained in 2011 by roditty and williams on the other hand no algorithm working in o wn time was previously known for undirected graphs with negative weights furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time computing diameter and radius we give a simple o wn time algorithm for computing a diameter and radius of an undirected or directed graphs to the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights finding minimum weight perfect matchings we present an o wn time algorithm for finding minimum weight perfect matchings in undirected graphs this resolves an open problem posted by sankowski 2009 who presented such an algorithm but only in the case of bipartite graphs these three problems that are solved in the full generality demonstrate the utility of this framework hence we believe that it can find applications for solving larger spectra of related problems as an illustrative example we apply it to the problem of computing a set of vertices that lie on cycles of length at most t for some given t we give a simple o wn time algorithm for this problem that improves over the o wn t time algorithm given by yuster in 2011 besides giving this flexible framework the other main contribution of this article is the development of a novel combinatorial interpretation of the dual solution for the minimum weight perfect matching problem despite the long history of the matching problem such a combinatorial interpretation was not known previously this result sheds a new light on the problem as there exist many structural theorems about unweighted matchings but almost no results that could cope with the weighted case

global minimization using an augmented lagrangian method with variable lower level constraints a novel global optimization method based on an augmented lagrangian framework is introduced for continuous constrained nonlinear optimization problems at each outer iteration k the method requires the varepsilon k global minimization of the augmented lagrangian with simple constraints where varepsilon k to varepsilon global convergence to an varepsilon global minimizer of the original problem is proved the subproblems are solved using the bb method numerical experiments are presented

greatest common divisors of polynomials given by straight line programs algorithms on multivariate polynomials represented by straight line programs are developed first it is shown that most algebraic algorithms can be probabilistically applied to data that are given by a straight line computation testing such rational numeric data for zero for instance is facilitated by random evaluations modulo random prime numbers then auxiliary algorithms that determine the coefficients of a multivariate polynomial in a single variable are constructed the first main result is an algorithm that produces the greatest common divisor of the input polynomials all in straight line representation the second result shows how to find a straight line program for the reduced numerator and denominator from one for the corresponding rational function both the algorithm for that construction and the greatest common divisor algorithm are in random polynomial time for the usual coefficient fields and output a straight line program which with controllably high probability correctly determines the requested answer the running times are polynomial functions in the binary input size the input degrees as unary numbers and the logarithm of the inverse of the failure probability the algorithm for straight line programs for the numerators and denominators of rational functions implies that every degree bounded rational function can be computed fast in parallel that is in polynomial size and polylogarithmic depth

tensor rank and lower bounds for arithmetic formulas we show that any explicit example for a tensor a n r f with tensor rank nr 1 o 1 where r r n log n log log n is super constant implies an explicit super polynomial lower bound for the size of general arithmetic formulas over f this shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply super polynomial lower bounds for the size of general arithmetic formulas one component of our proof is a new approach for homogenization and multilinearization of arithmetic formulas that gives the following results we show that for any n variate homogeneous polynomial f of degree r if there exists a fanin 2 formula of size s and depth d for f then there exists a homogeneous formula of size o d r 1 r s for f in particular for any r o log n if there exists a polynomial size formula for f then there exists a polynomial size homogeneous formula for f this refutes a conjecture of nisan and wigderson 1996 and shows that super polynomial lower bounds for homogeneous formulas for polynomials of small degree imply super polynomial lower bounds for general formulas we show that for any n variate set multilinear polynomial f of degree r if there exists a fanin 2 formula of size s and depth d for f then there exists a set multilinear formula of size o d 2 r s for f in particular for any r o log n log log n if there exists a polynomial size formula for f then there exists a polynomial size set multilinear formula for f this shows that super polynomial lower bounds for set multilinear formulas for polynomials of small degree imply super polynomial lower bounds for general formulas

process planning in a fuzzy environment abstract mixed integer optimization models for chemical process planning typically assume that model parameters can be accurately predicted as precise forecasts are difficult to obtain process planning usually involves uncertainty and ambiguity in the data this paper presents an application of fuzzy programming to process planning the forecast parameters are assumed to be fuzzy with a linear or triangular membership function the process planning problem is then formulated in terms of decision making in a fuzzy environment with fuzzy constraints and fuzzy net present value goals the model is transformed to a deterministic mixed integer linear program or mixed integer nonlinear program depending on the type of uncertainty involved in the problem for the nonlinear case a global optimization algorithm is developed for its solution this algorithm is applicable to general possibilistic programs and can be used as an alternative to the commonly used bisection method illustrative examples and computational results for a petrochemical complex with 38 processes and 24 products illustrate the applicability of the developed models and algorithms

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

change of order for regular chains in positive dimension we discuss changing the variable order for a regular chain in positive dimension this quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations we propose a modular method reducing the problem to computations in dimension zero and one the problems raised by the choice of the specialization points and the lack of the crucial information of what are the free and algebraic variables for the new order are discussed strong but not unusual hypotheses for the initial regular chain are required the main required subroutines are change of order in dimension zero and a formal newton iteration

nearly optimal computations with structured matrices we estimate the boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices we study four basic and most popular classes that is toeplitz hankel cauchy and vandermonde matrices for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation the boolean cost estimates for the latter problems have been obtained by kirrinnis in 10 except for rational interpolation we supply them now as well as the boolean complexity estimates for the important problems of multiplication of transposed vandermonde matrix and its inverse by a vector all known boolean cost estimates from 10 for such problems rely on using kronecker product this implies the d fold precision increase for the d th degree output but we avoid such an increase by relying on distinct techniques based on employing fft furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions this also enables further extensions of our estimates to cover trummer s important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes as well as the transposed vandermonde matrices it is known that the solution of toeplitz hankel cauchy vandermonde and transposed vandermonde linear systems of equations is generally prone to numerical stability problems and numerical problems arise even for multiplication of cauchy vandermonde and transposed vandermonde matrices by a vector thus our fft based results on the boolean complexity of these important computations could be quite interesting because our estimates are reasonable even for more general classes of structured matrices showing rather moderate growth of the complexity as the input size increases

reformulation of the multiperiod milp model for capacity expansion of chemical processes the problem of selecting processes and capacity expansion policies for a chemical complex consisting of continuous chemical processes can be formulated as a multiperiod mixed integer linear programming milp problem based on a variable disaggregation technique which exploits lot sizing substructures we propose two reformulations of the conventional milp model the first one is an nlp reformulation which very quickly yields good suboptimal solutions the second is an milp reformulation for exact solutions which leads to up to an order of magnitude faster computational results for large problems due to its tighter linear programming relaxation

solving multi item capacitated lot sizing problems using variable redefinition mixed integer programming models are typically not used to solve realistic sized production scheduling problems because they require exorbitant solution times we impose a useful taxonomy on production scheduling problems and develop alternative formulations for a wide variety of problems within the taxonomy the linear programming relaxation of the new models is very effective in generating bounds we show that these bounds are equal to those that could be generated using lagrangian relaxation or column generation the linear programming bounds increase in effectiveness as the problems become larger perhaps of greatest significance is that practitioners can obtain our results using only standard off the shelf codes such as lindo or mpsx 370 we report computational experience in several computing environments hardware and software on problems with up to 200 products and 10 time periods 2000 0 1 variables