discontinuous piecewise linear optimization a theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known a penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions it is straightforwardly extendable using standard nonlinear programming techniques tononlinear discontinuous piecewise differentiable functions the descent algorithm which is elaborated uses active set and projected gradient approaches it is a generalization of the ideas used by conn to deal with nonsmoothness in thel1 exact penalty function and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part the constrained case is reduced to the unconstrained minimization of a piecewise linear l1 exact penalty function we also discuss how the algorithm is modified when it encounters degenerate points preliminary numerical results are presented the algorithm is applied to discontinuous optimization problems from models in industrial engineering 1998 the mathematical programming society inc published by elsevier science b v

markov chain of distances between parked cars we describe the distribution of distances between parked cars as a solution of certain markov processes and show that its solution is obtained with the help of a distributional fixed point equation under certain conditions the process is solved explicitly the resulting probability density is compared with the actual parking data measured in the city

a network penalty method we consider the minimum cost network flow problem and describe how the non linear penalty function methods of conn and bartels can be specialized to a combinatorial algorithm for this problem we report on preliminary computational results which show that this method can require fewer pivots than the simplex method while the amount of work required for each pivot is comparable the algorithm can be proven finite using a modification of cunningham s strongly feasible basis pivoting rule

a branch and bound method for the fixed charge transportation problem in this paper we develop a new conditional penalty for the fixed charge transportation problem this penalty is stronger than both the driebeek penalties and the lagrangean penalties of cabot and erenguc computational testing shows that the use of these penalties leads to significant reductions in enumeration and solution times for difficult problems in the size range tested we also study the effect of problem parameters on the difficulty of the problem the ratio of fixed charges to variable costs the shape of the problem arc density in the underlying network and fixed charge arc density are shown to have a significant effect on problem difficulty for problems involving up to 40 origins and 40 destinations

an efficient newton barrier method for minimizing a sum of euclidean norms a new newton barrier method is proposed for minimizing a sum of euclidean norms msn f x sum i 1 k a i t x b i 2 msn is a nonsmooth problem because f is not differentiable at any point x where any of the norms is zero the method used is based on approximating f with a smooth function which in the limit has the same optimal value as f msn is shown to have a dual problem with properties very similar to duality theory in linear programming this is used in the development of the method and to give a proof of when an optimal solution for the smooth approximation is epsilon optimal measured in the duality gap for the original problem an implementation of the algorithm is described for large sparse problems and numerical results are presented for problems with more than 270 000 nonlinear variables these problems arise from plastic collapse analysis

genetic algorithms for modelling and optimisation genetic algorithms gas are a heuristic search and optimisation technique inspired by natural evolution they have been successfully applied to a wide range of real world problems of significant complexity this paper is intended as an introduction to gas aimed at immunologists and mathematicians interested in immunology we describe how to construct a ga and the main strands of ga theory before speculatively identifying possible applications of gas to the study of immunology an illustrative example of using a ga for a medical optimal control problem is provided the paper also includes a brief account of the related area of artificial immune systems

embedding formulations and complexity for unions of polyhedra it is well known that selecting a good mixed integer programming mip formulation is crucial for effectively obtaining a solution with state of the art solvers although best practices and guidelines for constructing good formulations abound there is rarely a single systematic construction approach that leads to the best possible formulation here we introduce embedding formulations and complexity as a new mip formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size more specifically they yield the smallest possible ideal formulation i e one whose lp relaxation has integral extreme points among all formulations that only use 0 1 auxiliary variables we use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables we also show that the resultant formulations can provide a significant computational advantage over all known formulations for pi

a diagrammatic approach to hecke algebras of the reflection equation hecke algebra deformations of the wreath product cd sn arise in solutions to the yang baxter equations with boundary the reflection equation we use a simple diagrammatic approach to construct bases for these algebras we hence introduce quotient algebras suitable for constructing physical solutions to the reflection equation with a well defined thermodynamic large n limit for all d and n we determine the generic representation theory of these algebras developing a formalism suitable for the analysis of corresponding q spin chain hamiltonian and transfer matrix spectra

coal a new heuristic approach for solving the fixed charge problem computational results abstract computational results are presented for a new heuristic algorithm for solving large general fixed charge problems based on lagrangian relaxation and cost allocation heuristics these results are contrasted with results using adjacent extreme point heuristics and branch and bound mixed integer programming general facility location and capacitated warehouse location problems as well as production planning problems with fixed and variable work force considerations are used the many test problems evaluated include problems with a large number of fixed charge and continuous variables the new algorithm demonstrates superiority over alternative methods for solving these problems

the teacher assignment problem a special case of the fixed charge transportation problem a basic model of the task of assigning classes to professors in such a way that the average number of distinct subjects assigned to each professor is minimized is formulated as a mixed integer program it turns out that the problem is a special case of the fixed charge transportation problem which in some cases corresponds to finding a basic solution of a transportation problem which is as degenerate as possible we present an equivalent alternative formulation of the problem which makes it easy to prove that it is np hard in the strong sense and an exact branch and bound algorithm for its solution based on this alternative formulation is outlined computational experiments with data from a concrete problem concludes the paper

