new developments in the primal dual column generation technique abstract the optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and consequently originates an unnecessarily large number of iterations of the method to overcome this drawback variations of the standard approach use interior points of the dual feasible set instead of optimal solutions in this paper we focus on a variation known as the primal dual column generation technique which uses a primal dual interior point method to obtain well centered non optimal solutions of the restricted master problems we show that the method converges to an optimal solution of the master problem even though non optimal solutions are used in the course of the procedure also computational experiments are presented using linear relaxed reformulations of three classical integer programming problems the cutting stock problem the vehicle routing problem with time windows and the capacitated lot sizing problem with setup times the numerical results indicate that the appropriate use of a primal dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times on average furthermore the results show that the larger the instance the better the relative performance of the primal dual column generation technique

an infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter global convergence in constrained optimization algorithms has traditionally been enforced by the use of parametrized penalty functions recently the filter strategy has been introduced as an alternative at least part of the motivation for using filter methods consists of avoiding the need for estimating a suitable penalty parameter which is often a delicate task in this paper we demonstrate that the use of a parametrized penalty function in nonsmooth convex optimization can be avoided without using the relatively complex filter methods we propose an approach which appears to be more direct and easier to implement in the sense that it is closer in spirit and structure to the well developed unconstrained bundle methods preliminary computational results are also reported

shallow deep and very deep cuts in the analytic center cutting plane method in this paper we show that the cut does not need to go through the query point it can be deep or shallow the primal framework leads to a simple analysis of the potential variation which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step

solving large scale linear multicommodity flow problems with an active set strategy and proximal accpm in this paper we propose to solve the linear multicommodity flow problem using a partial lagrangian relaxation the relaxation is restricted to the set of arcs that are likely to be saturated at the optimum this set is itself approximated by an active set strategy the partial lagrangian dual is solved with proximal accpm a variant of the analytic center cutting plane method the new approach makes it possible to solve huge problems when few arcs are saturated at the optimum as appears to be the case in many practical problems

large momentum properties and wilson short distance expansion in non perturbative field theory large momentum properties and wilson zimmerman short distance expansion are established via phase space analysis for the weakly coupled massive gross neveu model in dimension 2 methods are applicable more generally

a doubly stabilized bundle method for nonsmooth convex optimization we propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations most bundle algorithms use a cutting plane model of the objective function to formulate a subproblem whose solution gives the next iterate proximal bundle methods employ the model in the objective function of the subproblem while level methods put the model in the subproblem s constraints the proposed algorithm defines new iterates by solving a subproblem that employs the model in both the objective function and in the constraints one advantage when compared to the proximal approach is that the level set constraint provides a certain lagrange multiplier which is used to update the proximal parameter in a novel manner we also show that in the case of inexact function and subgradient evaluations no additional procedure needs to be performed by our variant to deal with inexactness as opposed to the proximal bundle methods that require special modifications numerical experiments on almost 1 000 instances of different types of problems are presented our experiments show that the doubly stabilized bundle method inherits useful features of the level and the proximal versions and compares favorably to both of them

a cross decomposition scheme with integrated primal dual multi cuts for two stage stochastic programming investment planning problems we describe a cross decomposition algorithm that combines benders and scenario based lagrangean decomposition for two stage stochastic programming investment planning problems with complete recourse where the first stage variables are mixed integer and the second stage variables are continuous the algorithm is a novel cross decomposition scheme and fully integrates primal and dual information in terms of primal dual multi cuts added to the benders and the lagrangean master problems for each scenario the potential benefits of the cross decomposition scheme are demonstrated with numerical experiments on a number of instances of a facility location problem under disruptions in the original formulation where the underlying lp relaxation is weak the cross decomposition method outperforms multi cut benders decomposition if the formulation is improved with the addition of tightening constraints the performance of both decomposition methods improves but cross decomposition clearly remains the best method for large scale problems

