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

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

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

piecewise linear programming the compact cplp algorithm a compact algorithm is presented for solving the convex piecewise linear programming problem formulated by means of a separable convex piecewise linear objective function to be minimized and a set of linear constraints this algorithm consists of a finite sequence of cycles derived from the simplex method characteritic of linear programming and the line search characteristic of nonlinear programming both the required storage and amount of calculation are reduced with respect to the usual approach based on a linear programming formulation with an expanded tableau the tableau dimensions arem n 1 wherem is the number of constraints andn the number of the original structural variables and they do not increase with the number of breakpoints of the piecewise linear terms constituting the objective function

a branch and cut algorithm for capacitated network design problems we present a branch and cut algorithm to solve capacitated network design problems given a capacitated network and point to point traffic demands the objective is to install more capacity on the edges of the network and route traffic simultaneously so that the overall cost is minimized we study a mixed integer programming formulation of the problem and identify some new facet defining inequalities these inequalities together with other known combinatorial and mixed integer rounding inequalities are used as cutting planes to choose the branching variable we use a new rule called knapsack branching we also report on our computational experience using real life data

a cost function approach to stabilization by output feedback we present a simple cost function approach to the stabilizability of linear time invariant systems by output feedback it makes use of a method of berger 1982 for obtaining the largest hypersphere containing only polynomials that are stable

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

large scale portfolio optimization this paper describes a practical algorithm for large scale mean variance portfolio optimization the emphasis is on developing an efficient computational approach applicable to the broad range of portfolio models employed by the investment community what distinguishes these from the usual quadratic program is i the form of the covariance matrix arising from the use of factor and scenario models of return and ii the inclusion of transactions limits and costs a third aspect is the question of whether the problem should be solved parametrically in the risk reward trade off parameter or separately for several discrete values of we show how the parametric algorithm can be made extremely efficient by sparsifying the covariance matrix with the introduction of a few additional variables and constraints and by treating the transaction cost schedule as an essentially nonlinear nondifferentiable function then we show how these two seemingly unrelated approaches can be combined to yield good approximate solutions when minimum trading size restrictions buy or sell at least a certain amount or not at all are added in combination these approaches make possible the parametric solution of problems on a scale not heretofore possible on computers where cpu time and storage are the constraining factors

on lipschitz optimization based on gray box piecewise linearization we address the problem of minimizing objectives from the class of piecewise differentiable functions whose nonsmoothness can be encapsulated in the absolute value function they possess local piecewise linear approximations with a discrepancy that can be bounded by a quadratic proximal term this overestimating local model is continuous but generally nonconvex it can be generated in its abs normal form by a minor extension of standard algorithmic differentiation tools here we demonstrate how the local model can be minimized by a bundle type method which benefits from the availability of additional gray box information via the abs normal form in the convex case our algorithm realizes the consistent steepest descent trajectory for which finite convergence was established earlier specifically covering counterexamples where steepest descent with exact line search famously fails the analysis of the abs normal representation and the design of the optimization algorithm are geared toward the general case whereas the convergence proof so far covers only the convex case

a decomposition based pricing procedure for large scale linear programs an application to the linear multicommodity flow problem we propose and test a new pricing procedure for solving large scale structured linear programs the procedure interactively solves a relaxed subproblem to identify potential entering basic columns the subproblem is chosen to exploit special structure rendering it easy to solve the effect of the procedure is the reduction of the number of pivots needed to solve the problem our approach is motivated by the column generation approach of dantzig wolfe decomposition we test our procedure on two sets of multicommodity flow problems one group of test problems arises in routing telecommunications traffic and the second group is a set of logistics problem which have been widely used to test multicommodity flow algorithms

the complexity of subgraph isomorphism for classes of partial k trees we present a clear demarcation between classes of bounded tree width graphs for which the subgraph isomorphism problem is np complete and those for which it can be solved in polynomial time in previous work it has been shown that this problem is solvable in polynomial time if the source graph either has bounded degree or is k connected for k the tree width of the two graphs as well it has also been shown that for certain specific connectivity or degree conditions the problem becomes np complete here we give a complete characterization of the complexity of this problem on bounded tree width graphs for all possible connectivity conditions of the two input graphs specifically we show that when the source graph is not k connected or has more than k vertices of unbounded degree the problem is np complete this answers an open question of matousek and thomas many of our reductions are restricted to using a subset of graphs of bounded tree width namely graphs of bounded path width as a direct result of our constructions we also show that when the source and target graphs have connectivity less than k or at least one has k vertices of unbounded degree the subgraph isomorphism problem for bounded path width graphs is np complete

