long time regularity of solutions of the hele shaw problem abstract in this paper we study the long time behavior of solutions of the one phase hele shaw problem without surface tension we show that after a finite time solutions of the hele shaw problem become starshaped and lipschitz continuous in space based on this observation we then prove that the free boundary become smooth in space and time with nondegenerate free boundary speed

commitment under uncertainty two stage stochastic matching problems we define and study two versions of the bipartite matching problem in the framework of two stage stochastic optimization with recourse in one version the uncertainty is in the second stage costs of the edges and in the other version the uncertainty is in the set of vertices that needs to be matched we prove lower bounds and analyze efficient strategies for both cases these problems model real life stochastic integral planning problems such as commodity trading reservation systems and scheduling under uncertainty

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

a potential reduction algorithm allowing column generation using the dantzig wolfe decomposition technique a potential reduction algorithm allowing column generation for the linear feasibility lf problem is developed the point of departure is a simple containing polytope and its analytic center in each iteration an inequality violated at the current center is selected used to cut the polytope and then used to find the new center for the shrunken polytope the potential value associated with the containing polytope is reduced by a constant at each step and the algorithm is terminated in a polynomial time that depends only on the number and data length of the inequalities generated in the iterative process

two stage stochastic matching and spanning tree problems polynomial instances and approximation abstract this article deals with the two stage stochastic model which aims at explicitly taking into account uncertainty in optimization problems that kong and schaefer have recently studied for the maximum weight matching problem n kong a j schaefer a factor 1 2 approximation algorithm for two stage stochastic matching problems european journal of operational research 172 3 2006 740 746 they have proved that the problem is np hard and they have provided a factor 1 2 approximation algorithm we further study this problem and strengthen the hardness results slightly improve the approximation ratio and exhibit some polynomial cases we similarly tackle the maximum weight spanning tree problem in the two stage setting finally we make numerical experiments on randomly generated instances to compare the quality of several interesting heuristics

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

analysis of sparse cutting planes for sparse milps with applications to stochastic milps in this paper we present an analysis of the strength of sparse cutting planes for mixed integer linear programs milp with sparse formulations we examine three kinds of problems packing problems covering problems and more general milps with the only assumption that the objective function is nonnegative given an milp instance of one of these three types assume that we decide on the support of cutting planes to be used and the strongest inequalities on these supports are added to the linear programming relaxation we present bounds on the ratio of optimal value of the lp relaxation after adding cuts and the optimal value of the milp that depends only on the sparsity structure of the constraint matrix and the support of sparse cuts selected that is these bounds are completely data independent these results also shed light on the strength of single scenario cuts for two stage stochastic milps

a near optimal solution to a two dimensional cutting stock problem we present an asymptotic fully polynomial approximation scheme for strip packing or packing rectangles into a rectangle of fixed width and minimum height a classicalnp hard cutting stock problem the algorithm based on a new linear programming relaxation finds a packing ofn rectangles whose total height is within a factor of 1 e of optimal up to an additive term and has running time polynomial both in n and in 1 e

complexity analysis of an interior cutting plane method for convex feasibility problems we further analyze the convergence and the complexity of a dual column generation algorithm for solving general convex feasibility problems defined by a separation oracle the oracle is called at an approximate analytic center of the set given by the intersection of the linear inequalities which are the previous answers of the oracle we show that the algorithm converges in finite time and is in fact a fully polynomial approximation algorithm provided that the feasible region has a nonempty interior

computational experience with an interior point cutting plane algorithm there has been a great deal of success in the last 20 years with the use of cutting plane algorithms to solve specialized integer programming problems generally these algorithms work by solving a sequence of linear programming relaxations of the integer programming problem and they use the simplex algorithm to solve the relaxations in this paper we describe experiments using a predictor corrector interior point method to solve the relaxations for some problems the interior point code requires considerably less time than a simplex based cutting plane algorithm

totally unimodular stochastic programs we consider totally unimodular tu stochastic programs that is two stage stochastic programs whose extensive form constraint matrix is tu we generalize the notion of total unimodularity to apply to sets of matrices and provide properties of such sets we provide several sufficient conditions on stochastic programs to be tu when solving tu stochastic problems using the l shaped method it is not clear whether the integrality restrictions should be imposed on the master problem such restrictions will make each master problem more difficult to solve on the other hand solving the linear relaxation of the master typically means sending fractional and unlikely optimal solutions to the subproblems perhaps leading to more iterations our computational results investigate this trade off and provide insight into which strategy is preferable