simplex and interior point specialized algorithms for solving nonoriented multicommodity flow problems multicommodity network flow models arise in a wide variety of contexts typical among which is the dimensioning of telecommunication networks in this paper we present various approaches based on specialization of the simplex algorithm and interior point methods to solve nonoriented multicommodity flowproblems algorithms are tested with data from the france telecom paris district transmission network first we focus on a specialization for the node arc formulation of the problem a primal simplex and dual affine scaling algorithms exploiting the particular structure of the constraint matrix are presented and compared numerical results are provided for problems up to about 800 000 constraints and 6 000 000 variables however much more powerful approaches based on specialized decomposition methods can be implemented for solving the problem a dantzig wolfe decomposition method is designed and compared with a specialized implementation of the analytic center cutting plane method accpm partitioning techniques are used to exploit the structure of the master programs involved in those methods

computational experience with a bundle approach for semidefinite cutting plane relaxations of max cut and equipartition we propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities our approach is based on lagrangian duality where the inequalities are dualized and only a basic set of constraints is maintained explicitly this leads to function evaluations requiring to solve a relatively simple semidefinite program our approach provides accurate solutions to semidefinite relaxations of the max cut and the equipartition problem which are not achievable by direct approaches based only on interior point methods

convex proximal bundle methods in depth a unified analysis for inexact oracles the last few years have seen the advent of a new generation of bundle methods capable to handle inexact oracles polluted by noise proving convergence of a bundle method is never simple and coping with inexact oracles substantially increases the technicalities besides several variants exist to deal with noise each one needing an ad hoc proof to show convergence we state a synthetic convergence theory in which we highlight the main arguments and specify which assumption is used to establish each intermediate result the framework is comprehensive and generalizes in various ways a number of algorithms proposed in the literature based on the ingredients of our synthetic theory we consider various bundle methods adapted to oracles for which high accuracy is possible yet it is preferable not to make exact calculations often because they are too time consuming

a log barrier method with benders decomposition for solving two stage stochastic linear programs an algorithm incorporating the logarithmic barrier into the benders decomposition technique is proposed for solving two stage stochastic programs basic properties concerning the existence and uniqueness of the solution and the underlying path are studied when applied to problems with a finite number of scenarios the algorithm is shown to converge globally and to run in polynomial time

ergodic primal convergence in dual subgradient schemes for convex programming lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large structured optimization prob lems the dual subgradient scheme does not automatically produce primal feasible solutions there is an abundance of techniques for computing such solutions via penalty functions tangential approximation schemes or the solution of auxiliary primal programs all of which require a fair amount of computational effort we consider a subgradient optimization scheme applied to a lagrangean dual formulation of a convex program and construct at minor cost an ergodic sequence of subproblem solutions which converges to the primal solution set numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained compared to those generated in the basic subgradient scheme

an interior point cutting plane method for the convex feasibility problem with second order cone inequalities the convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set we assume that the outer set is described by second order cone inequalities and propose an analytic center cutting plane technique to solve this problem we discuss primal and dual settings simultaneously two complexity results are reported the complexity of restoration procedure and complexity of the overall algorithm we prove that an approximate analytic center is updated after adding a second order cone cut socc ino 1 newton step and that the analytic center cutting plane method accpm with socc is a fully polynomial algorithm

approximate max min resource sharing for structured concave optimization we present a lagrangian decomposition algorithm which uses logarithmic potential reduction to compute an varepsilon approximate solution of the general max min resource sharing problem with m nonnegative concave constraints on a convex set b we show that this algorithm runs in o m varepsilon 2 rm ln m iterations a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy varepsilon in 0 1 in the block angular case b is the product of k convex sets blocks and each constraint function is block separable for such models an iteration of our method requires a theta varepsilon approximate solution of k independent block maximization problems which can be computed in parallel

coordination complexity of parallel price directive decomposition the general block angular convex resource sharing problem in k blocks and m nonnegative block separable coupling constraints is considered applications of this model are in combinatorial optimization network flows scheduling communication networks engineering design and finance this paper studies the coordination complexity of approximate price directive decomposition pdd for this problem i e the number of iterations required to solve the problem to a fixed relative accuracy as a function of k and m first a simple pdd method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in m in the class of all pdd methods that work with the original unrestricted blocks it is then shown that logarithmic and exponential potentials generate a polylogarithmically optimal algorithm for a wider class of pdd methods which can restrict the blocks by the coupling constraints as an application the fastest currently known deterministic approximation algorithm for minimum cost multicommodity flows is obtained

interior point methods an old and new approach to nonlinear programming in this paper we discuss the main concepts of structural optimization a field of nonlinear programming which was formed by the intensive development of modern interior point schemes