an effective linear approximation method for separable programming problems among the numerous applications of piecewise linearization methods include data fitting network analysis logistics and statistics in the early 1950s a concave function was found to be able to be linearized by introducing 0 1 variables most textbooks in operations research offer such methods for expressing linear approximations various methods of linearization have also been developed in recent literature nevertheless the transformed linear scheme has a severe shortcoming most standard procedures for linearizing typically involve a large increase in the number of binary variables consequently the gains to be derived from dealing with linear functions are quite likely to be nullified by the increase in the size of the problem conventional methods for linearizing a concave function with m break points require m 1 binary variables however when m becomes large the computation will be very time consuming and may cause a heavy computational burden this study proposes an effective approach in which only log 2 m 1 binary variables are used the proposed method has the following features i it offers more convenient and efficient means of expressing a piecewise linear function ii fewer 0 1 variables are used iii the computational results show that the proposed method is much more efficient and faster than the conventional one especially when the number of break points becomes large

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

an interior point method for block angular optimization an interior point method for block angular optimization is developed and the convergence properties of the method are described a major motivation for such a method is that most of the computation is easily parallelized computational results are presented for a class of large scale linear programming models these models are multicommodity flow problems that arise from an air force military airlift command application and generate problems as large as 100 000 rows and 300 000 columns

approximating the longest cycle problem in sparse graphs we consider the problem of finding long paths and cycles in hamiltonian graphs the focus of our work is on sparse graphs e g cubic graphs that satisfy some property known to hold for hamiltonian graphs e g k cyclability we first consider the problem of finding long cycles in 3 connected cubic graphs whose edges have weights w i geq 0 we find cycles of weight at least sum w i a frac 1 a for a log 2 3 based on this result we develop an algorithm for finding a cycle of length at least m log 3 2 2 approx m 0 315 in 3 cyclable graphs with vertices of degree at most 3 and with m edges as a corollary of this result for arbitrary graphs with vertices of degree at most 3 that have a cycle of length l or more generally a 3 cyclable minor with degrees at most 3 and with l edges we find a cycle of length at least l log 3 2 2 we consider the graph property of 1 toughness that is common to hamiltonian graphs and 3 connected cubic graphs and we try to determine if 1 toughness implies the existence of long cycles we show that 2 connectivity and 1 toughness for constant degree graphs may give cycles that are only of logarithmic length however we exhibit a class of 3 connected 1 tough graphs with degrees up to 6 where we can find cycles of length at least m log 3 2 2

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

an implementation of linear and nonlinear multicommodity network flows abstract this work presents a new code for solving the multicommodity network flow problem with a linear or nonlinear objective function considering additional linear side constraints that link arcs of the same or different commodities for the multicommodity network flow problem through primal partitioning the code implements a specialization of murtagh and saunders strategy of dividing the set of variables into basic nonbasic and superbasic several tests are reported using random problems obtained from different network generators and real problems arising from the fields of long and short term hydrothermal scheduling of electricity generation and traffic assignment with sizes of up to 150000 variables and 45 000 constraints the performance of the code developed is compared to that of alternative methodologies for solving the same problems a general purpose linear and nonlinear constrained optimization code a specialised linear multicommodity network flow code and a primal dual interior point code

output feedback and generic stabilizability we consider questions of pole placement and stabilization for generic linear systems with prescribed state input and output dimensions where the controller must be implemented by linear memoryless output feedback we present a criterion in terms of a special pole placement property for generic stabilizability and apply this to describe constraints on the dimensions which are consistent with generic stabilizability we also discuss the rationality and solvability by radicals of stabilizing or pole positioning gains and we describe how decision algebra can theoretically handle existence questions for generic systems

the gas transmission problem solved by an extension of the simplex algorithm the problem of distributing gas through a network of pipelines is formulated as a cost minimization subject to nonlinear flow pressure relations material balances and pressure bounds the solution method is based on piecewise linear approximations of the nonlinear flow pressure relations the approximated problem is solved by an extension of the simplex method the solution method is tested on real world data and compared with alternative solution methods