a simplex algorithm for piecewise linear programming i derivation and proof the simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise linear objective subject to linear constraints this three part paper develops and analyzes a general computationally practical simplex algorithm for piecewiselinear programming

on the set covering polytope i all the facets with coefficients in 0 1 2 while the set packing polytope through its connection with vertex packing has lent itself to fruitful investigations little is known about the set covering polytope we characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0 1 or 2 and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining we show that all inequalities in the above class are contained in the elementary closure of the constraint set and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure we point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix finally we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation

quantification of the level of crowdedness for pedestrian movements abstract within the realm of pedestrian research numerous measures have been proposed to estimate the level of crowdedness experienced by pedestrians however within the field of pedestrian traffic flow modelling there does not seem to be consensus on the question which of these measures performs best this paper shows that the shape and scatter within the resulting fundamental diagrams differs a lot depending on the measure of crowdedness used the main aim of the paper is to establish the advantages and disadvantages of the currently existing measures to quantify crowdedness in order to evaluate which measures provide both accurate and consistent results the assessment is not only based on the theoretical differences but also on the qualitative and quantitative differences between the resulting fundamental diagrams computed using the crowdedness measures on one and the same data set the qualitative and quantitative functioning of the classical grid based measure is compared to with the x t measure an exponentially weighted distance measure and a voronoi diagram measure the consistency of relating these measures for crowdedness to the two macroscopic flow variables velocity and flow the computational efficiency and the amount of scatter present within the fundamental diagrams produced by the implementation of the different measures are reviewed it is found that the voronoi diagram and x t measure are the most efficient and consistent measures for crowdedness

a tabu search heuristic procedure for the fixed charge transportation problem abstract we develop a tabu search approach for the fixed charge transportation fct problem using recency based and frequency based memories together with two strategies for each of the intermediate and long term memory processes making use of a network based implementation of the simplex method as the local search method our approach is evaluated computationally on randomly generated problems of different sizes and of different ranges of magnitude of fixed costs relative to variable costs comparisons are made with two leading methods previously proposed one in the category of exact methods and one in the category of heuristic methods our tabu search procedure obtains optimal or near optimal solutions more than a thousand times faster than the exact solution algorithm for simple problems and thoroughly dominates the exact algorithm on all dimensions for more complex problems compared to the heuristic approach on very small and easy test problems the tabu search procedure required about the same amount of solution time and found solutions at least as good however for larger problems and for problems with higher fixed relative to variable costs the tabu search procedure was 3 4 times faster than the competing heuristic and found significantly better solutions in all cases

a projected newton method for l p norm location problems this paper is concerned with the numerical solution of continuous minisum multifacility location problems involving thel p norm where 1 p x this class of problems is potentially difficult to solve because the objective function is not everywhere diflerentiable after developing conditions that characterize the minimum of the problems under consideration a second order algorithm is presented this algorithm is based on the solution of a finite sequence of linearly constrained subproblems descent directions for these subproblems are obtained by projecting the newton direction onto the corresponding constraint manifold univariate minimization is achieved via a specialized linesearch which recognizes the possibility of first derivative discontinuity and second derivative unboundedness at points along the search direction the algorithm motivated by earlier works of calamai and conn and related to methods recently described by overton and dax is shown to possess both global and quadratic convergence properties degeneracy can complicate the numerical solution of the subproblems this degeneracy is identified and a method for handling it is outlined an implementation of the algorithm that exploits the intrinsic structure of the location problem formulation is then described along with a discussion of numerical results

combined lp and quasi newton methods for minimax optimization we present an algorithm for minimax optimization that combines lp methods and quasi newton methods the quasi newton algorithm is used only if an irregular solution is detected in which case second order derivative information is needed in order to obtain a fast final rate of convergence we prove that the algorithm can converge only to a stationary point and that normally the final rate of convergence will be either quadratic or superlinear the performance is illustrated through some numerical examples

the fixed charge transportation problem an exact algorithm based on a new integer programming formulation the fixed charge transportation problem generalizes the well known transportation problem where the cost of sending goods from a source to a sink is composed of a fixed cost and a continuous cost proportional to the amount of goods sent in this paper we describe a new integer programming formulation with exponentially many variables corresponding to all possible flow patterns to sinks we show that the linear relaxation of the new formulation is tighter than that of the standard mixed integer programming formulation we describe different classes of valid inequalities for the new formulation and a column generation method to compute a valid lower bound embedded into an exact branch and price algorithm computational results on test problems from the literature show that the new algorithm outperforms the state of the art exact methods from the literature and can solve instances with up to 70 sources and 70 sinks data as supplemental material are available at http dx doi org 10 1287 mnsc 2014 1947 this paper was accepted by dimitris bertsimas optimization