2 particle asymptotic completeness and bound states in weakly coupled quantum field theories rigorous results on poles of the 2 and 4 point functions which yield 2 particle asymptotic completeness and give information on the presence or absence of 2 particle bound states and resonances are presented for weakly coupled even and non even field theories with mass gap in space time dimensiond 2 3 and for related hypothetical theories in dimension 4 methods used are more convenient and more general than those used previously with more limited results forp 2 theories

bethe salpeter kernel and short distance expansion in the massive gross neveu model the bethe salpeter kernel is defined non perturbatively for the weakly coupled massive gross neveu model its large momentum properties are established they are used to justify subtracted bethe salpeter equations initially proposed for 44 by k symanzik and in turn to give non perturbative proofs of the wilson short distance expansion at first order and of 2 particle asymptotic completeness and related results

using central prices in the decomposition of linear programs abstract this paper deals with a decomposition technique for linear programs which proposes a new treatment of the master program in the classical dantzig wolfe algorithm this approach exploits a the relationship between the master program and the minimization of a convex piecewise linear function and b a recently proposed cutting plane algorithm for convex nondifferentiable optimization the new algorithm seeks to achieve deep cuts by generating them at or near to the analytic centre of a set of localization containing the solution therefore each iteration entails a sizeable decrease of the set of localization and the overall algorithm shows fast convergence this algorithm has been used to solve reputedly difficult convex nondifferentiable optimization problems its implementation as a decomposition algorithm for large scale structured linear program seems to be a promising alternative to the standard dantzig wolfe approach which suffers from very slow convergence in many practical problems the new approach differs from the dantzig wolfe algorithm through the fact that it does not use prices corresponding to an optimal combination of the past proposals of the subproblem but rather central prices i e analytic centres which balance them all this property seems to explain the good behaviour of the new algorithm

proximal level bundle methods for convex nondifferentiable optimization saddle point problems and variational inequalities we study proximal level methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we show that they enjoy almost optimal efficiency estimates we give extensions for solving convex constrained problems convex concave saddle point problems and variational inequalities with monotone operators we present several variants establish their efficiency estimates and discuss possible implementations in particular our methods require bounded storage in contrast to the original level methods of lemarechal nemirovskii and nesterov

nodal aggregation of resource constraints in a shortest path problem the shortest path problem with resource constraints consists of finding the minimum cost path between two specified points while respecting constraints on resource consumption its solving by a dynamic programming algorithm requires a computation time increasing with the number of resources with the aim of producing rapidly a good heuristic solution we propose to reduce the state space by aggregating resources our approach consists of projecting the resources on a vector of smaller dimension and then to dynamically adjust the projection matrix to get a better approximation of the optimal solution we propose an adjustment based on lagrangian and surrogate relaxations in a column generation framework in which the sub problems are shortest path problems with resource constraints we adjust the multipliers only one time at each column generation iteration this permit to obtain good solutions of the scheduling problem in few time

using an interior point method for the master problem in a decomposition approach abstract we address some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach the main points concem the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration the particular structure is the presence of gub constraints and the natural partitioning of the constraint matrix into blocks built of cuts generated by different subproblems the method can be used in a fairly general case i e in any decomposition approach whenever the master is solved by an interior point method in which the normal equations are used to compute orthogonal projections computational results demonstrate its advantages for one particular decomposition approach analytic center cutting plane method accpm is applied to solve large scale nonlinear multicommodity network flow problems up to 5000 arcs and 10000 commodities

comparison of bundle and classical column generation when a column generation approach is applied to decomposable mixed integer programming problems it is standard to formulate and solve the master problem as a linear program seen in the dual space this results in the algorithm known in the nonlinear programming community as the cutting plane algorithm of kelley and cheney goldstein however more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard one of them is the bundle method our aim is to illustrate its differences with kelley s method in the process we review alternative stabilization techniques used in column generation comparing them from both primal and dual points of view numerical comparisons are presented for five applications cutting stock which includes bin packing vertex coloring capacitated vehicle routing multi item lot sizing and traveling salesman we also give a sketchy comparison with the volume algorithm

