stochastic programming with integer variables including integer variables into traditional stochastic linear programs has considerable implications for structural analysis and algorithm design starting from mean risk approaches with different risk measures we identify corresponding two and multi stage stochastic integer programs that are large scale block structured mixed integer linear programs if the underlying probability distributions are discrete we highlight the role of mixed integer value functions for structure and stability of stochastic integer programs when applied to the block structures in stochastic integer programming well known algorithmic principles such as branch and bound lagrangian relaxation or cutting plane methods open up new directions of research we review existing results in the field and indicate departure points for their extension

the ancestral benders cutting plane algorithm with multi term disjunctions for mixed integer recourse decisions in stochastic programming this paper focuses on solving two stage stochastic mixed integer programs smips with general mixed integer decision variables in both stages we develop a decomposition algorithm in which the first stage approximation is solved by a branch and bound algorithm with its nodes inheriting benders cuts that are valid for their ancestor nodes in addition we develop two closely related convexification schemes which use multi term disjunctive cuts to obtain approximations of the second stage mixed integer programs we prove that the proposed methods are finitely convergent one of the main advantages of our decomposition scheme is that we use a benders based branch and cut approach in which linear programming approximations are strengthened sequentially moreover as in many decomposition schemes these subproblems can be solved in parallel we also illustrate these algorithms using several variants of an smip example from the literature as well as a new set of test problems which we refer to as stochastic server location and sizing finally we present our computational experience with previously known examples as well as the new collection of smip instances our experiments reveal that our algorithm is able to produce provably optimal solutions within an hour of cpu time even in instances for which a highly reliable commercial mip solver is unable to provide an optimal solution within an hour of cpu time

stochastic set packing problem abstract in this paper a stochastic version of the set packing problem spp is studied via scenario analysis we consider a one stage recourse approach to deal with the uncertainty in the coefficients it consists of maximizing in the stochastic spp a composite function of the expected value minus the weighted risk of obtaining a scenario whose objective function value is worse than a given threshold the splitting variable representation is decomposed by dualizing the nonanticipativity constraints that link the deterministic spp with a 0 1 knapsack problem for each scenario under consideration as a result a structured larger pure 0 1 model is created we present several procedures for obtaining good feasible solutions as well as a preprocessing approach for fixing variables the lagrange multipliers updating is performed by using the volume algorithm computational experience is reported for a broad variety of instances which shows that the new approach usually outperforms a state of the art optimization engine producing a comparable optimality gap with smaller several orders of magnitude computing time

a general procedure for learning mixtures of independent component analyzers this paper presents a new procedure for learning mixtures of independent component analyzers the procedure includes non parametric estimation of the source densities supervised unsupervised learning of the model parameters incorporation of any independent component analysis ica algorithm into the learning of the ica mixtures and estimation of residual dependencies after training for correction of the posterior probability of every class to the testing observation vector we demonstrate the performance of the procedure in the classification of ica mixtures of two three and four classes of synthetic data and in the classification of defective materials consisting of 3d finite element models and lab specimens in non destructive testing using the impact echo technique the application of the proposed posterior probability correction demonstrates an improvement in the classification accuracy semi supervised learning shows that unlabeled data can degrade the performance of the classifier when they do not fit the generative model comparative results of the proposed method and standard ica algorithms for blind source separation in one and multiple ica data mixtures show the suitability of the non parametric ica mixture based method for data modeling

decomposition of test sets in stochastic integer programming we study graver test sets for linear two stage stochastic integer programs and show that test sets can be decomposed into finitely many building blocks whose number is independent on the number of scenarios of the stochastic program we present a finite algorithm to compute the building blocks directly without prior knowledge of test set vectors once computed building blocks can be employed to solve the stochastic program by a simple augmentation scheme again without explicit knowledge of test set vectors finally we report preliminary computational experience

a case study of joint online truck scheduling and inventory management for multiple warehouses for a real world problem transporting pallets between warehouses to guarantee sufficient supply for known and additional stochastic demand we propose a solution approach via convex relaxation of an integer programming formulation suitable for online optimization the essential new element linking routing and inventory management is a convex piecewise linear cost function that is based on minimizing the expected number of pallets that still need transportation for speed the convex relaxation is solved approximately by a bundle approach yielding an online schedule in five to 12 minutes for up to three warehouses and 40 000 articles in contrast computation times of state of the art lp solvers are prohibitive for online application in extensive numerical experiments on a real world data stream the approximate solutions exhibit negligible loss in quality in long term simulations the proposed method reduces the average number of pallets needing transportation due to short term demand to less than half the number observed in the data stream

expectation and chance constrained models and algorithms for insuring critical paths in this paper we consider a class of two stage stochastic optimization problems arising in the protection of vital arcs in a critical path network a project is completed after a series of dependent tasks are all finished we analyze a problem in which task finishing times are uncertain but can be insured a priori to mitigate potential delays a decision maker must trade off costs incurred in insuring arcs with expected penalties associated with late project completion times where lateness penalties are assumed to be lower semicontinuous nondecreasing functions of completion time we provide decomposition strategies to solve this problem with respect to either convex or nonconvex penalty functions in particular for the nonconvex penalty case we employ the reformulation linearization technique to make the problem amenable to solution via benders decomposition we also consider a chance constrained version of this problem in which the probability of completing a project on time is sufficiently large we demonstrate the computational efficacy of our approach by testing a set of size and complexity diversified problems using the sample average approximation method to guide our scenario generation

a closed form representation of mixed integer program value functions we identify a class of formulas computable in polynomial time such that the functions defined by these formulas are precisely the value functions of mixed integer programs with rational constraint coefficients