improved penalties for fixed cost linear programs using lagrangean relaxation the most commonly used penalty in branch and bound approaches to integer programming is the driebeek tomlin penalty it has been used successfully in solving fixed cost linear programs by kennington and unger and by barr glover and klingman it is well known that the driebeek tomlin penalty can be derived from a lagrangean relaxation of the integer programming problem we show however that the lagrangean relaxation for fixed cost problems not only yields the driebeek tomlin penalty but two penalties which sometimes dominate it we show the strength of the new penalties by solving a series of text problems and comparing the number of nodes generated on the branch and bound tree and the total computer time needed to solve each problem

methods for measuring pedestrian density flow speed and direction with minimal scatter abstract the progress of image processing during recent years allows the measurement of pedestrian characteristics on a microscopic scale with low costs however density and flow are concepts of fluid mechanics defined for the limit of infinitely many particles standard methods of measuring these quantities locally e g counting heads within a rectangle suffer from large data scatter the remedy of averaging over large spaces or long times reduces the possible resolution and inhibits the gain obtained by the new technologies in this contribution we introduce a concept for measuring microscopic characteristics on the basis of pedestrian trajectories assigning a personal space to every pedestrian via a voronoi diagram reduces the density scatter similarly calculating direction and speed from position differences between times with identical phases of movement gives low scatter sequences for speed and direction finally we discuss the methods to obtain reliable values for derived quantities and new possibilities of an in depth analysis of experiments the resolution obtained indicates the limits of stationary state theory

on quantum spin chain spectra and the structure of hecke algebras and q groups at roots of unity we combine statistical mechanical and algebraic lie theory techniques to investigate the ordinary representation theory of the quotients of the hecke algebras when q is a root of unity we show how to determine the main invariants of these algebras the standard module contents of the indecomposable projective modules in general n we give complete explicit results in the cases n 2 3 these results are used to determine the q variation energy level convergences in invariant quantum spin chains

a structure exploiting algorithm for nonlinear minimax problems in this paper some basic concepts are generalised which characterise the best linear chebyshev approximation in one variable to general nonlinear minimax problems a new method for solving a nonlinear minimax problem is presented which exploits the structure and characterisation of the solution whenever possible the algorithm is globally convergent with a superlinear convergence rate numerical results indicate the efficacy of the new method

a genetic algorithm for the two stage supply chain distribution problem associated with a fixed charge this paper considers a two stage distribution problem of a supply chain that is associated with a fixed charge two kinds of cost are involved in this problem a continuous cost that linearly increases with the amount transported between a source and a destination and secondly a fixed charge that incurs whenever there exists a transportation of a non zero quantity between a source and a destination the objective criterion is the minimisation of the total cost of distribution a genetic algorithm ga that belongs to evolutionary search heuristics is proposed and illustrated the proposed methodology is evaluated for its solution quality by comparing it with the approximate and lower bound solutions thus the comparison reveals that the ga generates better solution than the approximation method and is capable of providing solution either equal or closer to the lower bound solution of the problem

the lagrangian coordinate system and what it means for two dimensional crowd flow models abstract a continuum crowd flow model is solved using the lagrangian coordinate system the system has proven to give computational advantages over the traditional eulerian coordinate system for one dimensional road traffic flow our extension of the model and simulation method to two dimensional crowd flow paves the way to explore the advantages for crowd flow simulation detailed analysis of the advantages is left for future research however this paper provides a first exploration and shows that a model and simulation method for two dimensional crowd flow can be developed using lagrangian numerical techniques and that it leads to accurate simulation results

a new method for optimal truss topology design truss topology optimization formulated in terms of displacements and bar volumes results in a large nonconvex optimization problem for the case of maximization of stiffness for a prescribed volume this paper presents a new equivalent an unconstrained and convex minimization problem in displacements only where the function to be minimized is the sum of terms each of which is the maximum of two convex quadratic functions existence of solutions is proved as is the convergence of a nonsmooth steepest descent type algorithm for solving the topology optimization problem the algorithm is computationally attractive and has been tested on a large number of examples some of which are presented

solving zero one mixed integer programming problems using tabu search we describe a tabu search ts approach for solving general zero one mixed integer programming mip problems that exploits the extreme point property of zero one solutions specialized choice rules and aspiration criteria are identified for the problems expressed as functions of integer infeasibility measures and objective function values the first level ts mechanisms are then extended with advanced level strategies and learning we also look at probabilistic measures in this framework and examine how the learning tool target analysis ta can be applied to identify better control structures and decision rules computational results are reported on a portfolio of multiconstraint knapsack problems our approach is designed to solve thoroughly general 0 1 mip problems and thus contains no problem domain specific knowledge yet it obtains solutions for the multiconstraint knapsack problem whose quality rivals and in some cases surpasses the best solutions obtained by special purpose methods that have been created to exploit the special structure of these problems

