implicit finite difference approximation for time fractional diffusion equations time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical brownian motion model in this paper we develop an implicit unconditionally stable numerical method to solve the one dimensional linear time fractional diffusion equation formulated with caputo s fractional derivative on a finite slab several numerical examples of interest are also included

duality and statistical tests of optimality for two stage stochastic programs we present alternative methods for verifying the quality of a proposed solution to a two stage stochastic program with recourse our methods revolve around implications of a dual problem in which dual multipliers on the nonanticipativity constraints play a critical role using randomly sampled observations of the stochastic elements we introduce notions of statistical dual feasibility and sampled error bounds additionally we use the nonanticipativity multipliers to develop connections to reduced gradient methods finally we propose a statistical test based on directional derivatives we illustrate the applicability of these tests via some examples

duality gaps in nonconvex stochastic optimization we consider multistage stochastic optimization models containing nonconvex constraints e g due to logical or integrality requirements we study three variants of lagrangian relaxations and of the corresponding decomposition schemes namely scenario nodal and geographical decomposition based on convex equivalents for the lagrangian duals we compare the duality gaps for these decomposition schemes the first main result states that scenario decomposition provides a smaller or equal duality gap than nodal decomposition the second group of results concerns large stochastic optimization models with loosely coupled components the results provide conditions implying relations between the duality gaps of geographical decomposition and the duality gaps for scenario and nodal decomposition respectively

an aggregate subgradient method for nonsmooth convex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

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

two stage stochastic optimization problems with stochastic ordering constraints on the recourse abstract we consider two stage risk averse stochastic optimization problems with a stochastic ordering constraint on the recourse function two new characterizations of the increasing convex order relation are provided they are based on conditional expectations and on integrated quantile functions a counterpart of the lorenz function we propose two decomposition methods to solve the problems and prove their convergence our methods exploit the decomposition structure of the risk neutral two stage problems and construct successive approximations of the stochastic ordering constraints numerical results confirm the efficiency of the methods

a sequential sampling procedure for stochastic programming we develop a sequential sampling procedure for a class of stochastic programs we assume that a sequence of feasible solutions with an optimal limit point is given as input to our procedure such a sequence can be generated by solving a series of sampling problems with increasing sample size or it can be found by any other viable method our procedure estimates the optimality gap of a candidate solution from this sequence if the point estimate of the optimality gap is sufficiently small according to our termination criterion then we stop otherwise we repeat with the next candidate solution from the sequence under an increased sample size we provide conditions under which this procedure i terminates with probability one and ii terminates with a solution that has a small optimality gap with a prespecified probability

numerical identification of a robin coefficient in parabolic problems this paper studies a regularization approach for an inverse problem of estimating a spatially and temporally dependent robin coefficient arising in the analysis of convective heat transfer the parameter to state map is analyzed especially a differentiability result is established a regularization approach is proposed and the properties e g existence and optimality system of the functional are investigated a finite element method is adopted for discretizing the continuous optimization problem and the convergence of the finite element approximations as the mesh size and temporal step size tend to zero is established numerical results by the conjugate gradient method for oneand two dimensional problems are presented

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

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

finite master programs in regularized stochastic decomposition stochastic decomposition is a stochastic analog of benders decomposition in which randomly generated observations of random variables are used to construct statistical estimates of supports of the objective function in contrast to deterministic benders decomposition for two stage stochastic programs the stochastic version requires infinitely many inequalities to ensure convergence we show that asymptotic optimality can be achieved with a finite master program provided that a quadratic regularizing term is included our computational results suggest that the elimination of the cutting planes impacts neither the number of iterations required nor the statistical properties of the terminal solution

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

an adaptive partition based approach for solving two stage stochastic programs with fixed recourse we study an adaptive partition based approach for solving two stage stochastic programs with fixed recourse a partition based formulation is a relaxation of the original stochastic program and we study a finitely converging algorithm in which the partition is adaptively adjusted until it yields an optimal solution a solution guided refinement strategy is developed to refine the partition by exploiting the relaxation solution obtained from a partition in addition to refinement we show that in the case of stochastic linear programs it is possible to merge some components in a partition without weakening the corresponding relaxation bound thus allowing the partition size to be kept small we also show that for stochastic linear programs with simple recourse there exists a small partition that yields an optimal solution the size of this partition is independent of the number of scenarios used in the model our computational results show that the proposed adaptive partition based approach converges v

boundary value problems for the diffusion equation of the variable order in differential and difference settings abstract solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied it is shown that the method of the energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case the credibility of the obtained results is verified by performing numerical calculations for a test problem

adaptive multicut aggregation for two stage stochastic linear programs with recourse outer linearization methods for two stage stochastic linear programs with recourse such as the l shaped algorithm generally apply a single optimality cut on the nonlinear objective at each major iteration while the multicut version of the algorithm allows for several cuts to be placed at once in general the l shaped algorithm tends to have more major iterations than the multicut algorithm however the trade offs in terms of computational time are problem dependent this paper investigates the computational trade offs of adjusting the level of optimality cut aggregation from single cut to pure multicut specifically an adaptive multicut algorithm that dynamically adjusts the aggregation level of the optimality cuts in the master program is presented and tested on standard large scale instances from the literature computational results reveal that a cut aggregation level that is between the single cut and the multicut can result in substantial computational savings over the single cut method