exact nonlinear control of large angle rotational maneuvers the rigid body attitude control problem with external torques is transformed into an equivalent linear form implementable by three double integrators the linearizing transformations themselves are formulated in vector algebra requiring no integrators for implementation it is thereby shown that optimal command generation for fast slewing maneuvers can be carried out exactly in the transformed system together with regulator design without gain scheduling for correction of unmodeled disturbances in particular a general rest to rest maneuver is computed in closed form and coupled with a previously obtained exact detumbling maneuver so that arbitrary initial conditions can be accommodated an illustrative simulation is appended

probabilistic distance for mixtures of independent component analyzers independent component analysis ica is a blind source separation technique where data are modeled as linear combinations of several independent non gaussian sources the independence and linear restrictions are relaxed using several ica mixture models icamms obtaining a two layer artificial neural network structure this allows for dependence between sources of different classes and thus a myriad of multidimensional probability density functions can be accurate modeled this paper proposes a new probabilistic distance pdi between the parameters learned for two icamms the pdi is computed explicitly unlike the popular kullback leibler divergence kld and other similar metrics removing the need for numerical integration furthermore the pdi is symmetric and bounded within 0 and 1 which enables its use as a posterior probability in fusion approaches in this paper the pdi is employed for change detection by measuring the distance between two icamms learned in consecutive time windows the changes might be associated with relevant states from a process under analysis that are explicitly reflected in the learned icamm parameters the proposed distance was tested in two challenging applications using simulated and real data 1 detecting flaws in materials using ultrasounds and 2 detecting changes in electroencephalography signals from humans performing neuropsychological tests the results demonstrate that the pdi outperforms the kld in change detection capabilities

bfc a branch and fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0 1 programs we present a framework for solving some types of 0 1 multi stage scheduling planning problems under uncertainty in the objective function coe cients and the right hand side a scenario analysis scheme with full recourse is used the solution o ered for each scenario group at each stage takes into account all scenarios but without subordinating to any of them the constraints are modelled by a splitting variables representation via scenarios so a 0 1 model for each scenario is consid ered plus the non anticipativity constraints that equate the 0 1 variables from the same group of scenarios in each stage the mathematical representation of the model is very amenable for the proposed framework to deal with the 0 1 character of the variables a branch and fix coordi nation approach is introduced for coordinating the selection of the branching nodes and branching variables in the scenario subproblems to be jointly optimized some computational experience is reported for di erent types of problems

two stage quadratic integer programs with stochastic right hand sides we consider two stage quadratic integer programs with stochastic right hand sides and present an equivalent reformulation using value functions we propose a two phase solution approach the first phase constructs value functions of quadratic integer programs in both stages the second phase solves the reformulation using a global branch and bound algorithm or a level set approach we derive some basic properties of value functions of quadratic integer programs and utilize them in our algorithms we show that our approach can solve instances whose extensive forms are hundreds of orders of magnitude larger than the largest quadratic integer programming instances solved in the literature

on a level set characterization of the value function of an integer program and its application to stochastic programming we propose a level set approach to characterize the value function of a pure linear integer program with inequality constraints we study theoretical properties of our characterization and show how they can be exploited to optimize a class of stochastic integer programs through a value function reformulation specifically we develop algorithmic approaches that solve two stage multidimensional knapsack problems with random budgets yielding encouraging computational results

probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra we consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions they can be reformulated as large scale mixed integer programming problems with knapsack constraints using specific properties of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities for these mixed integer programming problems we also develop methods for lifting these inequalities these procedures are used in a general iterative algorithm for solving probabilistically constrained problems the results are illustrated with a numerical example

generalized likelihood ratios for testing the properness of quaternion gaussian vectors in a recent paper the second order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing which are associated with the two main types of quaternion properness in this paper we consider the problem of determining from a finite number of independent vector observations whether a quaternion gaussian vector is proper or not specifically we derive three generalized likelihood ratio tests glrts for testing the two main kinds of quaternion properness and show that the glrts reduce to the estimation of three previously proposed quaternion improperness measures interestingly the three glrt statistics improperness measures can be interpreted as an estimate of the entropy loss due to the quaternion improperness additionally we analyze the case in which the orthogonal basis for the representation of the quaternion vector is unknown which results in the problem of estimating the principal bbc properness direction i e the pure unit quaternion minimizing the bbc improperness measure although this estimation problem is not convex we propose a technique based on successive convex approximations which can be solved in closed form finally some simulation examples illustrate the performance and practical application of the proposed tests

the attitude control problem a general framework for the analysis of the attitude tracking control problem for a rigid body is presented a large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body the controllers share the common structure of a proportional derivative feedback plus some feedforward which can be zero the model independent case the coriolis torque compensation or an adaptive compensation these controller structures are compared in terms of the requirement on the a priori model information guaranteed transient performance and robustness the global stability of the luh walker paul robot end effector controller is also analyzed in this framework

integration of gibbs markov random field and hopfield type neural networks for unsupervised change detection in remotely sensed multitemporal images in this paper a spatiocontextual unsupervised change detection technique for multitemporal multispectral remote sensing images is proposed the technique uses a gibbs markov random field gmrf to model the spatial regularity between the neighboring pixels of the multitemporal difference image the difference image is generated by change vector analysis applied to images acquired on the same geographical area at different times the change detection problem is solved using the maximum a posteriori probability map estimation principle the map estimator of the gmrf used to model the difference image is exponential in nature thus a modified hopfield type neural network htnn is exploited for estimating the map in the considered hopfield type network a single neuron is assigned to each pixel of the difference image and is assumed to be connected only to its neighbors initial values of the neurons are set by histogram thresholding an expectation maximization algorithm is used to estimate the gmrf model parameters experiments are carried out on three multispectral and multitemporal remote sensing images results of the proposed change detection scheme are compared with those of the manual trial and error technique automatic change detection scheme based on gmrf model and iterated conditional mode algorithm a context sensitive change detection scheme based on htnn the gmrf model and a graph cut algorithm a comparison points out that the proposed method provides more accurate change detection maps than other methods

an approximation framework for two stage ambiguous stochastic integer programs under mean mad information abstract we consider two stage recourse models in which only limited information is available on the probability distributions of the random parameters in the model if all decision variables are continuous then we are able to derive the worst case and best case probability distributions under the assumption that only the means and mean absolute deviations of the random parameters are known contrary to most existing results in the literature these probability distributions are the same for every first stage decision the ambiguity set that we use in this paper also turns out to be particularly suitable for ambiguous recourse models involving integer decisions variables for such problems we develop a general approximation framework and derive error bounds for using these approximatons we apply this approximation framework to mixed ambiguous mixed integer recourse models in which some of the probability distributions of the random parameters are known and others are ambiguous to illustrate these results we carry out numerical experiments on a surgery block allocation problem

a heuristic procedure for stochastic integer programs with complete recourse in this paper we propose a successive approximation heuristic which solves large stochastic mixed integer programming problem with complete fixed recourse we refer to this method as the scenario updating method since it solves the problem by considering only a subset of scenarios which is updated at each iteration only those scenarios which imply a significant change in the objective function are added the algorithm is terminated when no such scenarios are available to enter in the current scenario subtree several rules to select scenarios are discussed bounds on heuristic solutions are computed by relaxing some of the non anticipativity constraints the proposed procedure is tested on a multistage stochastic batch sizing problem

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

a finite branch and bound algorithm for two stage stochastic integer programs this paper addresses a general class of two stage stochastic programs with integer recourse and discrete distributions we exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm the proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature

minimum cardinality non anticipativity constraint sets for multistage stochastic programming we consider multistage stochastic programs in which decisions can adapt over time i e at each stage in response to observation of one or more random variables uncertain parameters the case that the time at which each observation occurs is decision dependent known as stochastic programming with endogeneous observation of uncertainty presents particular challenges in handling non anticipativity although such stochastic programs can be tackled by using binary variables to model the time at which each endogenous uncertain parameter is observed the consequent conditional non anticipativity constraints form a very large class with cardinality in the order of the square of the number of scenarios however depending on the properties of the set of scenarios considered only very few of these constraints may be required for validity of the model here we characterize minimal sufficient sets of non anticipativity constraints and prove that their matroid structure enables sets of minimum cardinality to be found efficiently under general conditions on the structure of the scenario set

convexity and decomposition of mean risk stochastic programs traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criterion a common approach to addressing risk in decision making problems is to consider a weighted mean risk objective where some dispersion statistic is used as a measure of risk we investigate the computational suitability of various mean risk objective functions in addressing risk in stochastic programming models we prove that the classical mean variance criterion leads to computational intractability even in the simplest stochastic programs on the other hand a number of alternative mean risk functions are shown to be computationally tractable using slight variants of existing stochastic programming decomposition algorithms we propose decomposition based parametric cutting plane algorithms to generate mean risk efficient frontiers for two particular classes of mean risk objectives

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

quasi monte carlo methods for two stage stochastic mixed integer programs we consider randomized qmc methods for approximating the expected recourse in two stage stochastic optimization problems containing mixed integer decisions in the second stage it is known that the second stage optimal value function is piecewise linear quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets this structure is exploited to provide conditions implying that first and higher order terms of the integrand s anova decomposition math comp 79 2010 953 966 have mixed weak first order partial derivatives this leads to a good smooth approximation of the integrand and hence to good convergence rates of randomized qmc methods if the effective superposition dimension is low

bounds on risk averse mixed integer multi stage stochastic programming problems with mean cvar abstract risk averse mixed integer multi stage stochastic programming forms a class of extremely challenging problems since the problem size grows exponentially with the number of stages the problem is non convex due to integrality restrictions and the objective function is nonlinear in general we propose a scenario tree decomposition approach namely group subproblem approach to obtain bounds for such problems with an objective of dynamic mean conditional value at risk mean cvar our approach does not require any special problem structure such as convexity and linearity therefore it can be applied to a wide range of problems we obtain lower bounds by using different convolution of mean cvar risk measures and different scenario partition strategies the upper bounds are obtained through the use of optimal solutions of group subproblems using these lower and upper bounds we propose a solution algorithm for risk averse mixed integer multi stage stochastic problems with mean cvar risk measures we test the performance of the proposed algorithm on a multi stage stochastic lot sizing problem and compare different choices of lower bounds and partition strategies comparison of the proposed algorithm to a commercial solver revealed that on the average the proposed algorithm yields 1 13 stronger bounds the commercial solver requires additional running time more than a factor of five on the average to reach the same optimality gap obtained by the proposed algorithm

on the convergence of cutting plane algorithms for a class of nonconvex mathematical programs we discuss the convergence of cutting plane algorithms for a class of nonconvex programs called the generalized lattice point problems glpp a set of sufficient conditions which guarantee finite convergence are presented although these conditions are usually difficult to enforce in a practical implementation they do illustrate the various factors that must be involved in a convergent rudimentary cutting plane algorithm a striking example of nonconvergence in which no subsequence converges to a feasible solution even when seemingly strong cutting planes are used is presented to show the effect of neglecting one such factor we give an application of our analysis to problems with multiple choice constraints and finally discuss a modification of cutting plane algorithms so as to make finite convergence more readily implementable