revised modified penalties for fixed charge transportation problems conditional penalties are used to obtain lower bounds to subproblems in a branch and bound procedure that can be tighter than the lp relaxation of the subproblems for the fixed charge transportation problem fctp branch and bound algorithms have been implemented using conditional penalties proposed by driebeek driebeek n 1966 an algorithm for the solution of mixed integer programming problems management sci 12 576 587 cabot and erenguc cabot a v s s erenguc 1984 some branch and bound procedures for fixed cost transportation problems naval res logistics31 145 154 and palekar et al palekar v s m h karwan s zionts 1990 a branch and bound method for the fixed charge transportation problem management sci 36 1092 1105 the last conditional penalties are referred to as the modified penalties in this paper we show that the modified penalties are not valid conditional penalties in fact in nearly a quarter of the test problems examined the modified penalties prevented the branch and bound algorithm from properly identifying the optimal solution to the fctp a simple change which corrects a subcase in the penalty calculation restores the validity of the modified penalties while retaining their efficiency computational tests indicate that the revised modified penalties continue to dominate the driebeek and the cabot and erenguc penalties

a simplex algorithm for piecewise linear programming 11 finiteness feasibility and degeneracy the simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise linear objective subject to linear constraints part i of this paper has developed a general and direct simplex algorithm for piecewise linear programming under convenient assumptions that guarantee a finite number of basic solutions existence of basic feasible solutions and nondegeneracy of all such solutions part ii now shows how these assumptions can be weakened so that they pose no obstacle to effective use of the piecewise linear simplex algorithm the theory of piecewise linear programming is thereby extended and numerous features of linear programming are generalized or are seen in a new light an analysis of the algorithm s computational requirements and a survey of applications will be presented in part iii

exact algorithms for procurement problems under a total quantity discount structure in this paper we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so called total quantity discount policy this policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount moreover the discounted prices apply to all goods bought from the supplier not only to those goods exceeding the volume threshold we refer to this cost minimization problem as the total quantity discount tqd problem we give a mathematical formulation for this problem and argue that not only it is np hard but also that there exists no polynomial time approximation algorithm with a constant ratio unless p np apart from the basic form of the tqd problem we describe four variants in a first variant the market share that one or more suppliers can obtain is constrained another variant allows the buyer to procure more goods than strictly needed in order to reach a lower total cost we also consider a setting where the buyer needs to pay a disposal cost for the extra goods bought in a third variant the number of winning suppliers is limited both in general and per product finally we investigate a multi period variant where the buyer not only needs to decide what goods to buy from what supplier but also when to do this while considering the inventory costs we show that the tqd problem and its variants can be solved by solving a series of min cost flow problems finally we investigate the performance of three exact algorithms min cost flow based branch and bound linear programming based branch and bound and branch and cut on randomly generated instances involving 50 suppliers and 100 goods it turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time however we find that different algorithms perform best in terms of computation time for different variants

an efficient primal dual interior point method for minimizing a sum of euclidean norms the problem of minimizing a sum of euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature this nonsmooth optimization problem arises in many different kinds of modern scientific applications we derive a primal dual interior point algorithm for the problem by applying newton s method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition the main work at each step consists of solving a system of linear equations the schur complement equations this schur complement matrix is not symmetric unlike in linear programming we incorporate a mehrotra type predictor corrector scheme and present some experimental results comparing several variations of the algorithm including as one option explicit symmetrization of the schur complement with a skew corrector term we also present results obtained from a code implemented to solve large sparse problems using a symmetrized schur complement this has been applied to problems arising in plastic collapse analysis with hundreds of thousands of variables and millions of nonzeros in the constraint matrix the algorithm typically finds accurate solutions in less than 50 iterations and determines physically meaningful solutions previously unobtainable

a globally and superlinearly convergent algorithm for convex quadratic programs with simple bbounds a globally and superlinearly convergent algorithm for solving convex quadratic programs with simple bounds is presented the algorithm is developed using a new formulation of the problem the minimization of an unconstrained piecewise quadratic function that has the same optimality conditions as the original problem the major work at each iteration is the cholesky factorization of a positive definite matrix with the size and structure of the hessian of the quadratic hence the algorithm is suitable for solving large sparse problems and for implementation on parallel computers the numerical results indicate that the new approach has promise

a simplex algorithm for piecewise linear programming iii computational analysis and application the first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise linear functions subject to linear constraints this concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program the advantages are shown to be implicit in relationships between the linear and piecewise linear algorithms and to be independent of many details of implementation two sets of computational results serve to illustarate these arguments the piecewise linear simplex algorithm is observed to run 2 6 times faster than a comparable linear algorithm not including any additional expense that might be incurred in setting up the equivalent linear program further support for the practical value of a good piecewise linear programming algorithm is provided by a survey of many varied applications