a nonlinear analytic center cutting plane method for a class of convex programming problems a cutting plane algorithm for minimizing a convex function subject to constraints defined by a separation oracle is presented the algorithm is based on approximate analytic centers the nonlinearity of the objective function is taken into account yet the feasible region is approximated by a containing polytope this containing polytope is regularly updated by adding a new cut through a test point each test point is an approximate analytic center of the intersection of a containing polytope and a level set of the nonlinear objective function we establish the complexity of the algorithm our complexity estimate is given in terms of the problem dimension the desired accuracy of an approximate solution and other parameters that depend on the geometry of a specific instance of the problem

asymptotic completeness and multiparticle structure in field theories ii theories with renormalization the gross neveu model the ideas developed in part i ref 1 are applied to the recently constructed massive gross neveu model we define in this case an irreducible kernel satisfying a regularized bethe salpeter equation which is convenient to derive asymptotic completeness in the 2 particle region as in part i the method ailows direct graphical definition of general irreducible kernels and is well suited to the analysis of asymptotic completeness and related results in more general energy regions a large part of the paper is devoted to a new self contained construction via phase space expansion of the gross neveu model the presentation is somewhat simpler than previous ones is more complete on some points and is best suited to our purposes

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

solving combinatorial optimization problems using karmakar s algorithm we describe a cutting plane algorithm for solving combinatorial optimization problems the primal projective standard form variant of karmarkar s algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem

a matrix generation approach for eigenvalue optimization we study the extension of a column generation technique to nonpolyhedral models in particular we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices at each step of the algorithm a restricted master problem in the primal space corresponding to the relaxed dual original problem is formed a query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem the original objective function is evaluated at the query point and depending on its differentiability a column or a matrix is added to the restricted master problem we discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix the computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal dual interior point methods on dense and sparse problems using the software sdpt3 our numerical results illustrate that the matrix generation algorithm outperforms primal dual interior point methods on dense problems with no structure and also on a class of sparse problems

a cutting plane method from analytic centers for stochastic programming the stochastic linear programming problem with recourse has a dual block angular structure it can thus be handled by benders decomposition or by kelley s method of cutting planes equivalently the dual problem has a primal block angular structure and can be handled by dantzig wolfe decomposition the two approaches are in fact identical by duality here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations the only significant difference form the aforementioned methods is that new cutting planes or columns by duality will be generated not from the optimum of the linear programming relaxation but from the analytic center of the set of localization

j means a new local search heuristic for minimum sum of squares clustering abstract a new local search heuristic called j m eans is proposed for solving the minimum sum of squares clustering problem the neighborhood of the current solution is defined by all possible centroid to entity relocations followed by corresponding changes of assignments moves are made in such neighborhoods until a local optimum is reached the new heuristic is compared with two other well known local search heuristics k and h m eans as well as with h m eans an improved version of the latter in which degeneracy is removed moreover another heuristic which fits into the variable neighborhood search metaheuristic framework and uses j m eans in its local search step is proposed too results on standard test problems from the literature are reported it appears that j m eans outperforms the other local search methods quite substantially when many entities and clusters are considered

two time scale controlled markov chains a decomposition and parallel processing approach this correspondence deals with a class of ergodic control problems for systems described by markov chains with strong and weak interactions these systems are composed of a set of subchains that are weakly coupled using results already available in the literature one formulates a limit control problem the solution of which can be obtained via an associated nondifferentiable convex programming ndcp problem the technique used to solve the ndcp problem is the analytic center cutting plane method accpm which implements a dialogue between on one hand a master program computing the analytical center of a localization set containing the solution and on the other hand an oracle proposing cutting planes that reduce the size of the localization set at each main iteration the interesting aspect of this implementation comes from two characteristics i the oracle proposes cutting planes by solving reduced sized markov decision problems mdp via a linear program lp or a policy iteration method ii several cutting planes can be proposed simultaneously through a parallel implementation on processors the correspondence concentrates on these two aspects and shows on a large scale mdp obtained from the numerical approximation ldquoa la kushner dupuisrdquo of a singularly perturbed hybrid stochastic control problem the important computational speed up obtained