optimization in quaternion dynamic systems gradient hessian and learning algorithms the optimization of real scalar functions of quaternion variables such as the mean square error or array output power underpins many practical applications solutions typically require the calculation of the gradient and hessian however real functions of quaternion variables are essentially nonanalytic which are prohibitive to the development of quaternion valued learning systems to address this issue we propose new definitions of quaternion gradient and hessian based on the novel generalized hamilton real ghr calculus thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field rather than using the isomorphism with the real domain as is current practice in addition unlike the existing quaternion gradients the ghr calculus allows for the product and chain rule and for a one to one correspondence of the novel quaternion gradient and hessian with their real counterparts properties of the quaternion gradient and hessian relevant to numerical applications are also introduced opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms the proposed ghr calculus is shown to yield the same generic algorithm forms as the corresponding real and complex valued algorithms advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks

lagrangian solution techniques and bounds for loosely coupled mixed integer stochastic programs many production problems involve facility setups that lead to integer variables production decisions that are continuous and demands that are likely to be random while these problems can be quite difficult to solve we propose a model and an efficient solution technique for this basic class of stochastic mixed integer programs we use a set of scenarios to reflect uncertainty the resulting mathematical model is solved using lagrangian relaxation we show that the duality gap of our relaxation is bounded above by a constant that depends on the cost function and the number of branching points in the scenario tree we apply our technique to the problem of generating electric power numerical results indicate significant savings when the stochastic model is used instead of a deterministic one

decomposition algorithms with parametric gomory cuts for two stage stochastic integer programs we consider a class of two stage stochastic integer programs with binary variables in the rst stage and general integer variables in the second stage we develop decomposition algorithms akin to the l shaped or benders methods by utilizing gomory cuts to obtain iteratively tighter approximations of the second stage integer programs we show that the proposed methodology is exible in that it allows several modes of implementation all of which lead to nitely convergent algorithms we illustrate our algorithms using examples from the literature we report computational results using the stochastic server location problem instances which suggest that our decomposition based approach scales better with increases in the number of scenarios than a state of the art solver which was used to solve the deterministic equivalent formulation

ica mixture models for unsupervised classification of non gaussian classes and automatic context switching in blind signal separation an unsupervised classification algorithm is derived by modeling observed data as a mixture of several mutually exclusive classes that are each described by linear combinations of independent non gaussian densities the algorithm estimates the density of each class and is able to model class distributions with non gaussian structure the new algorithm can improve classification accuracy compared with standard gaussian mixture models when applied to blind source separation in nonstationary environments the method can switch automatically between classes which correspond to contexts with different mixing properties the algorithm can learn efficient codes for images containing both natural scenes and text this method shows promise for modeling non gaussian structure in high dimensional data and has many potential applications

a class of stochastic programs with decision dependent uncertainty we address a class of problems where decisions have to be optimized over a time horizon given that the future is uncertain and that the optimization decisions influence the time of information discovery for a subset of the uncertain parameters the standard approach to formulate stochastic programs is based on the assumption that the stochastic process is independent of the optimization decisions which is not true for the class of problems under consideration we present a hybrid mixed integer disjunctive programming formulation for the stochastic program corresponding to this class of problems and hence extend the stochastic programming framework a set of theoretical properties that lead to reduction in the size of the model is identified a lagrangean duality based branch and bound algorithm is also presented

l shaped decomposition of two stage stochastic programs with integer recourse we consider two stage stochastic programming problems with integer recourse the l shaped method of stochastic linear programming is generalized to these problems by using generalized benders decomposition nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques finite convergence of the method is established when gomory s fractional cutting plane algorithm or a branch and bound algorithm is applied

a class of quaternion kalman filters the existing kalman filters for quaternion valued signals do not operate fully in the quaternion domain and are combined with the real kalman filter to enable the tracking in 3 d spaces using the recently introduced hr calculus we develop the fully quaternion valued kalman filter qkf and quaternion extended kalman filter qekf allowing for the tracking of 3 d and 4 d signals directly in the quaternion domain to consider the second order noncircularity of signals we employ the recently developed augmented quaternion statistics to derive the widely linear qkf wl qkf and widely linear qekf wl qekf to reduce computational requirements of the widely linear algorithms their efficient implementation are proposed and it is shown that the quaternion widely linear model can be simplified when processing 3 d data further reducing the computational requirements simulations using both synthetic and real world circular and noncircular signals illustrate the advantages offered by widely linear over strictly linear quaternion kalman filters

solving stochastic programs with integer recourse by enumeration a framework using gro bner basis reductions in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

two stage integer programs with stochastic right hand sides a superadditive dual approach we consider two stage pure integer programs with discretely distributed stochastic right hand sides we present an equivalent superadditive dual formulation that uses the value functions in both stages we give two algorithms for finding the value functions to solve the reformulation after obtaining the value functions we develop a global branch and bound approach and a level set approach to find an optimal tender we show that our method can solve randomly generated instances whose extensive forms are several orders of magnitude larger than the extensive forms of those instances found in the literature

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

continuity properties of expectation functions in stochastic integer programming sufficient conditions for the lipschitz continuity of the expectation of second stage costs are given for two stage stochastic programs where the optimization problem in the second stage is a mixed integer linear program we also present counterexamples to show that in general the results can no longer be maintained when relaxing assumptions as well as multivariate probability distributions for which the theory works

decomposition algorithms for two stage distributionally robust mixed binary programs in this paper we introduce and study a two stage distributionally robust mixed binary problem tsdr mbp in which the random parameters follow the worst case distribution belonging to an uncertain

model based classification methods of global patterns in dermoscopic images in this paper different model based methods of classification of global patterns in dermoscopic images are proposed global patterns identification is included in the pattern analysis framework the melanoma diagnosis method most used among dermatologists the modeling is performed in two senses first a dermoscopic image is modeled by a finite symmetric conditional markov model applied to l a b color space and the estimated parameters of this model are treated as features in turn the distribution of these features are supposed that follow different models along a lesion a gaussian model a gaussian mixture model and a bag of features histogram model for each case the classification is carried out by an image retrieval approach with different distance metrics the main objective is to classify a whole pigmented lesion into three possible patterns globular homogeneous and reticular an extensive evaluation of the performance of each method has been carried out on an image database extracted from a public atlas of dermoscopy the best classification success rate is achieved by the gaussian mixture model based method with a 78 44 success rate in average in a further evaluation the multicomponent pattern is analyzed obtaining a 72 91 success rate

risk aversion via excess probabilities in stochastic programs with mixed integer recourse we consider linear two stage stochastic programs with mixed integer recourse instead of basing the selection of an optimal first stage solution on expected costs alone we include into the objective a risk term reflecting the probability that a preselected cost threshold is exceeded after we have put the resulting mean risk model into perspective with stochastic dominance we study further structural properties of the model and derive some basic stability results in the algorithmic part of the paper we propose a scenario decomposition method and report initial computational experience

a comparison between hmlp and hrbf for attitude control in this paper the problem of controlling the attitude of a rigid body such as a spacecraft in three dimensional space is approached by introducing two new control strategies developed in hypercomplex algebra the proposed approaches are based on two parallel controllers both derived in quaternion algebra the first is a feedback controller of the proportional derivative pd type while the second is a feedforward controller which is implemented either by means of a hypercomplex multilayer perceptron hmlp neural network or by means of a hypercomplex radial basis function hrbf neural network several simulations show the performance of the two approaches the results are also compared with a classical pd controller and with an adaptive controller showing the improvements obtained by using neural networks especially when an external disturbance acts on the rigid body in particular the hmlp network gave better results when considering trajectories not presented during the learning phase

a spectral bundle method with bounds semidefinite relaxations of quadratic 0 1 programming or graph partitioning problems are well known to be of high quality however solving them by primal dual interior point methods can take much time even for problems of moderate size the recent spectral bundle method of helmberg and rendl can solve quite efficiently large structured equality constrained semidefinite programs if the trace of the primal matrix variable is fixed as happens in many applications we extend the method so that it can handle inequality constraints without seriously increasing computation time in addition we introduce inexact null steps this abolishes the need of computing exact eigenvectors for subgradients which brings along significant advantages in theory and in practice encouraging preliminary computational results are reported

stochastic programming with simple integer recourse stochastic integer programs are notoriously difficult very few properties are known and solution algorithms are very scarce in this paper we introduce the class of stochastic programs with simple integer recourse a natural extension of the simple recourse case extensively studied in stochastic continuous programs analytical as well as computational properties of the expected recourse function of simple integer recourse problems are studied this includes sharp bounds on this function and the study of the convex hull finally a finite termination algorithm is obtained that solves two classes of stochastic simple integer recourse problems

cutting planes for the multistage stochastic unit commitment problem as renewable energy penetration rates continue to increase in power systems worldwide new challenges arise for system operators in both regulated and deregulated electricity markets to solve the security constrained coal fired unit commitment problem with intermittent generation due to renewables and uncertain load in order to ensure system reliability and maintain cost effectiveness in this paper we study a security constrained coal fired stochastic unit commitment model which we use to enhance the reliability unit commitment process for day ahead power system operations in our approach we first develop a deterministic equivalent formulation for the problem which leads to a large scale mixed integer linear program then we verify that the turn on off inequalities provide a convex hull representation of the minimum up down time polytope under the stochastic setting next we develop several families of strong valid inequalities mainly through lifting schemes in particular by exploring sequence independent lifting and subadditive approximation lifting properties for the lifting schemes we obtain strong valid inequalities for the ramping and general load balance polytopes finally branch and cut algorithms are developed to employ these valid inequalities as cutting planes to solve the problem our computational results verify the effectiveness of the proposed approach

stochastic survivable network design problems theory and practice abstract we study survivable network design problems with edge connectivity requirements under a two stage stochastic model with recourse and finitely many scenarios for the formulation in the natural space of edge variables we show that facet defining inequalities of the underlying polytope can be derived from the deterministic counterparts moreover by using graph orientation properties we introduce stronger cut based formulations for solving the proposed mixed integer programing models we suggest a two stage branch cut algorithm based on a decomposed model in order to accelerate the computations we suggest a new technique for strengthening the decomposed l shaped optimality cuts which is computationally fast and easy to implement a computational study shows the benefit of the decomposition and the cut strengthening which significantly reduces the number of master iterations and the computational running time moreover we evaluate the stability of the scenario generation method and analyze the value of the stochastic solution

dantzig wolfe decomposition for solving multistage stochastic capacity planning problems we describe a multistage stochastic mixed integer programming model for planning capacity expansion of production facilities a scenario tree represents uncertainty in the model a general mixed integer program defines the operational submodel at each scenario tree node and capacity expansion decisions link the stages we apply variable splitting to two model variants and solve those variants using dantzig wolfe decomposition the dantzig wolfe master problem can have a much stronger linear programming relaxation than is possible without variable splitting over 700 stronger in one case the master problem solves easily and tends to yield integer solutions obviating the need for a full branch and price solution procedure for each scenario tree node the decomposition defines a subproblem that may be viewed as a single period deterministic capacity planning problem an effective solution procedure results as long as the subproblems solve efficiently and the procedure incorporates a good duals stabilization method we present computational results for a model to plan the capacity expansion of an electricity distribution network in new zealand given uncertain future demand the largest problem we solve to optimality has six stages and 243 scenarios and corresponds to a deterministic equivalent with a quarter of a million binary variables

modeling the amplitude statistics of ultrasonic images in this paper a new statistical model for representing the amplitude statistics of ultrasonic images is presented the model is called the rician inverse gaussian riig distribution due to the fact that it is constructed as a mixture of the rice distribution and the inverse gaussian distribution the probability density function pdf of the riig model is given in closed form as a function of three parameters some theoretical background on this new model is discussed and an iterative algorithm for estimating its parameters from data is given then the appropriateness of the riig distribution as a model for the amplitude statistics of medical ultrasound images is experimentally studied it is shown that the new distribution can fit to the various shapes of local histograms of linearly scaled ultrasound data better than existing models a log likelihood cross validation comparison of the predictive performance of the riig the k and the generalized nakagami models turns out in favor of the new model furthermore a maximum a posteriori map filter is developed based on the riig distribution experimental studies show that the riig map filter has excellent filtering performance in the sense that it smooths homogeneous regions and at the same time preserves details

disjunctive decomposition for two stage stochastic mixed binary programs with random recourse this paper introduces disjunctive decomposition for two stage mixed 0 1 stochastic integer programs sips with random recourse disjunctive decomposition allows for cutting planes based on disjunctive programming to be generated for each scenario subproblem under a temporal decomposition setting of the sip problem a new class of valid inequalities for mixed 0 1 sip with random recourse is presented in particular we derive valid inequalities that allow for scenario subproblems for sip with random recourse but deterministic technology matrix and right hand side vector to share cut coefficients the valid inequalities are used to derive a disjunctive decomposition method whose derivation has been motivated by real life stochastic server location problems with random recourse which find many applications in operations research computational results with large scale instances to demonstrate the potential of the method are reported

tight second stage formulations in two stage stochastic mixed integer programs we study two stage stochastic mixed integer programs tss mips with integer variables in the second stage we show that under suitable conditions the second stage mips can be convexified by adding parametric cuts a priori as special cases we extend the results of miller and wolsey math program 98 2003 pp 73 88 to tss mips furthermore we consider second stage programs that are generalizations of the well known mixing and continuous mixing set or certain piecewise linear convex objective integer programs these results allow us to relax the integrality restrictions on the second stage integer variables without effecting the integrality of the optimal solution of the tss mip we also use four variants of the two stage stochastic capacitated lot sizing problems as test problems for computational experiments and present tight second stage formulations for these problems our computational results show that adding parametric inequalities that a priori convexify the second stage formulation si

a convex approximation for two stage mixed integer recourse models with a uniform error bound we develop a convex approximation for two stage mixed integer recourse models and we derive an error bound for this approximation that depends on the total variations of the probability density functions of the random variables in the model we show that the error bound converges to zero if all these total variations converge to zero our convex approximation is a generalization of the one in romeijnders van der vlerk and klein haneveld math program to appear restricted to totally unimodular integer recourse models for this special case it has the best worst case error bound possible the error bound in this paper is the first in the general setting of mixed integer recourse models as main building blocks in its derivation we generalize the asymptotic periodicity results of gomory linear algebra appl 2 1969 pp 451 558 for pure integer programs to the mixed integer case and we use the total variation error bounds on the expectation of periodic functions derived in romeijnders van der v

k adaptability in two stage robust binary programming over the last two decades robust optimization has emerged as a computationally attractive approach to formulate and solve single stage decision problems affected by uncertainty more recently robust optimization has been successfully applied to multistage problems with continuous recourse this paper takes a step toward extending the robust optimization methodology to problems with integer recourse which have largely resisted solution so far to this end we approximate two stage robust binary programs by their corresponding k adaptability problems in which the decision maker precommits to k second stage policies here and now and implements the best of these policies once the uncertain parameters are observed we study the approximation quality and the computational complexity of the k adaptability problem and we propose two mixed integer linear programming reformulations that can be solved with off the shelf software we demonstrate the effectiveness of our reformulations for stylized instances o

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

joint estimation of inverse covariance matrices lying in an unknown subspace we consider the problem of joint estimation of inverse covariance matrices lying in an unknown subspace of the linear space of symmetric matrices we perform the estimation using groups of measurements with different covariances assuming the inverse covariances span a low dimensional subspace our aim is to determine this subspace and to exploit this knowledge in order to improve the estimation we develop a novel optimization algorithm discovering and exploiting the underlying low dimensional subspace we provide a computationally efficient algorithm and derive a tight upper performance bound numerical simulations on synthetic and real world data are presented to illustrate the performance benefits of the algorithm

convex approximations for totally unimodular integer recourse models a uniform error bound we consider a class of convex approximations for totally unimodular tu integer recourse models and derive a uniform error bound by exploiting properties of the total variation of the probability density functions involved for simple integer recourse models this error bound is tight and improves the existing one by a factor 2 whereas for tu integer recourse models this is the first nontrivial error bound available the bound ensures that the performance of the approximations is good as long as the total variations of the densities of all random variables in the model are small enough

a binary decision diagram based algorithm for solving a class of binary two stage stochastic programs we consider a special class of two stage stochastic integer programming problems with binary variables appearing in both stages the class of problems we consider constrains the second stage variables to belong to the intersection of sets corresponding to first stage binary variables that equal one our approach seeks to uncover strong dual formulations to the second stage problems by transforming them into dynamic programming dp problems parameterized by first stage variables we demonstrate how these dps can be formed by use of binary decision diagrams which then yield traditional benders inequalities that can be strengthened based on observations regarding the structure of the underlying dps we demonstrate the efficacy of our approach on a set of stochastic traveling salesman problems

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

quaternion ica from second order statistics this paper addresses the independent component analysis ica of quaternion random vectors in particular we focus on the gaussian case and therefore only consider the quaternion second order statistics sos which are given by the covariance matrix and three complementary covariance matrices first we derive the necessary and sufficient conditions for the identifiability of the quaternion ica model which are based on the definition of the properness profile of a quaternion random variable and more specifically on the concept of rotationally equivalent properness profiles second we show that the maximum likelihood ml approach to the quaternion ica problem reduces to the approximated joint diagonalization ajd of the sample mean estimates of the covariance and complementary covariance matrices unlike the complex case these four matrices cannot be simultaneously diagonalized in general and we have to resort to a particular ajd algorithm the proposed technique which can be seen as a quasi newton method is based on the local approximation of the nonconvex ml ica cost function a measure of the entropy loss due to the residual correlation among the estimated quaternion sources and it provides a satisfactory solution of the quaternion ica model the performance of the proposed quaternion ml ica algorithm as well as its relationship to the identifiability conditions are illustrated by means of several numerical examples

decomposition with branch and cut approaches for two stage stochastic mixed integer programming decomposition has proved to be one of the more effective tools for the solution of large scale problems especially those arising in stochastic programming a decomposition method with wide applicability is benders decomposition which has been applied to both stochastic programming as well as integer programming problems however this method of decomposition relies on convexity of the value function of linear programming subproblems this paper is devoted to a class of problems in which the second stage subproblem s may impose integer restrictions on some variables the value function of such integer subproblem s is not convex and new approaches must be designed in this paper we discuss alternative decomposition methods in which the second stage integer subproblems are solved using branch and cut methods one of the main advantages of our decomposition scheme is that stochastic mixed integer programming smip problems can be solved by dividing a large problem into smaller mip subproblems that can be solved in parallel this paper lays the foundation for such decomposition methods for two stage stochastic mixed integer programs

nondifferential reverse convex programs and facetial convexity cuts via a disjuntive characterization disjunctive programs can often be transcribed as reverse convex constrained problems with nondifferentiable constraints and unbounded feasible regions we consider this general class of nonconvex programs called reverse convex programs rcp and show that under quite general conditions the closure of the convex hull of the feasible region is polyhedral this development is then pursued from a more constructive standpoint in that for certain special reverse convex sets we specify a finite linear disjunction whose closed convex hull coincides with that of the special reverse convex set when interpreted in the context of convexity intersection cuts this provides the capability of generating any negative edge extension facet cut although this characterization is more clarifying than computationally oriented our development shows that if certain bounds are available then convexity intersection cuts can be strengthened relatively inexpensively

robust attitude stabilization of spacecraft using nonlinear quaternion feedback this paper considers the problem of three axis attitude stabilization of a rigid spacecraft a nonlinear control law which uses the feedback of the unit quaternion and the measured angular velocities is proposed and is shown to provide global asymptotic stability the control law does not require the knowledge of the system parameters and is therefore robust to modeling errors the significance of the control law is that it can be used for large angle maneuvers with guaranteed stability

a geometric buchberger algorithm for integer programming let ip a c denote the family of integer programs of the form min cx ax b x nn obtained by varying the right hand side vector b but keeping a and c fixed a test set for ipa c is a set of vectors in zn such that for each nonoptimal solution to a program in this family there is at least one element g in this set such that g has an improved cost value as compared to we describe a unique minimal test set for this family called the reduced grobner basis of ip a c an algorithm for its construction is presented which we call a geometric buchberger algorithm for integer programming and we show how an integer program may be solved using this test set the reduced grobner basis is then compared with some other known test sets from the literature we also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix a zn 2 n of full row rank

a loose benders decomposition algorithm for approximating two stage mixed integer recourse models we propose a new class of convex approximations for two stage mixed integer recourse models the so called generalized alpha approximations the advantage of these convex approximations over existing ones is that they are more suitable for efficient computations indeed we construct a loose benders decomposition algorithm that solves large problem instances in reasonable time to guarantee the performance of the resulting solution we derive corresponding error bounds that depend on the total variations of the probability density functions of the random variables in the model the error bounds converge to zero if these total variations converge to zero we empirically assess our solution method on several test instances including the sizes and sslp instances from siplib we show that our method finds near optimal solutions if the variability of the random parameters in the model is large moreover our method outperforms existing methods in terms of computation time especially for large problem instances

the value function of an integer program we consider integer programs in which the objective function and constraint matrix are fixed while the right hand side varies the value function gives for each feasible right hand side the criterion value of the optimal solution we provide a precise characterization of the closed form expression for any value function

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

locally most powerful invariant tests for the properness of quaternion gaussian vectors previous works have addressed the second order statistical characterization of quaternion random vectors introducing different properness definitions and presenting the generalized likelihood ratio tests glrts for determining the kind of quaternion properness this paper considers the more challenging problem of deriving the locally most powerful invariant tests lmpits which can be obtained even without an explicit expression for the maximal invariants thanks to the wijsman s theorem specifically we consider three binary hypothesis testing problems involving the two main kinds of quaternion properness and show that the lmpit statistics are given by the frobenius norm of three previously defined sample coherence matrices the proposed detectors exhibit interesting connections with the problem of testing for the properness of a complex vector and with the problems of testing for the sphericity of a four dimensional real or two dimensional complex proper vector additionally some numerical examples show that in general the proposed lmpits outperform their glrt counterparts and in some cases the performance gap is very noticeable