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

use of penalties in a branch and bound procedure for the fixed charge transportation problem abstract the fixed charge transportation problem fctp is an extension of the classic transportation problem in which a fixed cost is incurred for every supply point that is used in the solution a branch and bound procedure which requires the use of two subproblems a classic transportation problem and a knapsack problem can be used to solve the fctp unlike fixed change problems with the fctp there is a many to one correspondence between the continuous variables of the transportation problem and the fixed charge variables of the knapsack problem this paper incorporates penalties into the branch and bound procedure in order to improve the solution times also an adjacent extreme point heuristic is presented that finds a near optimal solution quickly

discontinuous optimization by smoothing we present a simple tool for solving many discontinuous optimization problems the basic idea is to express discontinuities by means of a step function and then to approximate the step function by a smooth one this way a smooth once or twice continuously differentiable approximate problem is obtained this problem can be solved by any gradient technique the approximations introduced contain a single parameter which controls their accuracy so that the original problem is replaced only in some neighborhoods of the points of discontinuity some convergence properties are established and numerical experiments with some test problems are reported

on the solution of highly degenerate linear programs the solution of scheduling problems often gives rise to highly degenerate linear programmes which cause significant computational difficulties for the revised simplex method wolfe s highly effective ad hoc method for overcoming the cycling or stalling problems associated with degeneracy is described here it is given a geometric interpretation in terms of finding a direction of recession for a reduced problem which is fundamental to a full understanding of the procedure an example of an aircrew scheduling problem is used to illustrate the effectiveness of the method

genetic algorithms for a supply management problem mip recombination vs greedy decoder two variants of genetic algorithm ga for solving the supply management problem with lower bounded demands smpld are proposed and experimentally tested the smpld problem consists in planning the shipments from a set of suppliers to a set of customers minimizing the total cost given lower and upper bounds on shipment sizes lower bounded consumption and linear costs for opened deliveries the first variant of ga uses the standard binary representation of solutions and a new recombination operator based on the mixed integer programming mip techniques the second ga is based on the permutation representation and a greedy decoder our experiments indicate that the ga with mip recombination compares favorably to the other ga and to the mip solver cplex 9 0 in terms of cost of obtained solutions the ga based on greedy decoder is shown to be the most robust in finding feasible solutions

on quantum spin chain spectra and the representation theory of hecke algebras by augmented braid diagrams we use simple diagrammatic techniques to analyse the ordinary representation theory of the hecke algebras and to construct modules resp representations which are generically simple irreducible and well defined in every specialization of q including roots of unity we determine several physically important properties of these modules generalizing properties of the temperley lieb algebra and its diagrams which have proved useful for lattice models we show how these results can be used to locate energy level crossings in invariant quantum spin chains and locate a new crossing of the thermodynamic limit spin chain at as an example

reduction methods of type 2 uncertain variables and their applications to solid transportation problem uncertainty theory is a branch of mathematics for dealing with realistic uncertainties arising out of complexity changeability and non decidability of practical environments an uncertain variable is defined as a function from the uncertainty space to the set of real numbers and is characterized by an uncertainty distribution this paper proposes the definition of type 2 uncertain variables within the framework of uncertainty theory through introduction of generalized uncertain measures and focuses on more complex twofold uncertainties some uncertainty reduction methods associated with type 2 uncertain variables are also proposed for convenience of applicability including reduction of optimistic value pessimistic value and expected value moreover four classes of type 2 uncertain variables are reduced to type 1 uncertain variables with specific uncertainty distributions type 2 uncertain optimization methods are applied to solving the fixed charge solid transportation problem with the type 2 uncertain parameters where the solution methods are also provided for the proposed models finally numerical experiments are implemented to demonstrate application and sensitivity analysis of the proposed approaches

primal methods are better than dual methods for solving overdetermined linear systems in the l sense it is generally believed that methods based upon solving the dual of the linear programming formulation of the linear l infty problem are superior to methods based upon the primal formulation a primal approach is presented along with evidence that for random problems primal methods are to be preferred it is then shown that the method generally considered to be the best algorithm for discrete linear approximation in the l infty norm actually owes much of its efficiency to effective choice of a starting point this phenomenon is explained with the aid of results from classical minimax theory which are then used to determine a suitable initial point for a primal method with this enhancement it would appear that the primal approach is also preferable for problems arising from function approximations

a trust region algorithm for nonsmooth optimization a trust region algorithm is proposed for minimizing the nonsmooth composite functionf x h f x wheref is smooth andh is convex the algorithm employs a smoothing function which is closely related to fletcher s exact differentiable penalty functions global and local convergence results are given considering convergence to a strongly unique minimizer and to a minimizer satisfying second order sufficiency conditions