a probability metrics approach for reducing the bias of optimality gap estimators in two stage stochastic linear programming monte carlo sampling based estimators of optimality gaps for stochastic programs are known to be biased when bias is a prominent factor estimates of optimality gaps tend to be large on average even for high quality solutions this diminishes our ability to recognize high quality solutions in this paper we present a method for reducing the bias of the optimality gap estimators for two stage stochastic linear programs with recourse via a probability metrics approach motivated by stability results in stochastic programming we apply this method to the averaged two replication procedure a2rp by partitioning the observations in an effort to reduce bias which can be done in polynomial time in sample size we call the resulting procedure the averaged two replication procedure with bias reduction a2rp b we provide conditions under which a2rp b produces strongly consistent point estimators and an asymptotically valid confidence interval we illustrate the effectiveness of our approach analytically on a newsvendor problem and test the small sample behavior of a2rp b on a number of two stage stochastic linear programs from the literature our computational results indicate that the procedure effectively reduces bias we also observe variance reduction in certain circumstances

the identification of a robin coefficient by a conjugate gradient method this paper investigates a non linear inverse problem associated with the heat conduction problem of identifying a robin coefficient from boundary temperature measurement the variational formulation of the problem is given the conjugate gradient method combining with the discrepancy principle for choosing the suitable stop step are proposed for solving the optimization problem in which the finite difference method is used to solve the direct problems the performance of the method is verified by simulating four examples the convergence with respect to the grid refinement and the amount of noise in the data is also investigated copyright 2008 john wiley sons ltd

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

risk averse two stage stochastic linear programming modeling and decomposition we formulate a risk averse two stage stochastic linear programming problem in which unresolved uncertainty remains after the second stage the objective function is formulated as a composition of conditional risk measures we analyze properties of the problem and derive necessary and sufficient optimality conditions next we construct a new decomposition method for solving the problem that exploits the composite structure of the objective function we illustrate its performance on a portfolio optimization problem

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

equivalence of some quadratic programming algorithms we formulate a general algorithm for the solution of a convex but not strictly convex quadratic programming problem conditions are given under which the iterates of the algorithm are uniquely determined the quadratic programming algorithms of fletcher gill and murray best and ritter and van de panne and whinston dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points the various methods are shown to differ only in the manner in which they solve the linear equations expressing the kuhn tucker system for the associated equality constrained subproblems equivalence results have been established by goldfarb and djang for the positive definite hessian case our analysis extends these results to the positive semi definite case

assessing solution quality in stochastic programs determining whether a solution is of high quality optimal or near optimal is fundamental in optimization theory and algorithms in this paper we develop monte carlo sampling based procedures for assessing solution quality in stochastic programs quality is defined via the optimality gap and our procedures output is a confidence interval on this gap we review a multiple replications procedure that requires solution of say 30 optimization problems and then we present a result that justifies a computationally simplified single replication procedure that only requires solving one optimization problem even though the single replication procedure is computationally significantly less demanding the resulting confidence interval might have low coverage probability for small sample sizes for some problems we provide variants of this procedure that require two replications instead of one and that perform better empirically we present computational results for a newsvendor problem and for two stage stochastic linear programs from the literature we also discuss when the procedures perform well and when they fail and we propose using optimal solutions to strengthen the performance of our procedures

a weighted gram schmidt method for convex quadratic programming range space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases in this paper we describe a range space method based upon updating a weighted gram schmidt factorization of the constraints in the active set the updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active set strategy

uniqueness and numerical scheme for the robin coefficient identification of the time fractional diffusion equation abstract we study an inverse problem of determining the robin coefficient of fractional diffusion equation from a nonlocal boundary condition based on the property of caputo fractional derivative the uniqueness is proved the numerical schemes for the direct problem and the inverse problem are developed three examples are given to show the effectiveness of the presented methods

a bayesian inference approach to identify a robin coefficient in one dimensional parabolic problems this paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a robin coefficient from boundary temperature measurement a bayesian inference approach is presented for the solution of this problem the prior modeling is achieved via the markov random field mrf the use of a hierarchical bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed the markov chain monte carlo mcmc algorithm is used to explore the posterior state space numerical results indicate that mrf provides an effective prior regularization and the bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem

threshold probability of non terminal type in finite horizon markov decision processes abstract we consider a class of problems concerned with maximizing probabilities given stage wise targets which generalizes the standard threshold probability problem in markov decision processes the objective function is the probability that at all stages the associatively combined accumulation of rewards earned up to that point takes its value in a specified stage wise interval it is shown that this class reduces to the case of the nonnegative valued multiplicative criterion through an invariant imbedding technique we derive a recursive formula for the optimal value function and an effective method for obtaining the optimal policies

associative dynamic programs abstract in this paper we consider finite stage stochastic dynamic programs with associative reward systems an invariant imbedding technique yields a dynamic program with a terminal reward system on an augmented state space according to the associativity we clarify the validity of three possible recursive equations one parametrized equation and two non parametrized ones each of the additive multiplicative and multiplicative additive dynamic programs admits the three recursive equations further the corresponding composite relations among three optimum value functions hold however minimum maximum and fractional dynamic programs admit only the parametrized recursive equation they do not admit the two non parametrized equations further the corresponding relations among the three optimum value functions do not hold