a multiple cut analytic center cutting plane method for semidefinite feasibility problems we consider the problem of finding a point in a nonempty bounded convex body gamma in the cone of symmetric positive semidefinite matrices cal s m assume that gamma is defined by a separating oracle which for any given m ti m symmetric matrix hat y either confirms that hat y in gamma or returns several selected cuts i e a number of symmetric matrices ai i 1 p p le p max such that gamma is in the polyhedron y in cal s m mid a i bullet y le a i bullet hat y i 1 ldots p we present a multiple cut analytic center cutting plane algorithm starting from a trivial initial point the algorithm generates a sequence of positive definite matrices which are approximate analytic centers of a shrinking polytope in cal s m the algorithm terminates with a point in gamma within o m 3p max epsilon 2 newton steps to leading order where epsilon is the maximum radius of a ball contained in gamma

on the approximability of robust spanning tree problems in this paper the minimum spanning tree problem with uncertain edge costs is discussed in order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution the min max min max regret and 2 stage min max versions of the problem are discussed the complexity and approximability of all these problems are explored it is proved that the min max and min max regret versions with nonnegative edge costs are hard to approximate within o log 1 en for any e 0 unless the problems in np have quasi polynomial time algorithms similarly the 2 stage min max problem cannot be approximated within o logn unless the problems in np have quasi polynomial time algorithms in this paper randomized lp based approximation algorithms with performance bound of o log 2n for min max and 2 stage min max problems are also proposed

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

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

a spectral bundle method for semidefinite programming a central drawback of primal dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices this restricts applicability to problems of small dimension typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order we present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems these are convex nonsmooth programming problems and can be solved by bundle methods we propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems convergence follows from the traditional approach but a proof is included for completeness we present numerical examples demonstrating the efficiency of the approach on combinatorial examples

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

regularity of the free boundary in parabolic phase transition problems in this paper we start the study of the regularity properties of the free boundary for parabolic two phase free boundary problems may be the best known example of a parabolic two phase free boundary problem is the stefan problem a simplified model describing the melting or solidification of a material with a solid liquid interphase the concept of solution can be stated in several ways classical solution weak so lution on divergence form or viscosity solution and as usual one would like to prove that the weak solutions that may be constructed are in fact as smooth and classical as possible locally a classical solution of the stefan problem may be described as following on the unit cylinder q1 b1 1 1 we have two complementary domains and qi separated by a smooth surface s oi2 nq1 in fl and qi we have two smooth solutions ul and u2 of the heat equations

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

complexity analysis of the analytic center cutting plane method that uses multiple cuts we analyze the complexity of the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle the oracle is called at the analytic center of a polytope which contains a solution set and is given by the intersection of the linear inequalities previously generated from the oracle if the center is not in the solution set separating hyperplanes will be placed through the center to shrink the containing polytope while the complexity result has been recently established for the algorithm when one cutting plane is placed in each iteration the result remains open when multiple cuts are added moreover adding multiple cuts actually is a key to practical effectiveness in solving many problems and it presents theoretical difficulties in analyzing cutting plane methods in this paper we show that the analytic center cutting plane algorithm with multiple cuts added in each iteration still is a fully polynomial approximation algorithm

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

an analytic center quadratic cut method for the convex quadratic feasibility problem we consider a quadratic cut method based on analytic centers for two cases of convex quadratic feasibility problems in the first case the convex set is defined by a finite yet large number n of convex quadratic inequalities we extend quadratic cut algorithm of luo and sun 3 for solving such problems by placing or translating the quadratic cuts directly through the current approximate center we show that in terms of total number of addition and translation of cuts our algorithm has the same polynomial worst case complexity as theirs 3 however the total number of steps where steps consist of damped newton steps function evaluations and arithmetic operations required to update from one approximate center to another is where e is the radius of the largest ball contained in the feasible set in the second case the convex set is defined by an infinite number of certain strongly convex quadratic inequalities we adapt the same quadratic cut method for the first case to the second one we show that in this case the quadratic cut algorithm is a fully polynomial approximation scheme furthermore we show that at each iteration k the total number steps as described above required to update from one approximate center to another is at most with e as defined above

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

performance bounds for orthogonal oriented two dimensional packing algorithms two orthogonal oriented two dimensional packing algorithms are presented and their behavior is analyzed under several assumptions for one of the algorithms we show that the ratio of the height used by the algorithm to the optimal height is asymptotically bounded by frac 4 3 this bound is an improvement over similar bounds for previously proposed algorithms

confidence based reasoning in stochastic constraint programming in this work we introduce a novel approach based on sampling for finding assignments that are likely to be solutions to stochastic constraint satisfaction problems and constraint optimisation problems our approach reduces the size of the original problem being analysed by solving this reduced problem with a given confidence probability we obtain assignments that satisfy the chance constraints in the original model within prescribed error tolerance thresholds to achieve this we blend concepts from stochastic constraint programming and statistics we discuss both exact and approximate variants of our method the framework we introduce can be immediately employed in concert with existing approaches for solving stochastic constraint programs a thorough computational study on a number of stochastic combinatorial optimisation problems demonstrates the effectiveness of our approach

an analytic center based column generation algorithm for convex quadratic feasibility problems we consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities at each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut defined by a convex quadratic inequality is placed near the approximate center as the number of cuts increases the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem previously similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem our method differs from these early methods in that we use quadratic cuts in the computation instead of linear cuts moreover our method has a polynomial worst case complexity of o n ln frac 1 varepsilon on the total number of cuts to be used where n is the number of convex quadratic polynomial inequalities in the problem and varepsilon is the radius of the largest ball contained in the feasible set in contrast the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time

a hele shaw problem for tumor growth abstract we consider weak solutions to a problem modeling tumor growth under certain conditions on the initial data solutions can be obtained by passing to the stiff incompressible limit in a porous medium type problem with a lotka volterra source term describing the evolution of the number density of cancerous cells we prove that such limit solutions solve a free boundary problem of hele shaw type we also obtain regularity properties both for the solution and for its free boundary the main new difficulty arises from the competition between the growth due to the source which keeps the initial singularities and the free boundary which invades the domain with a regularizing effect new islands can be generated at singular times

approximating combinatorial optimization problems with the ordered weighted averaging criterion abstract this paper deals with combinatorial optimization problems with k cost scenarios inducing k linear objectives the popular ordered weighted averaging owa criterion is used to aggregate the objectives and compute a solution it is well known that minimizing owa for most basic combinatorial problems is weakly np hard even if the number of scenarios k equals two and strongly np hard when k is a part of the input in this paper the problem with nonincreasing weights in the owa criterion and a large k is first considered a method of reducing the number of scenarios by appropriately aggregating the costs before solving the problem is proposed it is shown that an optimal solution to the reduced problem has a guaranteed worst case approximation ratio some new approximation results for the hurwicz criterion which is a special case of owa are also presented

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

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

a strip packing algorithm with absolute performance bound 2 this paper proposes a new approximation algorithm m for packing rectangles into a strip with unit width and unbounded height so as to minimize the total height of the packing it is shown that for any list l of rectangles m l leq 2 cdot mbox opt l where m l is the strip height actually used by the algorithm m when applied to l and opt l is the minimum possible height within which the rectangles in l can be packed

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

an analytic center cutting plane approach for conic programming we analyze the problem of finding a point strictly interior to a bounded convex and fully dimensional set from a finite dimensional hilbert space we generalize the results obtained for the linear programming lp semidefinite programming sdp and second order core programming socp cases the cuts added by our algorithm are central and conic in our analysis we find an upper bound for the number of newton steps required to compute an approximate analytic center also we provide an upper bound for the total number of cuts added to solve the problem this bound depends on the quality of the cuts the dimensionality of the problem and the thickness of the set we are considering

regularity of the free boundary for the one phase hele shaw problem abstract in this paper we prove that in a local neighborhood lipschitz continuous free boundary of a solution of the one phase hele shaw problem is indeed smooth if the solution is lipschitz continuous and non degenerate in the neighborhood