exact penalty functions in constrained optimization in this paper formal definitions of exactness for penalty functions are introduced and sufficient conditions for a penalty function to be exact according to these definitions are stated thus providing a unified framework for the study of both nondifferentiable and continuously differentiable penalty functions in this framework the best known classes of exact penalty functions are analyzed and new results are established concerning the correspondence between the solutions of the constrained problem and the unconstrained minimizers of the penalty functions

exact dynamics of stochastic linear delayed systems application to spatiotemporal coordination of comoving agents delayed dynamics result from finite transmission speeds of a signal in the form of energy mass or information in stochastic systems the resulting lagged dynamics challenge our understanding due to the rich behavioral repertoire encompassing monotonic oscillatory and unstable evolution despite the vast literature quantifying this rich behavior is limited by a lack of explicit analytic studies of high dimensional stochastic delay systems here we fill this gap for systems governed by a linear langevin equation of any number of delays and spatial dimensions with additive gaussian noise by exploiting laplace transforms we are able to derive an exact time dependent analytic solution of the langevin equation by using characteristic functionals we are able to construct the full time dependence of the multivariate probability distribution of the stochastic process as a function of the delayed and nondelayed random variables as an application we consider interactions in animal collective movement that go beyond the traditional assumption of instantaneous alignment we propose models for coordinated maneuvers of comoving agents applicable to recent empirical findings in pigeons and bats whereby individuals copy the heading of their neighbors with some delay we highlight possible strategies that individual pairs may adopt to reduce the variance in their velocity difference and or in their spatial separation we also show that a minimum in the variance of the spatial separation at long times can be achieved with certain ratios of measurement to reaction delay

constant net time headway as a key mechanism behind pedestrian flow dynamics we show that keeping a constant lower limit on the net time headway is the key mechanism behind the dynamics of pedestrian streams there is a large variety in flow and speed as functions of density for empirical data of pedestrian streams obtained from studies in different countries the net time headway however stays approximately constant over all these different data sets by using this fact we demonstrate how the underlying dynamics of pedestrian crowds naturally follows from local interactions this means that there is no need to come up with an arbitrary fit function with arbitrary fit parameters as has traditionally been done further by using not only the average density values but the variance as well we show how the recently reported stop and go waves helbing et al phys rev e 75 046109 2007 emerge when local density variations take values exceeding a certain maximum global average density which makes pedestrians stop

a globally and quadratically convergent affine scaling method for linear e 1 problems recently various interior point algorithms related to the karmarkar algorithm have been developed for linear programming in this paper we first show how this interior point philosophy can be adapted to the linear l1 problem in which there are no feasibility constraints to yield a globally and linearly convergent algorithm we then show that the linear algorithm can be modified to provide aglobally and ultimatelyquadratically convergent algorithm this modified algorithm appears to be significantly more efficient in practise than a more straightforward interior point approach via a linear programming formulation we present numerical results to support this claim

regenerator location problem polyhedral study and effective branch and cut algorithms abstract in this paper we study the regenerator location problem rlp this problem arises in optical networks where an optical signal can only travel a certain maximum distance called the optical reach before its quality deteriorates needing regenerations by regenerators deployed at network nodes the rlp is to determine a minimal number of network nodes for regenerator placement such that for each node pair there exists a path of which no sub path without internal regenerators has a length greater than the optical reach starting with a set covering formulation involving an exponential number of constraints reported and studied in rahman 2012 and aneja 2012 we study the facial structure of the polytope arising from this formulation significantly extending known results making use of these polyhedral results we present a new branch and cut b c solution approach to solve the rlp to optimality we present a series of computational experiments to evaluate two versions of the proposed b c approach over 400 benchmark rlp instances we first compare them with an available exact method for the rlp in the literature because of the equivalence among the rlp the minimum connected dominating set problem mcdsp and the maximum leaf spanning tree problem mlstp we further compare our approaches with other available exact algorithms using 47 benchmark mcdsp mlstp instances

solving the fixed charge problem with lagrangian relaxation and cost allocation heuristics abstract a new heuristic algorithm is presented for solving large general fixed charge problems based on langrangian relaxation and cost allocation heuristics limited computational results indicate the new algorithm provides good solutions to large problems with reasonable computational requirements it demonstrates significant improvement over current methods for solving large general fixed charge problems