a dynamic programming algorithm for the local access telecommunication network expansion problem abstract in this paper we consider the local access telecommunication network expansion problem in which growing demand can be satisfied by expanding cable capacities and or installing concentrators in the network the problem is known to be np hard we prove that the problem is weakly np hard and present a pseudo polynomial dynamic programming algorithm for the problem with time complexity o nb 2 and storage requirements o nb where n refers to the size of the network and b to an upper bound on concentrator capacity the cost structure in the network is assumed to be decomposable but may be non convex non concave and node and edge dependent otherwise this allows for incorporation of many aspects occurring in practical planning problems computational results indicate that the algorithm is very efficient and can solve medium to large scale problems to optimality within fractions of seconds to minutes

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

dual algorithms for pure network problems this paper reports the development of a new algorithmic implementation of the dual algorithm for the capacitated minimum cost network flow problem furthermore it introduces dual reoptimization procedures and compares primal and dual algorithms for the optimization and reoptimization of network problems the implementation makes use of cut sets for the efficient execution of the entering variable selection and the selection of the leaving variable is tied to the basis structure at an iteration empirical testing of the dual algorithm for optimization shows that it dominates the primal procedure for assignment problems with 400 nodes transportation problems with more than 600 nodes and transshipment problems with more than 1500 nodes computational testing with typical changes found in decomposition and relaxation procedures indicates the superiority of dual reoptimization over primal reoptimization for a sequence of parametric changes as would be typical in large scale programming techniques on the

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

a comparison of mixed integer programming models for nonconvex piecewise linear cost minimization problems we study a generic minimization problem with separable nonconvex piecewise linear costs showing that the linear programming lp relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope we also show a relationship between this result and classical lagrangian duality theory

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

a simplex algorithm for piecewise linear fractional programming problems generalizations of the well known simplex method for linear programming are available to solve the piecewise linear programming problem and the linear fractional programming problem in this paper we consider a further generalization of the simplex method to solve piecewise linear fractional programming problems unifying the simplex method for linear programs piecewise linear programs and the linear fractional programs computational results are presented to obtain further insights into the behavior of the algorithm on random test problems

nonconvex piecewise linear knapsack problems this paper considers the minimization version of a class of nonconvex knapsack problems with piecewise linear cost structure the items to be included in the knapsack have a divisible quantity and a cost function an item can be included partially in the given quantity range and the cost is a nonconvex piecewise linear function of quantity given a demand the optimization problem is to choose an optimal quantity for each item such that the demand is satisfied and the total cost is minimized this problem and its close variants are encountered in manufacturing planning supply chain design volume discount procurement auctions and many other contemporary applications two separate mixed integer linear programming formulations of this problem are proposed and are compared with existing formulations motivated by different scenarios in which the problem is useful the following algorithms are developed 1 a fast polynomial time near optimal heuristic using convex envelopes 2 exact pseudo polynomial time dynamic programming algorithms 3 a 2 approximation algorithm and 4 a fully polynomial time approximation scheme a comprehensive test suite is developed to generate representative problem instances with different characteristics extensive computational experiments show that the proposed formulations and algorithms are faster than the existing techniques

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

automatic decrease of the penalty parameter in exact penalty function methods this paper presents an analysis of the involvement of the penalty parameter in exact penalty function methods that yields modifications to the standard outer loop which decreases the penalty parameter typically dividing it by a constant the procedure presented is based on the simple idea of making explicit the dependence of the penalty function upon the penalty parameter and is illustrated on a linear programming problem with the l1 exact penalty function and an active set approach the procedure decreases the penalty parameter when needed to the maximal value allowing the inner minimization algorithm to leave the current iterate it moreover avoids unnecessary calculations in the iteration following the step in which the penalty parameter is decreased we report on preliminary computational results which show that this method can require fewer iterations than the standard way to update the penalty parameter this approach permits a better understanding of the performance of exact penalty methods

