mixed perfectly matched layers for direct transient analysis in 2d elastic heterogeneous media abstract we are concerned with elastic waves arising in plane strain problems in an elastic semi infinite arbitrarily heterogeneous medium specifically we discuss the development of a new mixed displacement stress formulation for forward elastic wave simulations in perfectly matched layer pml truncated heterogeneous media to date most pml formulations split the displacement and stress fields resulting in non physical components for each field in this work we favor unsplit schemes primarily for the relative ease by which the resulting forms can be incorporated into existing codes the ease by which the resulting semi discrete forms can be integrated in time and the ease by which they can be used in adjoint formulations arising in inverse problems contrary to most past and current developments we start by following classical lines and apply complex coordinate stretching to the governing equations in the frequency domain while retaining both displacements and stress quantities as unknowns with the aid of auxiliary variables the resulting mixed form is rendered second order in time thereby allowing the use of standard time integration schemes we report on numerical simulations demonstrating the stability and efficacy of the approach

a branch and cut decomposition algorithm for solving chance constrained mathematical programs with finite support we present a new approach for exactly solving chance constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints such problems have been notoriously difficult to solve due to nonconvexity of the feasible region and most available methods are only able to find provably good solutions in certain very special cases our approach uses both decomposition to enable processing subproblems corresponding to one possible outcome at a time and integer programming techniques to combine the results of these subproblems to yield strong valid inequalities computational results on a chance constrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixed integer programming formulation and a simple decomposition approach that does not use strong valid inequalities we also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels as would be done when solving for the efficient frontier of risk and cost

a hybrid breakout local search and reinforcement learning approach to the vertex separator problem abstract the vertex separator problem vsp is an np hard problem which emerges from a variety of important domains and applications in this paper we present an improved breakout local search for vsp named bls rle the distinguishing feature of this approach is a new parameter control mechanism that draws upon ideas from reinforcement learning theory to reach an interdependent decision on the number and on the type of perturbation moves the mechanism complies with the principle of first carrying out intensification and then employing minimal diversification only if needed it uses a dedicated sampling strategy for a rapid convergence towards a limited set of parameter values that appear to be the most convenient for the given state of search extensive experimental evaluations and statistical comparisons on a wide range of benchmark instances show significant improvement in performance of the proposed algorithm over the existing bls algorithm for vsp indeed out of the 422 tested instances bls rle was able to attain the best known solution in 93 8 of the cases which is around 20 higher compared to the existing bls in addition we provide detailed analyses to evaluate the importance of the key elements of the proposed method and to justify the degree of diversification introduced during perturbation

an efficient trajectory method for probabilistic production inventory distribution problems we consider a supply chain operating in an uncertain environment the customers demand is characterized by a discrete probability distribution a probabilistic programming approach is adopted for constructing an inventory production distribution plan over a multiperiod planning horizon the plan does not allow the backlogging of the unsatisfied demand and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level it is a strategic plan that hedges against undesirable outcomes and that can be adjusted to account for possible favorable realizations of uncertain quantities a modular integrated and computationally tractable method is proposed for the solution of the associated stochastic mixed integer optimization problems containing joint probabilistic constraints with dependent right hand side variables the concept of p efficiency is used to construct a finite number of demand trajectories which in turn are employed to solve problems with joint probabilistic constraints we complement this idea by designing a preordered set based preprocessing algorithm that selects a subset of promising p efficient demand trajectories finally to solve the resulting disjunctive mixed integer programming problem we implement a special column generation algorithm that limits the risk of congestion in the resources of the supply chain the methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient it finds a solution with a minimal integrality gap and provides substantial cost savings

improved handling of uncertainty and robustness in set covering problems abstract set covering problems are one of the classical combinatorial optimization problems and part of many different practical applications e g the location of emergency service facilities in many real world problems the required statistical parameters are not precisely known and the obtained solutions may reveal a non adequate performance we introduce a robust formulation of the uncertain probabilistic set covering problem which combines the concepts of robust and probabilistic optimization by introducing robust covering constraints this robust uncertain set covering problem can be stated as a compact mixed integer linear programming model additionally two non compact integer linear model formulations are developed as the strength of these formulations is not known a priori we analyze the performance of these formulations in an extensive computational study a case study for the location of emergency service facilities highlights the benefits of our approach in comparison to a formulation neglecting these uncertainties

relaxations and approximations of chance constraints under finite distributions optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity in the setting of a finite distribution of the stochastic parameters an optimization problem with linear chance constraints can be formulated as a mixed integer linear program milp the natural milp formulation has a weak relaxation bound and is quite difficult to solve in this paper we review some recent results on improving the relaxation bounds and constructing approximate solutions for milp formulations of chance constraints we also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems this algorithm uses a relaxation to construct a solution whose constraint violation risk level may be larger than the pre specified threshold but is within a constant factor of it and whose objective value is also within a constant factor of the true optimal value finally we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios

probabilistic set covering with correlations we consider two variants of a probabilistic set covering psc problem the first variant assumes that there is uncertainty regarding whether a selected set can cover an item and the objective is to determine a minimum cost combination of sets so that each item is covered with a prespecified probability the second variant seeks to maximize the minimum probability that a selected set can cover all items to date literature on this problem has focused on the special case in which uncertainties are independent in this paper we formulate deterministic mixed integer programming models for distributionally robust psc problems with correlated uncertainties by exploiting the supermodularity of certain substructures and analyzing their polyhedral properties we develop strong valid inequalities to strengthen the formulations computational results illustrate that our modeling approach can outperform formulations in which correlations are ignored and that our algorithms can significantly reduce overall comput