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

fixed charge transportation problem with type 2 fuzzy variables this paper considers two fixed charge transportation problems with type 2 fuzzy parameters unit transportation costs fixed costs in the first problem and unit transportation costs fixed costs supplies and demands in the second problem are type 2 fuzzy variables for the first problem to get corresponding defuzzified values of the type 2 fuzzy cost parameters first critical value cv based reduction methods are applied to reduce type 2 fuzzy variables into type 1 fuzzy variables and then centroid method is used for complete defuzzification besides this we also apply geometric defuzzification method to the type 2 fuzzy cost parameters in the first problem to provide a comparison of the results coming to the second problem a chance constrained programming model is formulated using generalized credibility measure for the objective function as well as the constraints with the cv based reductions of corresponding type 2 fuzzy parameters next the reduced model is turned into equivalent parametric programming problem the deterministic problems so obtained are then solved by using the standard optimization solver lingo we have provided numerical examples illustrating the proposed models and techniques some sensitivity analyzes for the second model are also presented

hybrid extreme point tabu search abstract we develop a new hybrid tabu search method for optimizing a continuous differentiable function over the extreme points of a polyhedron the method combines extreme point tabu search epts with traditional descent algorithms based on linear programming the tabu search algorithm utilizes both recency based and frequency based memory and oscillates between local improvement and diversification phases the hybrid algorithm iterates between using a descent algorithm to find a local minimum and tabu search to improve locally and then move to a new area of the search space the descent algorithm acts as a form of intensification within the tabu search this algorithm can be used on many important classes of problems in global optimization including bilinear programming multilinear programming multiplicative programming concave minimization and complementarity problems the algorithm is applied to two practical problems the quasistatie multi rigid body contact problem qcp in robotics and the global tree optimization gto problem in machine learning we perform computational experiments on problems of significant real world dimensions the smallest machine learning problem tested has on the order of 10485 extreme points the results show our method obtains high quality solutions both by comparison with the embedded descent algorithm and by comparison to a version of tabu search that does not make use of descent

representations and traces of the hecke algebras hn q of type an 1 the notion of the connectivity class of minimal words in the algebra hn q is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented guided by these results the connection between the ocneanu trace on hn q and schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of hn q they appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to schur functions their evaluation involves the use of the littlewood richardson rule which is proved to apply to hn q just as it does to sn both representation matrices and characters are tabulated

local properties of algorithms for minimizing nonsmooth composite functions this paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionf x f x h c x wheref andc are smooth buth c may be nonsmooth local convergence at a second order rate is established for the generalized gauss newton method whenh is convex and globally lipschitz and the minimizer is strongly unique local convergence at a second order rate is established for a generalized newton method when the minimizer satisfies nondegeneracy strict complementarity and second order sufficiency conditions assuming the minimizer satisfies these conditions necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established necessary and sufficient conditions for a two step superlinear rate of convergence are also established when only reduced curvature information is available all these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems

nonlinear programming via an exact penalty function global analysis in this paper we motivate and describe an algorithm to solve the nonlinear programming problem the method is based on an exact penalty function and possesses both global and superlinear convergence properties we establish the global qualities here the superlinear nature is proven in 7 the numerical implementation techniques are briefly discussed and preliminary numerical results are given

the multi vehicle traveling purchaser problem with pairwise incompatibility constraints and unitary demands a branch and price approach abstract in this work we study a supplier selection and routing problem where a fleet of homogeneous vehicles with a predefined capacity is available for procuring different products from different suppliers with the aim to satisfy demand at the minimum traveling and purchasing cost decisions are further complicated by the presence of pairwise incompatibility constraints among products implying the impossibility of loading two incompatible products on the same vehicle the problem is known as the multi vehicle traveling purchaser problem with pairwise incompatibility constraints we study the special case in which the demand for each product is unitary and propose a column generation approach based on a dantzig wolfe reformulation of the problem where each column represents a feasible vehicle route associated with a compatible purchasing plan to solve the pricing problem we propose a hybrid strategy exploiting the advantages of two alternative exact methods a labeling algorithm solving a resource constrained elementary shortest path problem on an expanded graph and a tailored branch and cut algorithm due to the integrality request on variables we embed the column generation in a branch and bound framework and propose different branching rules extensive tests carried out on a large set of instances show that our branch and price method performs well improving on average both in quality and in computational time solutions obtained by a state of art branch and cut approach applied to a three index connectivity constraints based formulation

the transportation problem with exclusionary side constraints and two branch and bound algorithms abstract the transportation problem with exclusionary side constraints a practical distribution and logistics problem is formulated as a 0 1 mixed integer programming model two branch and bound b b algorithms are developed and implemented in this study to solve this problem both algorithms use the driebeek penalties to strengthen the lower bounds so as to fathom some of the subproblems to peg variables and to guide the selection of separation variables one algorithm also strongly exploits the problem structure in selecting separation variables in order to find feasible solutions sooner to take advantage of the underlying network structure of the problem the algorithms employ the primal network simplex method to solve network relaxations of the problem a computational experiment was conducted to test the performance of the algorithms and to characterize the problem difficulty the commercial mixed integer programming software cplex and an existing special purpose algorithm specifically designed for this problem were used as benchmarks to measure the performance of the algorithms computational results show that the new algorithms completely dominate the existing special purpose algorithm and run from two to three orders of magnitude faster than cplex