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

on the convexity of the multiplicative version of karmarkar s potential function karmarkar s potential function is quasi convex but not convex this note investigates the multiplicative version of the potential function and shows that it is not necessarily convex in general but is strictly convex when the corresponding feasible region is bounded this implies that the multiplicative version of the potential function in karmarkar s algorithm is convex since it works on a simplex

sensitivity analysis in linear programming and semidefinite programming using interior point methods we analyze perturbations of the right hand side and the cost parameters in linear programming lp and semidefinite programming sdp we obtain tight bounds on the perturbations that allow interior point methods to recover feasible and near optimal solutions in a single interior point iteration for the unique nondegenerate solution case in lp we show that the bounds obtained using interior point methods compare nicely with the bounds arising from using the optimal basis we also present explicit bounds for sdp using the monteiro zhang family of search directions and specialize them to the aho h k m and nt directions

dimension estimates for the set of points with non dense orbit in homogeneous spaces let x g gamma where g is a lie group and gamma is a lattice in g and let u be a subset of x whose complement is compact we use the exponential mixing results for diagonalizable flows on x to give upper estimates for the hausdorff dimension of the set of points whose trajectories miss u this extends a recent result of kadyrov et al dyn syst 30 2 149 157 2015 and produces new applications to diophantine approximation such as an upper bound for the hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms generalizing an estimate due to broderick and kleinbock int j number theory 11 7 2037 2054 2015

warm start of the primal dual method applied in the cutting plane scheme a practical warm start procedure is described for the infeasible primal dual interior point method ipm employed to solve the restricted master problem within the cutting plane method in contrast to the theoretical developments in this field the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow moreover it treats systematically the case when a large number of cuts are added at one time the technique proposed in this paper has been implemented in the context of hopdm the state of the art yet public domain interior point code numerical results confirm a high degree of efficiency of this approach regardless of the number of cuts added at one time can be thousands in the largest examples and regardless of the depth of the new cuts reoptimizations are usually done with a few additional iterations

improved branching disjunctions for branch and bound an analytic center approach abstract in classical branch and bound algorithms the branching disjunction is often based on a single variable which is a special case of the more general approach that involves multiple variables in this paper we present a new approach to generate good general branching disjunctions based on the shape of the polyhedron and interior point concepts the approach is based on approximating the feasible polyhedron using dikin s inscribed ellipsoid calculated using the analytic center from interior point methods we use the fact that the width of the ellipsoid in a given direction has a closed form expression to formulate a quadratic problem whose optimal solution is a thin direction of the ellipsoid while solving a quadratic problem at each node of the branch and bound tree is impractical we use an efficient neighborhood search heuristic for its solution we report computational results on hard mixed integer problems from the literature showing that the proposed approach leads to smaller branch and bound trees and a reduction in the computational time as compared with classical branching and strong branching as the computation of the analytic center is a bottleneck we finally test the approach within a general interior point based benders decomposition where the analytic center is readily available and show clear dominance of the approach over classical branching

flows on homogeneous spaces and diophantine properties of matrices we generalize the notions of badly approximable resp singular systems of m linear forms in n variables and relate these generalizations to certain bounded resp divergent trajectories in the space of lattices in rm n

an improved column generation algorithm for minimum sum of squares clustering given a set of entities associated with points in euclidean space minimum sum of squares clustering mssc consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized a column generation algorithm for mssc was given by du merle et al in siam journal scientific computing 21 1485 1505 the bottleneck of that algorithm is the resolution of the auxiliary problem of finding a column with negative reduced cost we propose a new way to solve this auxiliary problem based on geometric arguments this greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2 300 entities i e more than 10 times as much as previously done