decomposition algorithms for two stage chance constrained programs we study a class of chance constrained two stage stochastic optimization problems where second stage feasible recourse decisions incur additional cost in addition we propose a new model where recovery decisions are made for the infeasible scenarios to obtain feasible solutions to a relaxed second stage problem we develop decomposition algorithms with specialized optimality and feasibility cuts to solve this class of problems computational results on a chance constrained resource planing problem indicate that our algorithms are highly effective in solving these problems compared to a mixed integer programming reformulation and a naive decomposition method

a lagrangian heuristic for the capacitated plant location problem with single source constraints abstract in this paper we look at the lagrangian relaxations of the capacitated plant location problem with single source constraints in terms of the bounds and solution techniques we propose a heuristic based on a lagrangian relaxation of the problem we also show how this heuristic can be used to solve other combinatorial problems that are related to this problem such as the concentrator location problem and the bin packing problem

regularization methods for optimization problems with probabilistic constraints we analyze nonlinear stochastic optimization problems with probabilistic constraints on nonlinear inequalities with random right hand sides we develop two numerical methods with regularization for their numerical solution the methods are based on first order optimality conditions and successive inner approximations of the feasible set by progressive generation of p efficient points the algorithms yield an optimal solution for problems involving concave probability distributions for arbitrary distributions the algorithms solve the convex hull problem and provide upper and lower bounds for the optimal value and nearly optimal solutions the methods are compared numerically to two cutting plane methods

the vertex separator problem a polyhedral investigation the vertex separator vs problem in a graph g v e asks for a partition of v into nonempty subsets a b c such that there is no edge between a and b and c is minimized subject to a bound on max a b we give a mixed integer programming formulation of the problem and investigate the vertex separator polytope vsp the convex hull of incidence vectors of vertex separators necessary and sufficient conditions are given for the vsp to be full dimensional central to our investigation is the relationship between separators and dominators several classes of valid inequalities are investigated along with the conditions under which they are facet defining for the vsp some of our proofs combine in new ways projection with lifting in a companion paper we develop a branch and cut algorithm for the vs problem based on the inequalities discussed here and report on computational experience with a wide variety of vs problems drawn from the literature and inspired by various applications

a heuristic lagrangean algorithm for the capacitated plant location problem abstract lagrangean techniques have been widely applied to the uncapacitated plant location problem and in some cases they have proven to be successfull even when capacitated problems with additional constraints are taken into account in our paper we study the application of these techniques to the capacitated plant location problem when the model considered is a pure integer one several lagrangean decompositions are considered and for some of them heuristic algorithms have been designed to solve the resulting lagrangean subproblems the heuristics consisting of a two phase procedure the first location phase defines a set of multipliers from the analysis of the dual lp relaxation and makes a choice of the plants considering the resulting subproblems as a particular case of the general assignment problems several heuristics have been studied for this second phase based either on a decomposition of knapsack type subproblems through a definition of a set of penalties or of looking into the duality gap and trying to reduce it computational experience is reported

convex relaxations of non convex mixed integer quadratically constrained programs extended formulations this paper addresses the problem of generating strong convex relaxations of mixed integer quadratically constrained programming miqcp problems miqcp problems are very difficult because they combine two kinds of non convexities integer variables and non convex quadratic constraints to produce strong relaxations of miqcp problems we use techniques from disjunctive programming and the lift and project methodology in particular we propose new methods for generating valid inequalities from the equation y x x t we use the non convex constraint y x x t preccurlyeq 0 to derive disjunctions of two types the first ones are directly derived from the eigenvectors of the matrix y x x t with positive eigenvalues the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction we also use the convex sdp constraint y x x t succcurlyeq 0 to derive convex quadratic cuts and we combine both approaches in a cutting plane algorithm we present computational results to illustrate our findings

genetic algorithm based technique for solving chance constrained problems management and measurement of risk is an important issue in almost all areas that require decisions to be made under uncertain information chance constrained programming ccp have been used for modelling and analysis of risks in a number of application domains however the resulting mathematical problems are non trivial to represent using algebraic modelling languages and pose significant computational challenges due to their non linear non convex and the stochastic nature we develop and implement c classes to represent such ccp problems we propose a framework consisting of genetic algorithm and monte carlo simulation in order to process the problems the non linear and non convex nature of the ccp problems are processed using genetic algorithm whereas the stochastic nature is addressed through simulation the computational investigations have shown that the framework can efficiently represent and obtain good solutions for seven test problems

pattern based modeling and solution of probabilistically constrained optimization problems we propose a new modeling and solution method for probabilistically constrained optimization problems the methodology is based on the integration of the stochastic programming and combinatorial pattern recognition fields it permits the fast solution of stochastic optimization problems in which the random variables are represented by an extremely large number of scenarios the method involves the binarization of the probability distribution and the generation of a consistent partially defined boolean function pdbf representing the combination f p of the binarized probability distribution f and the enforced probability level p we show that the pdbf representing f p can be compactly extended as a disjunctive normal form dnf the dnf is a collection of combinatorial p patterns each defining sufficient conditions for a probabilistic constraint to hold we propose two linear programming formulations for the generation of p patterns that can be subsequently used to derive a linear programming inner approximation of the original stochastic problem a formulation allowing for the concurrent generation of a p pattern and the solution of the deterministic equivalent of the stochastic problem is also proposed the number of binary variables included in the deterministic equivalent formulation is not an increasing function of the number of scenarios used to represent uncertainty results show that large scale stochastic problems in which up to 50 000 scenarios are used to describe the stochastic variables can be consistently solved to optimality within a few seconds

combining very large scale and ilp based neighborhoods for a two level location problem abstract in this paper we tackle a generalization of the single source capacitated facility location problem in which two sets of facilities called intermediate level and upper level facilities have to be located the dimensioning of the intermediate set the assignment of clients to intermediate level facilities and of intermediate level facilities to upper level facilities must be optimized as well such problem arises for instance in telecommunication network design in fact in hierarchical networks the traffic arising at client nodes often have to be routed through different kinds of facility nodes which provide different services we propose a heuristic approach based on very large scale neighborhood search to tackle the problem in which both ad hoc algorithms and general purpose solvers are applied to explore the search space we report on experimental results using datasets from the capacitated location literature such results show that the approach is promising and that integer linear programming based neighborhoods are significantly effective

a regularized decomposition method for minimizing a sum of polyhedral functions a problem of minimizing a sum of many convex piecewise linear functions is considered in view of applications to two stage linear programming where objectives are marginal values of lower level problems it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise linear induced feasibility constraints we propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component the master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts the method goes through nonbasic points in general and is finitely convergent without any nondegeneracy assumptions next we present a special technique for solving the regularized master problem that uses an active set strategy and qr factorization and exploits the structure of the master finally some numerical evidence is given

high order boundary conditions and finite elements for infinite domains a finite element method for the solution of linear elliptic problems in infinite domains is proposed the two dimensional laplace helmholtz and modified helmholtz equations outside an obstacle and in a semi infinite strip are considered in detail in the proposed method an artificial boundary b is first introduced to make the computational domain finite then the exact nonlocal dirichlet to neumann dtn boundary condition is derived on b this condition is localized and a sequence of local boundary conditions on b of increasing order is obtained the problem in with a localized dtn boundary condition on b is then solved using the finite element method the numerical stability of the scheme is discussed a hierarchy of special conforming finite elements is developed and used in the layer adjacent to b in conjunction with the local high order boundary condition applied on b an error analysis is given for both nonlocal and local boundary conditions numerical experiments are presented to demonstrate the performance of the method

multi objective probabilistically constrained programs with variable risk models for multi portfolio financial optimization abstract we consider a class of multi objective probabilistically constrained programs mopcp with a joint probabilistic constraint and a variable risk level we consider two cases with only a random right hand side vector or a multi row random technology matrix and propose a boolean modeling framework to derive new mixed integer linear programs milp that are either equivalent reformulations or inner approximations of mopcp respectively via testing randomly generated mopcp instances we demonstrate modeling insights pertaining to the most suitable milp to the trade offs between conflicting objectives of cost revenue and reliability and to the parameter scalarization determining relative importance of each objective we then focus on several mopcp variants of a multi portfolio financial optimization problem to implement a downside risk measure which can be used in a centralized or decentralized investment context we study the impact of modeling parameters on the portfolios show via a cross validation study robustness of mopcp and perform a comparative analysis of the optimal investment decisions

balance 0 1 matrices bicoloring and total dual integrality a 0 1 matrixa is balanced if in every submatrix with two nonzero entries per row and column the sum of the entries is a multiple of four this definition was introduced by truemper 1978 and generalizes the notion of a balanced 0 1 matrix introduced by berge 1970 in this paper we extend a bicoloring theorem of berge 1970 and total dual integrality results of fulkerson hoffman and oppenheim 1974 to balanced 0 1 matrices

single commodity stochastic network design under probabilistic constraint with discrete random variables single commodity networks are considered where demands at the nodes are random the problem is to find minimum cost optimal built in capacities at the nodes and arcs subject to the constraint that all demands should be met on a prescribed probability level reliability constraint and some deterministic constraints should be satisfied the reliability constraint is formulated in terms of the gale hoffman feasibility inequalities but their number is reduced by elimination technique the concept of a p efficient point is used in a smart way to convert and then relax the problem into an lp the p efficient points are simultaneously generated with the solution of the lp the joint distribution of the demands is used to obtain the p efficient points for all stochastic inequalities that were not eliminated and the solution of a multiple choice knapsack problem is used to generate p efficient points the model can be applied to planning in interconnected power systems flood control networks design of shelter

a branch reduce cut algorithm for the global optimization of probabilistically constrained linear programs we consider probabilistically constrained linear programs with general distributions for the uncertain parameters these problems involve non convex feasible sets we develop a branch and bound algorithm that searches for a global optimal solution to this problem by successively partitioning the non convex feasible region and by using bounds on the objective function to fathom inferior partition elements this basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partition elements and hence tighten bounds the proposed branch reduce cut algorithm exploits the monotonicity properties inherent in the problem and requires solving linear programming subproblems we provide convergence proofs for the algorithm some illustrative numerical results involving problems with discrete distributions are presented

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

explicit finite element perfectly matched layer for transient three dimensional elastic waves the use of a perfectly matched layer pml model is an efficient approach toward the bounded domain modelling of wave propagation on unbounded domains this paper formulates a three dimensional pml for elastic waves by building upon previous work by the author and implements it in a displacement based finite element setting the novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration thus making it practicable for use in large scale three dimensional dynamic analyses an efficient method of calculating the strain terms in the pml is developed in order to take advantage of the lack of the overhead of solving equations at each time step the pml formulation is studied and validated first for a semi infinite bar and then for the classical soil structure interaction problems of a square flexible footing on a i half space ii layer on a half space and iii layer on a rigid base numerical results for these problems demonstrate that the pml models produce highly accurate results with small bounded domains and at low computational cost and that these models are long time stable with critical time step sizes similar to those of corresponding fully elastic models copyright 2008 john wiley sons ltd

first order dominance stronger characterization and a bivariate checking algorithm how to determine whether one distribution first order dominates another is a fundamental problem that has many applications in economics finance probability theory and statistics nevertheless little is known about how to efficiently check first order dominance for finite multivariate distributions utilizing that this problem can be formulated as a transportation problem with a special structure we provide a stronger characterization of multivariate first order dominance and develop a linear time complexity checking algorithm for the bivariate case we illustrate the use of the checking algorithm when numerically assessing first order dominance among continuous bivariate distributions