fixed charge transportation problem facets of the projection polyhedron in this paper we consider the well known fixed charge transportation problem to send any flow from source si to destination tj we incur a unit variable shipping cost of cij and a fixed cost fij here we study the structure of the projection polyhedron of this problem in the space of 0 1 variables associated with fixed charges and we develop several classes of valid inequalities and derive conditions under which they are facet defining in some cases if the conditions are not satisfied we show how they can be lifted to define facets several heuristics for generating and adding these facets are presented using these results we develop a computationally effective algorithm for solving the problem the computational results clearly indicate the usefulness of this approach

valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems multicommodity extended formulations of single source uncapacitated fixed charge network flow problems have significantly sharper linear programming relaxations than the standard flow formulations however the tradeoff is the introduction of many new constraints and variables to accomodate a sink oriented flow disaggregation in this paper we introduce a new family of dicut collection inequalities and show that they completely describe the projection of the multicommodity formulation onto the original variables a simple subclass is seen to include a variety of known inequalities for particular models and combinatorial separation is examined for some special cases

lifted flow cover inequalities for mixed 0 1 integer programs we investigate strong inequalities for mixed 0 1 integer programs derived from flow cover inequalities flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities however because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients lifting of flow cover inequalities is computationally very demanding we present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show the effectiveness of our lifting procedures

a projection method for the uncapacitated facility location problem several algorithms already exist for solving the uncapacitated facility location problem the most efficient are based upon the solution of the strong linear programming relaxation the dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function this paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections the amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints wherem is the number of clients for comparison the underlying linear programming dual hasmn m n variables andmn n constraints wheren is the number of potential locations for the facilities the method is flexible as it can handle side constraints in particular when there is a duality gap the linear programming formulation can be strengthened by adding cuts numerical results for some classical test problems are included

a conjugate rosen s gradient projection method with global line search for piecewise linear concave optimization abstract the kelley cutting plane method is one of the methods commonly used to optimize the dual function in the lagrangian relaxation scheme usually the kelley cutting plane method uses the simplex method as the optimization engine it is well known that the simplex method leaves the current vertex follows an ascending edge and stops at the nearest vertex what would happen if one would continue the line search up to the best point instead as a possible answer we propose the face simplex method which freely explores the polyhedral surface by following the rosen s gradient projection combined with a global line search on the whole surface furthermore to avoid the zig zagging of the gradient projection we propose a conjugate gradient version of the face simplex method for our preliminary numerical tests we have implemented this method in matlab this implementation clearly outperforms basic matlab implementations of the simplex method in the case of state of the art simplex implementations in c or similar our matlab implementation is only competitive for the case of many cutting planes

a global and quadratically convergent method for linear l problems a new globally and quadratically convergent algorithm is proposed for the linear l infty problem this method works on the piecewise linear l infty problem directly by generating descent directions via a sequence of weighted least squares problems and using a piecewise linear line search to ensure a decrease in the l infty function at every step it is proven that ultimately full newton like steps are taken where the newton step is based on the complementary slackness condition holding at the solution numerical results suggest a very promising method the number of iterations required to achieve high accuracy is relatively insensitive to problem size

nonlinear programming via an exact penalty function asymptotic analysis in this paper we consider the final stage of a global method to solve the nonlinear programming problem we prove 2 step superlinear convergence in the process of analyzing this asymptotic behavior we compare our method theoretically to the popular successive quadratic programming approach

fixed charge multicommodity network design using p partition facets abstract we are given an undirected network g v e and a set of traffic demands to install a potential edge e e we incur a cost fe to provide a positive capacity ae the objective is to select edges at minimum cost so as to permit a feasible multicommodity flow of all traffic we study structure of the projection polytope of this problem in the space of binary variables associated with fixed charges by relating facets of a p node problem p 2 3 or 4 defined over a multi graph obtained by a p partition of v to the facets of the original problem inspired from the well known cover inequalities of the knapsack problem we develop the notion of p partition cover inequalities we present necessary and sufficient conditions for such inequalities to be facet defining for p 3 and 4 a simple heuristic approach for separating and adding such violated inequalities is presented and implemented for p values up to 10 we report optimal solutions for problems with 30 nodes 60 edges and fully dense demand matrices within a few minutes of cpu time for most instances some results for dense graph problems are also reported

a reduced cost iterated local search heuristic for the fixed charge transportation problem the fixed charge transportation problem fctp is a generalization of the transportation problem where an additional fixed cost is paid for sending a flow from an origin to a destination we propose an iterated local search heuristic based on the utilization of reduced costs for guiding the restart phase the reduced costs are obtained by applying a lower bounding procedure that computes a sequence of nondecreasing lower bounds by solving a three index mathematical formulation of the problem strengthened with valid inequalities the proposed method was tested on two sets of benchmark instances from the literature the first set was used to evaluate the state of the art heuristics for the problem the proposed heuristic was able to provide new best knownupper bounds on all 124 open instances on the second set of instances which was recently introduced for testing the currently best exact method for the problem the new heuristic was able to provide provably good upper bounds within short computing times