an efficient method for determining economical configurations of elementary packet switched networks the packet switched network design problem can be formulated as a capacity and flow assignment cfa problem the cfa problem is investigated for an elementary network consisting of one tandem switch and n local switches it is regarded as the structural unit of a hierarchical network it is assumed that any line speed is available and the cost of each line is a linear function of its speed with a fixed charge for installation this cfa problem is shown to be equivalent to a 0 1 integer programming problem with a discontinuous cost function a threshold rule and a row wise or column wise improvement rci iteration are proposed to solve the problem the threshold rule assigns all the traffic between two local switches to a direct route if the required traffic exceeds a predetermined threshold value and otherwise to a tandem route the rci iteration searches the vertices of the unit cube of 2n dimensional euclidean space by a procedure roughly like the simplex method whenever the network has external traffic direct application of the threshold rule ensures a global optimum when there is no external traffic a simple modification of the rci iteration yields almost a global optimum within 2n steps

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

optimal expansion of capacitated transshipment networks abstract in this paper we address the problem of allocating a given budget to increase the capacities of arcs in a transshipment network to minimize the cost of flow in the network the capacity expansion costs of arcs are assumed to be piecewise linear convex functions we use properties of the optimum solution to convert this problem into a parametric network flow problem the concept of optimum basis structure is used which allows us to consider piecewise linear convex functions without introducing additional arcs the resulting algorithm yields an optimum solution of the capacity expansion problem for all budget levels less than or equal to the given budget for integer data the algorithm performs almost all computations in integers detailed computational results are also presented

solving the staircase cost facility location problem with decomposition and piecewise linearization abstract facility location problems with staircase costs i e fixed costs that appear at several levels of production are large structured mixed integer programming problems which often are quite hard to solve in this paper solution methods based on convex piecewise linearization and benders decomposition are investigated using convex piecewise linearization only the integer variables that are needed to improve the approximation are introduced and computational tests show that in average only a few are needed benders decomposition can be used to solve the resulting problems and we show how to recalculate benders cuts when new variables are introduced so they still can be used when the approximation is improved

a polynomial time descent method for separable convex optimization problems with linear constraints we propose a polynomial algorithm for a separable convex optimization problem with linear constraints we do not make any additional assumptions about the structure of the objective function except for polynomial computability that is the objective function can be non differentiable the running time of our algorithm is polynomial in the size of the input consisting of an instance of the problem and the accuracy with which the problem is to be solved our algorithm uses an oracle for solving auxiliary systems of linear inequalities this oracle can be any polynomial algorithm for linear programming

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

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

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

algorithms for network piecewise linear programs a comparative study abstract piecewise linear programming plp is an important area of mathematical programming and concerns the minimisation of a convex separable piecewise linear objective function subject to linear constraints in this paper a subarea of plp called network piecewise linear programming nplp is explored the paper presents four specialised algorithms for nplp strongly feasible primal simplex dual method out of kilter and strongly polynomial cost scaling and their relative efficiency is studied a statistically designed experiment is used to perform a computational comparison of the algorithms the response variable observed in the experiment is the cpu time to solve randomly generated network piecewise linear problems classified according to problem class transportation transshipment and circulation problem size extent of capacitation and number of breakpoints per arc results and conclusions on performance of the algorithms are reported

approximating the maximum clique minor and some subgraph homeomorphism problems we consider the minor and homeomorphic analogues of the maximum clique problem i e the problems of determining the largest h such that the input graph on n vertices has a minor isomorphic to k h or a subgraph homeomorphic to k h respectively as well as the problem of finding the corresponding subgraphs we term them as the maximum clique minor problem and the maximum homeomorphic clique problem respectively we observe that a known result of kostochka and thomason supplies an o n bound on the approximation factor for the maximum clique minor problem achievable in polynomial time we also provide an independent proof of nearly the same approximation factor with explicit polynomial time estimation by exploiting the minor separator theorem of plotkin et al next we show that another known result of bollobas and thomason and of komlos and szemeredi provides an o n bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time on the other hand we show an w n 1 2 o 1 l o g n c lower bound for some constant c unless email protected zptime 2 l o g n o 1 on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem nearly matching our upper bound finally we derive an interesting trade off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth

piecewise linear multicriteria programs the continuous case and its discontinuous generalization in this paper we study piecewise linear multicriteria programs that is multicriteria programs with either a continuous or discontinuous piecewise linear objective function and a polyhedron set constraint we obtain an algebraic representation of a semi closed polyhedron and apply it to show that the image of a semi closed polyhedron under a continuous linear function is always one semi closed polyhedron we establish that the weak pareto solution point set of a piecewise linear multicriteria program is the union of finitely many semi closed polyhedra we propose an algorithm for finding the pareto point set of a continuous piecewise linear bi criteria program and generalize it to the discontinuous case we apply our algorithm to solve the discontinuous bi criteria portfolio selection problem with an l risk measure and transaction costs and show that this algorithm can be improved by using an ideal point strategy

degeneracy in the presence of roundoff errors abstract a technique is described for resolving degeneracy in the simplex method for linear programming it is shown that this technique enables a guarantee of termination to be given not only for exact arithmetic but also for inexact arithmetic when roundoff errors are present the method is recursive and each level of recursion is obtained by localizing and dualizing the problem at a lower level the existence of a cost function at any level is exploited to provide the guarantee a number of interesting theoretical properties of the method are given data structures are described which enable the method to be implemented with very little overhead beyond what is normally required for the simplex method there is no difficulty in using any of the currently popular techniques for representing and updating matrix factors the technique is extended to handle degeneracy in l1 linear programming problems including linear l1 approximation and it is shown that all the above properties can be preserved some comparative numerical experiments are described

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

a geometric approach to stabilization by output feedback in this paper we present a geometric approach to the stabilizability of linear systems by output feedback the geometric approach is completely analysed for both discrete time and continuous time single input single output systems two examples are worked out in detail to illustrate the theoretical development we also show how our approach can be extended to dyadic systems the simplicity of the geometric approach in the special case when the number of inputs or outputs is one less than the order of the system is illustrated with the help of examples present address bell laboratories south plainfield new jersey 07080 u s a

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

variable disaggregation in network flow problems with piecewise linear costs we study mixed integer programming formulations based upon variable disaggregation for generic multicommodity network flow problems with nonconvex piecewise linear costs a problem class that arises frequently in many application domains in telecommunications transportation and logistics we present several structural results for these formulations and we analyze the results of extensive experiments on a large set of instances with various characteristics in particular we show that the linear programming relaxation of an extended disaggregated model approximates the objective function by its lower convex envelope in the space of commodity flows together the theoretical and computational results allow us to suggest which formulation might be the most appropriate depending on the characteristics of the problem instances

a practical anti cycling procedure for linearly constrained optimization a procedure is described for preventing cycling in active set methods for linearly constrained optimization including the simplex method the key ideas are a limited acceptance of infeasibilities in all variables and maintenance of a working feasibility tolerance that increases over a long sequence of iterations the additional work per iteration is nominal and stalling cannot occur with exact arithmetic the method appears to be reliable based on computational results for the first 53 linear programming problems in thenetlib set

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

a milp formulation for generalized geometric programming using piecewise linear approximations abstract generalized geometric programming ggp problems are converted to mixed integer linear programming milp problems using piecewise linear approximations our approach is to approximate a multiple term log sum function of the form log x 1 x 2 x n in terms of a set of linear equalities or inequalities of log x 1 log x 2 and log x n where x 1 x n are strictly positive the advantage of this approach is its simplicity and readiness to implement and solve using commercial milp solvers while milp problems in general are no easier than ggp problems this approach is justified by the phenomenal progress of computing power of both personal computers and commercial milp solvers the limitation of this approach is discussed along with numerical tests

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

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

a minimax portfolio selection rule with linear programming solution a new principle for choosing portfolios based on historical returns data is introduced the optimal portfolio based on this principle is the solution to a simple linear programming problem this principle uses minimum return rather than variance as a measure of risk in particular the portfolio is chosen that minimizes the maximum loss over all past observation periods for a given level of return this objective function avoids the logical problems of a quadratic nonmonotone utility function implied by mean variance portfolio selection rules the resulting minimax portfolios are diversified for normal returns data the portfolios are nearly equivalent to those chosen by a mean variance rule framing the portfolio selection process as a linear optimization problem also makes it feasible to constrain certain decision variables to be integer or 0 1 valued this feature facilitates the use of more complex decision making models including models with fixed transaction charges and models with boolean type constraints on allocations