a branch and price algorithm for the capacitated facility location problem the capacitated facility location problem cflp is a well known combinatorial optimization problem with applications in distribution and production planning it consists in selecting plant sites from a finite set of potential sites and in allocating customer demands in such a way as to minimize operating and transportation costs a number of solution approaches based on lagrangean relaxation and subgradient optimization has been proposed for this problem subgradient optimization does not provide a primal fractional optimal solution to the corresponding master problem however in order to compute optimal solutions to large or difficult problem instances by means of a branch and bound procedure information about such a primal fractional solution can be advantageous in this paper a stabilized column generation method is therefore employed in order to solve a corresponding master problem exactly the column generation procedure is then employed within a branch and price algorithm for computing optimal solutions to the cflp computational results are reported for a set of larger and difficult problem instances

specialized fast algorithms for iqc feasibility and optimization problems the conventional way to treat integral quadratic constraint iqc problems is to transform them into semi definite programs sdps sdps can then be solved using interior point methods which have been proven efficient this approach however is not always the most efficient since it introduces additional decision variables to the sdp and the additional decision variables sometimes largely increase the complexity of the problem in this paper we demonstrate how to solve iqc problems by other alternatives more specifically we consider two cutting plane algorithms we will show that in certain cases these cutting plane algorithms can solve iqc problems much faster than the conventional approach numerical examples as well as some explanations from the point of view of computational complexity are provided to support our point

solving nonlinear multicommodity flow problems by the analytic center cutting plane method the paper deals with nonlinear multicommodity flow problems with convex costs a decomposition method is proposed to solve them the approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network it is thus formulated as a shortest path problem and solved with dijkstra s d heap algorithm an implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations computational results show the efficiency of this approach on well known nondifferentiable problems and also large scale randomly generated problems up to 1000 arcs and 5000 commodities

decomposition based interior point methods for two stage stochastic convex quadratic programs with recourse zhao showed that the log barrier associated with the recourse function of two stage stochastic linear programs behaves as a strongly self concordant barrier and forms a self concordant family on the first stage solutions in this paper we show that the recourse function is also strongly self concordant and forms a self concordant family for the two stage stochastic convex quadratic programs with recourse this allows us to develop bender s decomposition based linearly convergent interior point algorithms an analysis of such an algorithm is given in this paper

step decision rules for multistage stochastic programming a heuristic approach stochastic programming with step decision rules spsdr aims to produce efficient solutions to multistage stochastic optimization problems spsdr like plain multistage stochastic programming sp operates on a monte carlo computing sample of moderate size that approximates the stochastic process unlike sp spsdr does not strive to build a balanced event tree out of that sample rather it defines a solution as a special type of decision rule with the property that the decisions at each stage are piecewise constant functions on the sample of scenarios those pieces define a partition of the set of scenarios at each stage t but the partition at t 1 need not be refinement of the partition at t however the rule is constructed so that the non anticipativity condition is met a necessary condition to make the rules operational to validate the method we show how to extend a non anticipatory decision rule to arbitrary scenarios within a very large validation sample of scenarios we apply three methods spsdr sp and robust optimization to the same 12 stage problem in supply chain management and compare them relatively to different objectives and performance criteria it appears that spsdr performs better than sp in that it produces a more accurate estimate prediction of the value achieved by its solution on the validation sample and also that the achieved value is better

a cutting plane method for solving kyp sdps semidefinite programs originating from the kalman yakubovich popov lemma are convex optimization problems and there exist polynomial time algorithms that solve them however the number of variables is often very large making the computational time extremely long algorithms more efficient than general purpose solvers are thus needed to this end structure exploiting algorithms have been proposed based on the dual formulation in this paper a cutting plane algorithm is proposed in a comparison with a general purpose solver and a structure exploiting solver it is shown that the cutting plane based solver can handle optimization problems of much higher dimension