concavity and efficient points of discrete distributions in probabilistic programming we consider stochastic programming problems with probabilistic constraints involving integer valued random variables the concept of a p efficient point of a probability distribution is used to derive various equivalent problem formulations next we introduce the concept of r concave discrete probability distributions and analyse its relevance for problems under consideration these notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables the results are illustrated with numerical examples

absorbing boundary conditions for acoustic media by decomposing the acoustic wave equation into incoming and outgoing components an absorbing boundary condition can be derived to eliminate reflections from plane waves according to their direction of propagation this boundary condition is characterized by a first order differential operator the differential operator or absorbing boundary operator is the basic element from which more complicated boundary conditions can be constructed the absorbing boundary operator can be designed to absorb perfectly plane waves traveling in any two directions by combining two or more absorption operators boundary conditions can be created which absorb plane waves propagating in any number of directions absorbing boundary operators simplify the task of designing boundary conditions to reduce the detrimental effects of outgoing waves in many wave propagation problems

upper and lower bounds for the single source capacitated location problem abstract the single source capacitated location problem is considered given a set of potential locations and the plant capacities it must be decided where and how many plants must be open and which clients must be assigned to each open plant a lagrangean relaxation is used to obtain lower bounds for this problem upper bounds are given by lagrangean heuristics followed by search methods and by one tabu search metaheuristic computational experiments on different sets of problems are presented

exploiting unsupervised and supervised constraints for subspace clustering data in many image and video analysis tasks can be viewed as points drawn from multiple low dimensional subspaces with each subspace corresponding to one category or class one basic task for processing such kind of data is to separate the points according to the underlying subspace referred to as subspace clustering extensive studies have been made on this subject and nearly all of them use unconstrained subspace models meaning the points can be drawn from everywhere of a subspace to represent the data in this paper we attempt to do subspace clustering based on a constrained subspace assumption that the data is further restricted in the corresponding subspaces e g belonging to a submanifold or satisfying the spatial regularity constraint this assumption usually describes the real data better such as differently moving objects in a video scene and face images of different subjects under varying illumination a unified integer linear programming optimization framework is used to approach subspace clustering which can be efficiently solved by a branch and bound bb method we also show that various kinds of supervised information such as subspace number outlier ratio pairwise constraints size prior and etc can be conveniently incorporated into the proposed framework experiments on real data show that the proposed method outperforms the state of the art algorithms significantly in clustering accuracy the effectiveness of the proposed method in exploiting supervised information is also demonstrated

an exact method for the two echelon single source capacitated facility location problem facility location problems form an important class of integer programming problems with applications in the telecommunication distribution and transportation industries in this paper we are conc

the probabilistic set covering problem in a probabilistic set covering problem the right hand side is a random binary vector and the covering constraint has to be satisfied with some prescribed probability we analyze the structure of the set of probabilistically efficient points of binary random vectors develop methods for their enumeration and propose specialized branch and bound algorithms for probabilistic set covering problems

hybrid scatter search and path relinking for the capacitated p median problem in this paper we propose heuristic algorithms for the capacitated p median problem in the first one solutions are obtained with scatter search whereas the second algorithm uses path relinking a third approach that combines path relinking with scatter search is also analyzed the grasp methodology is used to generate the initial reference set both for scatter search and path relinking computational experiments have been carried out with different data sets in order to evaluate the performance of the approaches in general scatter search outperforms path relinking but the use of path relinking prior to scatter search has shown to enhance the performance of scatter search specially in terms of the computation times required due to the improvement of the initial reference set the obtained results that have been compared with previous approaches show the efficiency of the methods both in terms of the quality of the solutions found and of the required computation times even for very large instances

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

the vertex separator problem algorithms and computations this is a companion paper to our polyhedral study 1 of the vertex separator vs problem given an undirected graph g the vs problem consists in identifying a minimum weight vertex set whose removal disconnects g subject to bounds on the size of the resulting components in this paper we describe versions of a branch and cut algorithm based on the results of that polyhedral study it uses two families of cuts symmetric and asymmetric for which we develop polynomial time greedy separation routines a heuristic to generate feasible separators is also used a computational experiment on several data sets from the literature compares the performance of three versions of our algorithm to that of the commercial mip solver xpress this experiment throws a sharp light on the role of cut density known to software developers but never before documented in the literature it convincingly shows that the practical usefulness of cuts in integer programming depends not only on their strength but also on their sparsity everything else being equal the smaller the cut support the better the power of the inequalities proposed here is well illustrated by the computational tests on dense graphs this is in accordance with the previous observation since the support of these cuts tends to decrease with graph density

a multi exchange heuristic for the single source capacitated facility location problem we present a very large scale neighborhood vlsn search algorithm for the capacitated facility location problem with single source constraints the neighborhood structures are induced by customer multi exchanges and by facility moves we consider both traditional single customer multi exchanges detected on a suitably definedcustomer improvement graph and more innovative multicustomer multi exchanges detected on afacility improvement graph dynamically built through the use of a greedy scheme computational results for some benchmark instances are reported that demonstrate the effectiveness of the approach for solving large scale problems a further test on real data involving an italian factory is also presented

iis branch and cut for joint chance constrained stochastic programs and application to optimal vaccine allocation we present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data the problem can be reformulated as a large scale mixed 0 1 integer program we derive a new class of optimality cuts called iis cuts and apply them to our problem the cuts are based on irreducibly infeasible subsystems iis of an lp defined by requiring that all scenarios be satisfied we propose a method for improving the upper bound of the problem when no cut can be found we derive and implement a branch and cut algorithm based on iis cuts and refer to this algorithm as the iis branch and cut algorithm we report on computational results with several test instances from optimal vaccine allocation the computational results are promising as the iis branch and cut algorithm gives better results than a state of the art commercial solver on one class of problems

a cut and solve based algorithm for the single source capacitated facility location problem abstract in this paper we present a cut and solve cs based exact algorithm for the single source capacitated facility location problem sscflp at each level of cs s branching tree it has only two nodes corresponding to the sparse problem sp and the dense problem dp respectively the sp whose solution space is relatively small with the values of some variables fixed to zero is solved to optimality by using a commercial mip solver and its solution if it exists provides an upper bound to the sscflp meanwhile the resolution of the lp of dp provides a lower bound for the sscflp a cutting plane method which combines the lifted cover inequalities and fenchel cutting planes to separate the 0 1 knapsack polytopes is applied to strengthen the lower bound of sscflp and that of dp these lower bounds are further tightened with a partial integrality strategy numerical tests on benchmark instances demonstrate the effectiveness of the proposed cutting plane algorithm and the partial integrality strategy in reducing integrality gap and the effectiveness of the cs approach in searching an optimal solution in a reasonable time computational results on large sized instances are also presented

iii posedness of absorbing boundary conditions for migration absorbing boundary conditions for wave equation migration were introduced by clayton and engquist we show that one of these boundary conditions the b2 second order condition applied with the 45 third order migration equation is ill posed in fact this boundary condition is subject to two distinct mechanisms of ill posedness a kreiss mode with finite speed at one boundary and another mode of a new kind involving wave propagation at unbounded speed back and forth between two boundaries unlike b2 the third order clayton engquist boundary condition b3 is well posed however we show that it is impossible for any boundary condition of clayton engquist type of order higher than one to be well posed with a migration equation whose order is higher than three

the dtn nonreflecting boundary condition for multiple scattering problems in the half plane abstract the multiple dirichlet to neumann multiple dtn non reflecting boundary condition is adapted to acoustic scattering from obstacles embedded in the half plane the multiple dtn map is coupled with the method of images as an alternative model for multiple acoustic scattering in the presence of acoustically soft and hard plane boundaries as opposed to the current practice of enclosing all obstacles with a large semicircular artificial boundary that contains portion of the plane boundary the proposed technique uses small artificial circular boundaries that only enclose the immediate vicinity of each obstacle in the half plane the adapted multiple dtn condition is simultaneously imposed in each of the artificial circular boundaries as a result the computational effort is significantly reduced a computationally advantageous boundary value problem is numerically solved with a finite difference method supported on boundary fitted grids approximate solutions to problems involving two scatterers of arbitrary geometry are presented the proposed numerical method is validated by comparing the approximate and exact far field patterns for the scattering from a single and from two circular obstacles in the half plane

discrete dirichlet to neumann maps for unbounded domains abstract it is shown how to construct a discrete counterpart of the dirichlet to neumann dtn map for use on an artificial boundary b introduced in exterior boundary value problems following the idea of deakin and dryden this discrete map provides an approximate non reflecting or absorbing boundary condition for use in formulating a problem in the finite computational domain bounded by b different discrete maps are constructed for use with finite difference and finite element methods in the solution of some simple problems shows that use of the discrete dirichlet to neumann ddtn map yields nearly the same accuracy as that of using the continuous dtn map

scenario min max optimization and the risk of empirical costs we consider convex optimization problems in the presence of stochastic uncertainty the min max sample based solution is the solution obtained by minimizing the max of the cost functions corresponding to a finite sample of the uncertainty parameter the empirical costs are instead the cost values that the solution incurs for the various parameter realizations that have been sampled our goal is to evaluate the risks associated with the empirical costs where the risk associated with a cost is the probability that the cost is exceeded when a new realization of the uncertainty parameter is seen this task is accomplished without resorting to uncertainty realizations other than those used in optimization the theoretical result proved in this paper is that these risks form a random vector whose probability distribution is an ordered dirichlet distribution irrespective of the probability measure of the stochastic uncertainty parameter this result provides a distribution free characterization of the risks as

a polyhedral study on chance constrained program with random right hand side the essential structure of the mixed integer programming formulation for chance constrained program ccp is the intersection of multiple mixing sets with a 0 1 knapsack to improve our computational capacity on ccp an underlying substructure the single mixing set with a 0 1 knapsack has received substantial attentions recently in this study we consider a ccp problem with stochastic right hand side under a finite discrete distribution we first present a family of strong inequalities that subsumes known facet defining ones for that single mixing set due to the flexibility of our generalized inequalities we develop a new separation heuristic that has a complexity much less than existing one and guarantees generated cutting planes are facet defining for the polyhedron of ccp then we study lifting and superadditive lifting on knapsack cover inequalities and provide an implementable procedure on deriving another family of strong inequalities for the single mixing set finally different from the traditional approach that aggregates original constraints to investigate polyhedral implications due to their interactions we propose a novel blending procedure that produces strong valid inequalities for ccp by integrating those derived from individual mixing sets we show that under certain conditions they are the first type of facet defining inequalities describing intersection of multiple mixing sets and design an efficient separation heuristic for implementation in the computational experiments we perform a systematic study and illustrate the efficiency of the proposed inequalities on solving chance constrained static probabilistic lot sizing problems

a provisioning problem with stochastic payments abstract we consider the problem of determining the minimal requirement one must establish in order to meet a series of future random payments it is shown in a very general setting that this problem can be recast as a chance constrained model and how the technique of sample average approximation can be employed to find solutions we also use comonotonic theory to analyze analytical approximations in a restricted gaussian setting our numerical illustrations demonstrate that the sample average approximation is a viable and efficient way to solve the stated problem generally and outperforms the analytical approximations in passing we present a result that is related to stein s famous lemma stein 1981 and is of interest in itself

a repeated matching heuristic for the single source capacitated facility location problem facility location models form an important class of integer programming problems with application in many areas such as the distribution and transportation industries an important class of solution methods for these problems are so called lagrangean heuristics which have been shown to produce high quality solutions and which are at the same time robust the general facility location problem can be divided into a number of special problems depending on the properties assumed in the capacitated location problem each facility has a specific capacity on the service it provides we describe a new solution approach for the capacitated facility location problem when each customer is served by a single facility the approach is based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied the method generates feasible solutions in each iteration in contrast to lagrangean heuristics where problem dependent heuristics must be used to construct a feasible solution numerical results show that the approach produces solutions which are of similar and often better than those produced using the best lagrangean heuristics

perfectly matched layers for time harmonic elastodynamics of unbounded domains theory and finite element implementation abstract one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies this paper develops the pml concept for time harmonic elastodynamics in cartesian coordinates utilising insights obtained with electromagnetics pmls and presents a novel displacement based symmetric finite element implementation of the pml for time harmonic plane strain or three dimensional motion the pml concept is illustrated through the example of a one dimensional rod on elastic foundation and through the anti plane motion of a two dimensional continuum the concept is explored in detail through analytical and numerical results from a pml model of the semi infinite rod on elastic foundation and through numerical results for the anti plane motion of a semi infinite layer on a rigid base numerical results are presented for the classical soil structure interaction problems of a rigid strip footing on a i half plane ii layer on a half plane and iii layer on a rigid base the analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by pml models even with small bounded domains

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

a sample approximation approach for optimization with probabilistic constraints we study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample we show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast this leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound we then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability once again we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem finally we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints

a heuristic for bilp problems the single source capacitated facility location problem abstract in the single source capacitated facility location problem sscflp each customer has to be assigned to one facility that supplies its whole demand the total demand of customers assigned to each facility cannot exceed its capacity an opening cost is associated with each facility and is paid if at least one customer is assigned to it the objective is to minimize the total cost of opening the facilities and supply all the customers in this paper we extend the kernel search heuristic framework to general binary integer linear programming bilp problems and apply it to the sscflp the heuristic is based on the solution to optimality of a sequence of subproblems where each subproblem is restricted to a subset of the decision variables the subsets of decision variables are constructed starting from the optimal values of the linear relaxation variants based on variable fixing are proposed to improve the efficiency of the kernel search framework the algorithms are tested on benchmark instances and new very large scale test problems computational results demonstrate the effectiveness of the approach the kernel search algorithm outperforms the best heuristics for the sscflp available in the literature it found the optimal solution for 165 out of the 170 instances with a proven optimum the error achieved in the remaining instances is negligible moreover it achieved on 100 new very large scale instances an average gap equal to 0 64 computed with respect to a lower bound or the optimum when available the variants based on variable fixing improved the efficiency of the algorithm with minor deteriorations of the solution quality

an empirical study of instability and improvement of absorbing boundary conditions for the elastic wave equation one of the persistent problems with numerical solutions to the elastic wave equation is the finite size of the numerical grid as with a physical body the grid boundaries will reflect incident energy if not eliminated or reduced substantially these reflections will invade the grid interior and interfere with the desired solution one method for eliminating reflections is creating a large and or expanding grid this method may be impractical since it can be quite costly in both computer time and memory another method is making the grid boundary transparent to outgoing energy this method is ideally done by designing absorbing or nonreflecting boundaries which are mathematically equivalent to a one way or outgoing elastic wave equation only in practice an outgoing elastic wave equation is an approximation since the wave equation is not generally separable into outgoing and incoming parts two absorbing boundary condition approximations commonly used are those from reynolds reynolds 1978 and cla

solving chance constrained optimization problems with stochastic quadratic inequalities we propose a new and systematic reformulation and algorithmic approach to solve a complex class of stochastic programming problems involving a joint chance constraint with random technology matrix and stochastic quadratic inequalities the method is general enough to apply to nonconvex as well as nonseparable quadratic terms we derive two new reformulations and give sufficient conditions under which the reformulated problem is equivalent the second reformulation provides a much sparser representation of the feasible set of the chance constraint and offers tremendous computational advantages this new reformulation method can be used for linear stochastic inequalities and will also significantly improve the solution of such joint chance constrained problems we provide general and easily identifiable conditions under which the base reformulations can be linearized we show that the size of the reformulated problems in particular their number of binary variables and quadratic mixed integer terms does n

on mixing sets arising in chance constrained programming the mixing set with a knapsack constraint arises in deterministic equivalent of chance constrained programming problems with finite discrete distributions we first consider the case that the chance constrained program has equal probabilities for each scenario we study the resulting mixing set with a cardinality constraint and propose facet defining inequalities that subsume known explicit inequalities for this set we extend these inequalities to obtain valid inequalities for the mixing set with a knapsack constraint in addition we propose a compact extended reformulation with polynomial number of variables and constraints that characterizes a linear programming equivalent of a single chance constraint with equal scenario probabilities we introduce a blending procedure to find valid inequalities for intersection of multiple mixing sets we propose a polynomial size extended formulation for the intersection of multiple mixing sets with a knapsack constraint that is stronger than the original mixing formulation we also give a compact extended linear program for the intersection of multiple mixing sets and a cardinality constraint for a special case we illustrate the effectiveness of the proposed inequalities in our computational experiments with probabilistic lot sizing problems

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

an exact algorithm for the capacitated facility location problems with single sourcing facility location problems are often encountered in many areas such as distribution transportation and telecommunication we describe a new solution approach for the capacitated facility location problem in which each customer is served by a single facility an important class of heuristic solution methods for these problems are lagrangian heuristics which have been shown to produce high quality solutions and at the same time be quite robust a primal heuristic based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied is incorporated into the lagrangian heuristic finally a branch and bound method based on the lagrangian heuristic is developed and compared computationally to the commercial code cplex the computational results indicate that the proposed method is very efficient

shadow price of information in discrete time stochastic optimization the shadow price of information has played a central role in stochastic optimization ever since its introduction by rockafellar and wets in the mid seventies this article studies the concept in an extended formulation of the problem and gives relaxed sufficient conditions for its existence we allow for general adapted decision strategies which enables one to establish the existence of solutions and the absence of a duality gap e g in various problems of financial mathematics where the usual boundedness assumptions fail as applications we calculate conjugates and subdifferentials of integral functionals and conditional expectations of normal integrands we also give a dual form of the general dynamic programming recursion that characterizes shadow prices of information

continuous quadratic programming formulations of optimization problems on graphs four np hard optimization problems on graphs are studied the vertex separator problem the edge separator problem the maximum clique problem and the maximum independent set problem we show that the vertex separator problem is equivalent to a continuous bilinear quadratic program this continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem the maximum clique problem and the maximum independent set problem all of these formulations when expressed as maximization problems are shown to follow from the convexity properties of the objective function along the edges of the feasible set an algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators computational results are given

on optimal allocation of indivisibles under uncertainty the optimal allocation of indivisible resources is formalized as a stochastic optimization problem involving discrete decision variables a general stochastic search procedure is proposed which develops the concept of the branch and bound method the main idea is to process large collections of possible solutions and to devote more attention to the most promising groups by gathering more information to reduce the uncertainty and by narrowing the search area the optimal solution can be found with probability one special techniques for calculating stochastic lower and upper bounds are discussed the results are illustrated by a computational experiment

mathematical programming approaches for generating p efficient points abstract probabilistically constrained problems in which the random variables are finitely distributed are non convex in general and hard to solve the p efficiency concept has been widely used to develop efficient methods to solve such problems those methods require the generation of p efficient points pleps and use an enumeration scheme to identify pleps in this paper we consider a random vector characterized by a finite set of scenarios and generate pleps by solving a mixed integer programming mip problem we solve this computationally challenging mip problem with a new mathematical programming framework it involves solving a series of increasingly tighter outer approximations and employs as algorithmic techniques a bundle preprocessing method strengthening valid inequalities and a fixing strategy the method is exact resp heuristic and ensures the generation of pleps resp quasi pleps if the fixing strategy is not resp is employed and it can be used to generate multiple pleps to the best of our knowledge generating a set of pleps using an optimization based approach and developing effective methods for the application of the p efficiency concept to the random variables described by a finite set of scenarios are novel we present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques

erratum to threshold boolean form for joint probabilistic constraints with random technology matrix this note reports a correction and clarifies the results presented in 2 the third author of this corrigendum is not an author of the paper 2 in the original publication theorem 28 sect 4 2 1 p 410 states that the probabilistically constrained problem pmrtm can be reformulated as the quadratic integer optimization problem qde this statementmust be corrected since problemqde is in fact an inner approximation of the chance constrained problem pmrtm and the equivalence between qde and pmrtm can only be claimed under the specific settings presented below in the modified theorem 28 in part ii of theorem 28 in 2 p 411 it is shown that any feasible solution for the chance constrained problem pmrtm defined by a p sufficient recombination k is also feasible for the reformulated problem qde while that statement is correct this along with the fact any feasible solution of qde is feasible for pmrtm part

mip reformulations of the probabilistic set covering problem in this paper we address the following probabilistic version psc of the set covering problem min cx mathbb p ax ge xi ge p x in 0 1 n where a is a 0 1 matrix xi is a random 0 1 vector and p in 0 1 is the threshold probability level we introduce the concepts of p inefficiency and polarity cuts while the former is aimed at deriving an equivalent mip reformulation of psc the latter is used as a strengthening device to obtain a stronger formulation simplifications of the mip model which result when one of the following conditions hold are briefly discussed a is a balanced matrix a has the circular ones property the components of xi are pairwise independent the distribution function of xi is a stationary distribution or has the disjunctive shattering property we corroborate our theoretical findings by an extensive computational experiment on a test bed consisting of almost 10 000 probabilistic instances this test bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location warehouse location and k median models our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve psc and in many cases can reduce hours of computing time to a fraction of a second

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

aircrew schedule generation using repeated matching we consider the aircrew scheduling problem of determining tours of duty tods for aircrews given a set of sectors or flights requiring regular crews a regular crew consists of two crew members but by including supplementary crew a third pilot on some sectors it is possible to extend duty periods to generate more cost efficient tods a related problem is thus to generate tods for these third pilots but the sectors requiring a third pilot are not known in advance to solve these two related problems simultaneously we apply a heuristic procedure that solves a sequence of matching problems i e a repeated matching algorithm numerical results based on the solution of a real problem show that this approach is a valid and efficient method for solving the aircrew scheduling problem especially when there is the option of using supplementary crew to extend duty periods

convex duality in stochastic optimization and mathematical finance this paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration the framework unifies many well known duality frameworks from operations research and mathematical finance the unification allows the extension of some useful techniques from these two fields to a much wider class of problems in particular combining certain finite dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance we are able to close the duality gap in some situations where traditional topological arguments fail

absorbing boundary conditions and surface waves one of the major problems in numerically simulating waves traveling in the earth is that an artificial boundary must be introduced to produce unique solutions to eliminate the spurious reflections introduced by this artificial boundary we use a damping expression based on analogies to shock absorbers this method can reduce the amplitude of the reflected wave to any pre specified value and is successful for waves at any angle of incidence the method can eliminate unwanted reflections from the surface reflections at the corners of the model and waves reflected off an interface that strike the artificial boundary many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary stability problems often occur with these boundary conditions the method we use at the artificial boundary allows use of stable dirichlet or von neumann condition

chance constrained problems and rare events an importance sampling approach we study chance constrained problems in which the constraints involve the probability of a rare event we discuss the relevance of such problems and show that the existing sampling based algorithms cannot be applied directly in this case since they require an impractical number of samples to yield reasonable solutions we argue that importance sampling is techniques combined with a sample average approximation saa approach can be effectively used in such situations provided that variance can be reduced uniformly with respect to the decision variables we give sufficient conditions to obtain such uniform variance reduction and prove asymptotic convergence of the combined saa is approach as it often happens with is techniques the practical performance of the proposed approach relies on exploiting the structure of the problem under study in our case we work with a telecommunications problem with bernoulli input distributions and show how variance can be reduced uniformly over a suitable approximation of the feasibility set by choosing proper parameters for the is distributions although some of the results are specific to this problem we are able to draw general insights that can be useful for other classes of problems we present numerical results to illustrate our findings

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

random convex programs random convex programs rcps are convex optimization problems subject to a finite number n of random constraints the optimal objective value j of an rcp is thus a random variable we study the probability with which j is no longer optimal if a further random constraint is added to the problem violation probability v it turns out that this probability rapidly concentrates near zero as n increases we first develop a theory for rcps leading to explicit bounds on the upper tail probability of v then we extend the setup to the case of rcps with r a posteriori violated constraints rcpvs a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control explicit and nonasymptotic bounds are derived also in this case the upper tail probability of v is upper bounded by a multiple of a beta distribution irrespective of the distribution on the random constraints all results are derived under no feasibility assumptions on the problem further the relation between rcpvs and chance constrained problems ccp is explored showing that the optimal objective j of an rcpv with the generic constraint removal rule provides with arbitrarily high probability an upper bound on the optimal objective of a corresponding ccp moreover whenever an optimal constraint removal rule is used in the rcpvs then appropriate choices of n and r exist such that j approximates arbitrarily well the objective of the ccp

a hypergraph multi exchange heuristic for the single source capacitated facility location problem abstract in this paper we introduce a large scale neighborhood search procedure for solving the single source capacitated facility location problem sscflp the neighborhood structures are induced by innovative split multi customer multi exchanges where clusters of customers assigned to one facility can be moved simultaneously to multiple destination facilities and vice versa to represent these exchanges we use two types of improvement hypergraphs the improvement hypergraphs are built dynamically and the moving customers associated with each hyperedge are selected by solving heuristically a suitably defined mixed integer program we develop a hypergraph search framework including forward and backward procedures to identify improving solutions efficiently our proposed algorithm can obtain improving moves more quickly and even find better solutions than a traditional multi exchange heuristic ahuja orlin pallottino scuttella 2004 in addition when compared with the kernel search algorithm guastaroba speranza 2014 which at present is the most effective for solving sscflp our algorithm is not only competitive but can find better solutions or even the best known solution to some very large scale benchmark instances from the literature

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

stochastic programs without duality gaps this paper studies dynamic stochastic optimization problems parameterized by a random variable such problems arise in many applications in operations research and mathematical finance we give sufficient conditions for the existence of solutions and the absence of a duality gap our proof uses extended dynamic programming equations whose validity is established under new relaxed conditions that generalize certain no arbitrage conditions from mathematical finance

artificial boundary conditions for 2d problems in geophysics abstract the numerical solution of a class of two dimensional problems in geophysics is considered the earth is modeled as a semi infinite acoustic or elastic medium in order to make the computational domain finite a semi circular artificial boundary ob is introduced thus truncating the original semi infinite domain an exact nonlocal relation is then derived on ob this relation is used as a boundary condition on ob which is combined with the finite element scheme employed in the computational domain the approach is applied to the helmholtz equation modeling time harmonic acoustic waves and to problems in elastostatics numerical examples validate the method and demonstrate its performance

bicriteria approximation of chance constrained covering problems a chance constrained optimization problem involves constraints with random data that can be violated with probability bounded from above by a prespecified small risk parameter such constraints are

a copositive programming approach to graph partitioning we consider 3 partitioning the vertices of a graph into sets s 1 s 2 and s 3 of specified cardinalities such that the total weight of all edges joining s 1 and s 2 is minimized this problem is closely related to several np hard problems like determining the bandwidth or finding a vertex separator in a graph we show that this problem can be formulated as a linear program over the cone of completely positive matrices leading in a natural way to semidefinite relaxations of the problem we show in particular that the spectral relaxation introduced by helmberg e 1995 can equivalently be formulated as a semidefinite program finally we propose a tightened version of this semidefinite program and show on some small instances that this new bound is a significant improvement over the spectral bound

optimizing over the split closure the polyhedron defined by all the split cuts obtainable directly i e without iterated cut generation from the lp relaxation p of a mixed integer program mip is termed the elementary or rank 1 split closure of p this paper deals with the problem of optimizing over the elementary split closure this is accomplished by repeatedly solving the following separation problem given a fractional point say x find a rank 1 split cut violated by x or show that none exists following caprara and letchford 17 we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program pmilp with a single parameter in the objective function and the right hand side we develop an algorithmic framework to deal with the resulting pmilp by creating and maintaining a dynamically updated grid of parameter values and use the corresponding mixed integer programs to generate rank 1 split cuts our approach was implemented in the coin or framework using cplex 9 0 as a general purpose mip solver we report our computational results on well known benchmark instances from miplib 3 0 and several classes of structured integer and mixed integer problems our computational results show that rank 1 split cuts close more than 98 of the duality gap on 15 out of 41 mixed integer instances from miplib 3 0 more than 75 of the duality gap can be closed on an additional 10 instances the average gap closed over all 41 instances is 72 78 in the pure integer case rank 1 split cuts close more than 75 of the duality gap on 13 out of 24 instances from miplib 3 0 on average rank 1 split cuts close about 72 of the duality gap on these 24 instances we also report results on several classes of structured problems capacitated versions of warehouse location single source facility location p median fixed charge network flow multi commodity network design with splittable and unsplittable flows and lot sizing the fraction of the integrality gap closed varies for these problem classes between 100 and 67 we also gathered statistics on the average coefficient size absolute value of the disjunctions generated they turn out to be surprisingly small