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

finding minimum cost circulations by successive approximation we develop a new approach to solving minimum cost circulation problems our approach combines methods for solving the maximum flow problem with successive approximation techniques based on cost scaling we measure the accuracy of a solution by the amount that the complementary slackness conditions are violated we propose a simple minimum cost circulation algorithm one version of which runs in on3lognc time on an n vertex network with integer arc costs of absolute value at most c by incorporating sophisticated data structures into the algorithm we obtain a time bound of onm logn2 mlognc on a network with m arcs a slightly different use of our approach shows that a minimum cost circulation can be computed by solving a sequence of on lognc blocking flow problems a corollary of this result is an on2log nlognc time m processor parallel minimum cost circulation algorithm our approach also yields strongly polynomial minimum cost circulation algorithms our results provide evidence that the minimum cost circulation problem is not much harder than the maximum flow problem we believe that a suitable implementation of our method will perform extremely well in practice

computational aspects of two segment separable programming recursive separable programming algorithms based on local two segment approximations are described for the solution of separable convex programs details are also given for the computation of lower bounds on the optimal value by both a primal and a dual approach and these approaches are compared computational comparisons of the methods are provided for a variety of test problems including a water supply application with more than 600 constraints and more than 900 variables and an econometric modelling problem with more than 200 variables

a lagrangean heuristic for the facility location problem with staircase costs in this paper we develop and investigate heuristic solution methods for the capacitated facility location problem with staircase shaped production cost functions these problems are linear structured mixed integer programming problems with a large proportion of integer variables lagrangean relaxation and subgradient optimization is used to obtain a dual solution and as a base for an efficient primal heuristic in the lagrangean scheme we use a convex piecewise linearization of the staircase shaped cost functions to get a good initial upper bound as well as an initial dual solution based on the subproblem solution we construct a transportation problem which yields primal feasible solutions for comparison we have generalized the well known add heuristic for the capacitated plant location model to the staircase cost facility location problem to improve this heuristic we have combined it with certain priority rules computational results are given indicating that the lagrangean heuristic is quite promising

a simple sufficient condition for the stability of linear discrete time systems the condition is found by finding the largest hypersphere which is interior to the stability boundaries in parameter space

weak sharp minima for piecewise linear multiobjective optimization in normed spaces abstract in a general normed space we consider a piecewise linear multiobjective optimization problem we prove that a cone convex piecewise linear multiobjective optimization problem always has a global weak sharp minimum property by a counter example we show that the weak sharp minimum property does not necessarily hold if the cone convexity assumption is dropped moreover under the assumption that the ordering cone is polyhedral we prove that a not necessarily cone convex piecewise linear multiobjective optimization problem always has a bounded weak sharp minimum property

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 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

on solving the sum of ratios problem this paper addresses the development of efficient global search methods for fractional programming problems such problems are in general nonconvex with numerous local extremums and belong to a class of global optimization problems first we reduce a rather general fractional programming problem with d c functions to solving an equation with a vector parameter that satisfies some nonnegativity assumption this theorem allows the justified use of the generalized dinkelbachs approach for solving fractional programming problems with a d c goal function based on solving of some d c minimization problem we developed a global search algorithm for fractional programming problems which was tested on a set of low dimensional test problems taken from the literature as well as on randomly generated problems with up to 200 variables or 200 terms in the sum

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

optimal objective function approximation for separable convex quadratic programming we present an optimal piecewise linear approximation method for the objective function of separable convex quadratic programs the method provides guidelines on how many grid points to use and how to position them for a piecewise linear approximation if the error induced by the approximation is to be bounded a priori

metric subregularity of piecewise linear multifunctions and applications to piecewise linear multiobjective optimization in this paper we consider the metric subregularity property for a piecewise linear multifunction with respect to a piecewise linear constraint as well as the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem of these properties we pay special attention to the global ones we first provide a result on a certain relationship between two nonnegative piecewise linear numerical functions for which the kernel of one of them is contained in the kernel of the other using this result we establish the bounded global metric subregularity results for a piecewise linear multifunction with respect to a piecewise linear set as applications we study the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem

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

finding a path of superlogarithmic length we consider the problem of finding a long simple path in an undirected graph we present a polynomial time algorithm that finds a path of length omega bigl log l log log l 2 bigr where l denotes the length of the longest simple path in the graph this establishes the performance ratio o n log log n log n 2 for the longest path problem where n denotes the number of vertices in the graph

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