multiple cuts in the analytic center cutting plane method we analyze the multiple cut generation scheme in the analytic center cutting plane method we propose an optimal primal and dual updating direction when the cuts are central the direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by dikin s primal and dual ellipsoids the new primal and dual directions use the variance covariance matrix of the normals to the new cuts in the metric given by dikin s ellipsoid we prove that the recovery of a new analytic center from the optimal restoration direction can be done in o plog p 1 damped newton steps where p is the number of new cuts added by the oracle which may vary with the iteration the results and the proofs are independent of the specific scaling matrix primal dual or primal dual that is used in the computations the computation of the optimal direction uses newton s method applied to a self concordant function of p variables the convergence result of ye math programming 78 1997 pp 85 104 holds here also the algorithm stops after o frac bar p 2n 2 varepsilon 2 cutting planes have been generated where bar p is the maximum number of cuts generated at any given iteration

polymers in a weak random potential in dimension four rigorous renormalization group analysis correlation functions of the edwards model of polymers at weak coupling are defined and studied at the critical point in dimension four by a rigorous renormalization group method which validates at any order perturbative renormalization group results on their behaviour at large distances remainders are controlled by a new argument which enlarges the use of methods of constructive field theory to models of statistical physics

a new warmstarting strategy for the primal dual column generation method this paper presents a new warmstarting technique in the context of a primal dual column generation method applied to solve a particular class of combinatorial optimization problems the technique relies on calculating an initial point and on solving auxiliary linear optimization problems to determine the step direction needed to fully restore primal and dual feasibilities after new columns arrive conditions on the maximum size of the cuts and on a suitable initial point are discussed additionally the strategy ensures that the duality gap of the warmstart is bounded by the old duality gap multiplied with a small constant which depends on the relation between the old and modified problems computational experiments demonstrate the gains achieved when compared to a coldstart approach

the analytic center cutting plane method with semidefinite cuts we analyze an analytic center cutting plane algorithm for convex feasibility problems with semidefinite cuts the problem of interest seeks a feasible point in a bounded convex set which contains a full dimensional ball with varepsilon 0 is a condition number of the field of cuts

the integration of an interior point cutting plane method within a branch and price algorithm this paper presents a novel integration of interior point cutting plane methods within branch and price algorithms unlike the classical method columns are generated at a central dual solution by applying the analytic centre cutting plane method accpm on the dual of the full master problem first we introduce some modifications to accpm we propose a new procedure to recover primal feasibility after adding cuts and use for the first time a dual newton s method to calculate the new analytic centre after branching second we discuss the integration of accpm within the branch and price algorithm we detail the use of accpm as the search goes deep in the branch and bound tree making full utilization of past information as a warm start we exploit dual information from accpm to generate incumbent feasible solutions and to guide branching finally the overall approach is implemented and tested for the bin packing problem and the capacitated facility location problem with single sourcing we compare against cplex mip 7 5 as well as a classical branch and price algorithm

irreducible kernels and nonperturbative expansions in a theory with purem m interaction recent results on the structure of thes matrix at them particle threshold m 2 in a simplifiedm m scattering theory with no subchannel interaction are extended to the green functionf on the basis of off shell unitarity through an adequate mathematical extension of some results of fredholm theory local two sheeted or infinite sheeted structure off arounds m 2 depending on the parity of m 1 1 where 0 is the mass and is the dimension of space time off shell definition of the irreducible kernelu which is the analogue of thek matrix in the two different parity cases m 1 1 odd or even and related local expansion off for m 1 1 even in powers of ln m 2 s it is shown that each term in this expansion is the dominant contribution to a feynman type integral in which each vertex is a kernelu the links between the kernelu and bethe salpeter type kernelsg of the theory are exhibited in both parity cases as also the links between the above expansion off and local expansions in the bethe salpeter type framework off in terms of feynman type integrals in which each vertex is a kernelg and which include both dominant and subdominant contributions

on the computation of weighted analytic centers and dual ellipsoids with the projective algorithm the primal projective algorithm for linear programs with unknown optimal objective function value is extended to the case where one uses a weighted karmarkar potential function this potential is defined with respect to a strict lower bound to the optimum the minimization of this potential when the lower bound is kept fixed yields a primal and a dual feasible solution the dual solution is the weighted analytic center of a certain dual polytope finally one exhibits a pair of homothetic dual ellipsoids that extends results obtained by sonnevend todd ye freund and anstreicher