on structure and stability in stochastic programs with random technology matrix and complete integer recourse for two stage stochastic programs with integrality constraints in the second stage we study continuity properties of the expected recourse as a function both of the first stage policy and the integrating probability measure

a modified lift and project procedure in recent years the lift and project approach has been used successfully within a branch and cut framework to solve large difficult pure and mixed 0 1 programs that have resisted solution efforts by pure branch and bound codes the approach uses a linear description in a higher dimensional space of the convex hull of the disjunctive set created by imposing one or several 0 1 conditions by solving a linear program derived from this higher dimensional representation the cut generating linear program cglp the standard lift and project procedure obtains a deepest cut in a well defined sense we propose a modification of cglp that allows us to generate not just one deepest cut but a class of cuts with desirable properties each at the cost of one extra pivot in the optimal tableau of the modified cglp

on solving discrete two stage stochastic programs having mixed integer first and second stage variables in this paper we propose a decomposition based branch and bound dbab algorithm for solving two stage stochastic programs having mixed integer first and second stage variables a modified benders decomposition method is developed where the benders subproblems define lower bounding second stage value functions of the first stage variables that are derived by constructing a certain partial convex hull representation of the two stage solution space this partial convex hull is sequentially generated using a convexification scheme such as the reformulation linearization technique rlt or lift and project process which yields valid inequalities that are reusable in the subsequent subproblems by updating the values of the first stage variables a branch and bound algorithm is designed based on a hyperrectangular partitioning process using the established property that any resulting lower bounding benders master problem defined over a hyperrectangle yields the same objective value as the original stochastic program over that region if the first stage variable solution is an extreme point of the defining hyperrectangle or the second stage solution satisfies the binary restrictions we prove that this algorithm converges to a global optimal solution some numerical examples and computational results are presented to demonstrate the efficacy of this approach

polyhedral risk measures in stochastic programming we consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures thereby we place emphasis on dynamic models i e multistage stochastic programs with multiperiod risk measures in this context we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right hand side of the dynamic constraints dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence as examples of polyhedral risk measures we propose multiperiod extensions of the conditional value at risk

on robust optimization of two stage systems robust optimization models belong to a special class of stochastic programs where the traditional expected cost minimization objective is replaced by one that explicitly addresses cost variability this paper explores robust optimization in the context of two stage planning systems we show that under arbitrary measures for variability the robust optimization approach might lead to suboptimal solutions to the second stage planning problem as a result the variability of the second stage costs may be underestimated thereby defeating the intended purpose of the model we propose sufficient conditions on the variability measure to remedy this problem under the proposed conditions a robust optimization model can be efficiently solved using a variant of the l shaped decomposition algorithm for traditional stochastic linear programs we apply the proposed framework to standard stochastic programming test problems and to an application that arises in auctioning excess electric power

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

a cutting plane approach to mixed 0 1 stochastic integer programs abstract we consider a mixed 0 1 integer programming problem with dual block angular structure arising in two stage stochastic programming a relaxation is proposed such that the problem is decomposed into subproblems each corresponding to the outcomes of the random variable the convex hull of feasible solutions for the relaxation is characterized using results from disjunctive programming and it is shown how lift and project cuts can be generated for one subproblem and made valid for different outcomes

the theory of quaternion matrix derivatives a systematic framework for the calculation of the derivatives of quaternion matrix functions with respect to quaternion matrix variables is introduced the proposed approach is equipped with the matrix product and chain rules and applies to both analytic and nonanalytic functions of quaternion variables this rectifies a mathematical shortcut in the existing methods which incorrectly use the traditional product rule we also show that within the proposed framework the derivatives of quaternion matrix functions can be calculated directly without using quaternion differentials or resorting to the isomorphism with real vectors illustrative examples show how the proposed quaternion matrix derivatives can be used as an important tool for solving optimization problems in signal processing applications

an exact algorithm for solving large scale two stage stochastic mixed integer problems some theoretical and experimental aspects abstract we present an algorithmic framework so called bfc tsmip for solving two stage stochastic mixed 0 1 problems the constraints in the deterministic equivalent model have 0 1 variables and continuous variables at any stage the approach uses the twin node family tnf concept within an adaptation of the algorithmic framework so called branch and fix coordination for satisfying the nonanticipativity constraints for the first stage 0 1 variables jointly we solve the mixed 0 1 submodels defined at each tnf integer set for satisfying the nonanticipativity constraints for the first stage continuous variables in these submodels the only integer variables are the second stage 0 1 variables a numerical example and some theoretical and computational results are presented to show the performance of the proposed approach

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

a branch and price algorithm for multistage stochastic integer programming with application to stochastic batch sizing problems in this paper we present a branch and price method to solve special structured multistage stochastic integer programming problems we validate our method on two different versions of a multistage stochastic batch sizing problem sbsp one version adopts a recourse formulation and the other is based on probabilistic constraints our algorithmic approach is applicable to both formulations our computational results suggest that both classes of problems can be solved using relatively few nodes of a branch and price tree the success of our approach calls for extensions in methodology as well as applications

modeling methods and a branch and cut algorithm for pharmaceutical clinical trial planning using stochastic programming we discuss methods for the solution of a multi stage stochastic programming formulation for the resource constrained scheduling of clinical trials in the pharmaceutical research and development pipeline first we present a number of theoretical properties to reduce the size and improve the tightness of the formulation focusing primarily on non anticipativity constraints second we develop a novel branch and cut algorithm where necessary non anticipativity constraints that are unlikely to be active are removed from the initial formulation and only added if they are violated within the search tree we improve the performance of our algorithm by combining different node selection strategies and exploring different approaches to constraint violation checking