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

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

a factor approximation algorithm for two stage stochastic matching problems abstract we introduce the two stage stochastic maximum weight matching problem and demonstrate that this problem is np complete we give a factor 1 2 approximation algorithm and prove its correctness we also provide a tight example to show the bound given by the algorithm is exactly 1 2 computational results on some two stage stochastic bipartite matching instances indicate that the performance of the approximation algorithm appears to be substantially better than its worst case performance

uniqueness and existence results on the hele shaw and the stefan problems in this paper we introduce a notion of viscosity solutions for the one phase hele shaw and stefan problems when there is no surface tension we prove the uniqueness and existence of the solutions for both problems and the uniform convergence of solutions of porous medium equation to those of the hele shaw problem

the analytic center quadratic cut method for strongly monotone variational inequality problems convergence of an algorithm for strongly monotone variational inequality problems vips is investigated at each iteration the algorithm adds a quadratic cut through the analytic center of the consequently shrinking convex set it is shown that the sequence of analytic centers converges to the unique solution in cal o 1 sqrt k where k is the number of iterations

finding a strict feasible solution of a linear semidefinite program abstract this study deals with the performance of projective interior point methods for linear semidefinite program we propose a modification in the initialization phases of the method in order to reduce the computation time this purpose is confirmed by numerical experiments showing the efficiency which are presented in the last section of the paper

the discrete sell or hold problem with constraints on asset values the discrete sell or hold problem dshp which is introduced in he et al 6 is studied under the constraint that each asset can only take a constant number of different values we show that if each asset can only take two different values the problem becomes polynomial time solvable however even if each asset can only take three different values the dshp is still np hard an approximation algorithm which could potentially achieve a better approximation ratio than the one proposed in 6 is also given under this setting

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

on recoverable and two stage robust selection problems with budgeted uncertainty in this paper the problem of selecting p out of n available items is discussed such that their total cost is minimized we assume that costs are not known exactly but stem from a set of possible outcomes robust recoverable and two stage models of this selection problem are analyzed in the two stage problem up to p items is chosen in the first stage and the solution is completed once the scenario becomes revealed in the second stage in the recoverable problem a set of p items is selected in the first stage and can be modified by exchanging up to k items in the second stage after a scenario reveals we assume that uncertain costs are modeled through bounded uncertainty sets i e the interval uncertainty sets with an additional linear budget constraint in their discrete and continuous variants polynomial algorithms for recoverable and two stage selection problems with continuous bounded uncertainty and compact mixed integer formulations in the case of discrete bounded uncertainty are constructed

bundle methods for sum functions with easy components applications to multicommodity network design we propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum function where some of the components are easy that is they are lagrangian functions of explicitly known compact convex programs this corresponds to a stabilized partial dantzig wolfe decomposition where suitably modified representations of the easy convex subproblems are inserted in the master problem as an alternative to iteratively inner approximating them by extreme points thus providing the algorithm with exact information about a part of the dual objective function the resulting master problems are potentially larger and less well structured than the standard ones ruling out the available specialized techniques and requiring the use of general purpose solvers for their solution this strongly favors piecewise linear stabilizing terms as opposed to the more usual quadratic ones which in turn may have an adverse effect on the convergence speed of the algorithm so that the overall performance may depend on appropriate tuning of all these aspects yet very good computational results are obtained in at least one relevant application the computation of tight lower bounds for fixed charge multicommodity min cost flow problems

hedging uncertainty approximation algorithms for stochastic optimization problems we study two stage finite scenario stochastic versions of several combinatorial optimization problems and provide nearly tight approximation algorithms for them our problems range from the graph theoretic shortest path vertex cover facility location to set theoretic set cover bin packing and contain representatives with different approximation ratios the approximation ratio of the stochastic variant of a typical problem is found to be of the same order of magnitude as its deterministic counterpart furthermore we show that common techniques for designing approximation algorithms such as lp rounding the primal dual method and the greedy algorithm can be adapted to obtain these results

an analytic center cutting plane method for semidefinite feasibility problems semidefinite feasibility problems arise in many areas of operations research the abstract form of these problems can be described as finding a point in a nonempty bounded convex body g in the cone of symmetric positive semidefinite matrices assume that g is defined by an oracle which for any givenm xm symmetric positive semidefinite matrix y either confirms that y e g or returns a cut i e a symmetric matrixa such that g is in the half space y a y a y we study an analytic center cutting plane algorithm for this problem at each iteration the algorithm computes an approximate analytic center of a working set defined by the cutting plane system generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise the new cutting plane returned by the oracle is added into the system as the number of iterations increases the working set shrinks and the algorithm eventually finds a solution to the problem all iterates generated by the algorithm are positive definite matrices the algorithm has a worst case complexity of o m3 e 2 on the total number of cuts to be used where e is the maximum radius of a ball contained by g