homogeneous analytic center cutting plane methods for convex problems and variational inequalities in this paper we consider a new analytic center cutting plane method in an extended space we prove the efficiency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem the variational inequality problem and the problem of constrained minimization our analysis is valid even for problems whose solution belongs to the boundary of the domain

a lagrangean relax and cut approach for the two stage capacitated facility location problem abstract the two stage capacitated facility location problem is to find the optimal locations of depots to serve customers with a given demand the optimal assignment of customers to depots and the optimal product flow from plants to depots in order to solve this problem a lagrangean heuristic is proposed which is based on the relaxation of the capacity constraints the resulting lagrangean subproblem is an uncapacitated facility location problem with an aggregate capacity constraint and can be solved efficiently by branch and bound methods the best bound available from this relaxation is computed by means of a weighted dantzig wolfe decomposition approach feasible solutions are constructed from those of the lagrangean subproblems by applying simple reassignment procedures furthermore the relaxation is strengthened by adding and dualizing valid inequalities which are violated by the fractional primal solution of the dual master program computational results and comparisons with some other bounds and heuristics for this problem are presented

efficient production distribution system design the production distribution system design is an integral part of the general supply chain design this paper proposes a novel solution methodology for this problem that is based on lagrangean relax

complexity of some cutting plane methods that use analytic centers we consider cutting plane methods for minimizing a convex possibly nondifferentiable function subject to box constraints at each iteration accumulated subgradient cuts define a polytope that localizes the minimum the objective and its subgradient are evaluated at the analytic center of this polytope to produce one or two cuts that improve the localizing set we give complexity estimates for several variants of such methods our analysis is based on the works of goffin luo and ye

an interior point constraint generation algorithm for semi infinite optimization with health care application we propose an interior point constraint generation ipcg algorithm for semi infinite linear optimization silo and prove that the algorithm converges to an e solution of silo after a finite number of constraints is generated we derive a complexity bound on the number of newton steps needed to approach the updated center after adding multiple violated constraints and a complexity bound on the total number of constraints that is required for the overall algorithm to converge we implement our algorithm to solve the sector duration optimization problem arising in leksell gamma knife perfexion elekta stockholm sweden treatment planning a highly specialized treatment for brain tumors using real patient data provided by the department of radiation oncology at princess margaret hospital in toronto ontario canada we show that our algorithm can efficiently handle problems in real life health care applications by providing a quality treatment plan in a timely manner comparing our computational results with mosek a commercial software package we show that the ipcg algorithm outperforms the classical primal dual interior point methods on sector duration optimization problem arising in perfexion treatment planning we also compare our results with that of a projected gradient method in both cases we show that ipcg algorithm obtains a more accurate solution substantially faster

asymptotic completeness and multiparticle structure in field theories the ideas developed in part i ref 1 are applied to the recently constructed massive gross neveu model we define in this case an irreducible kernel satisfying a regularized bethe salpeter equation which is convenient to derive asymptotic completeness in the 2 particle region as in part i the method allows direct graphical definition of general irreducible kernels and is well suited to the analysis of asymptotic completeness and related results in more general energy regions

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov

an interior point algorithm for minimum sum of squares clustering an exact algorithm is proposed for minimum sum of squares nonhierarchical clustering i e for partitioning a given set of points from a euclidean m space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong this problem is expressed as a constrained hyperbolic program in 0 1 variables the resolution method combines an interior point algorithm i e a weighted analytic center column generation method with branch and bound the auxiliary problem of determining the entering column i e the oracle is an unconstrained hyperbolic program in 0 1 variables with a quadratic numerator and linear denominator it is solved through a sequence of unconstrained quadratic programs in 0 1 variables to accelerate resolution variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region proved minimum sum of squares partitions are determined for the first time for several fairly large data sets from the literature including fisher s 150 iris

a renormalizable field theory the massive gross neveu model in two dimensions the euclidean massive gross neveu model in two dimensions is just renormalizable and asymptotically free thanks to the pauli principle bare perturbation theory with an ultra violet cut off and the correct ansatz for the bare mass is convergent in a disk whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant therefore the theory can be fully constructed in a perturbative way it satisfies the o s axioms and is the borel sum of the renormalized perturbation expansion of the model

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac