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

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

stable runge kutta integrations for differential systems with semi stable equilibria in this paper the stability of runge kutta methods for differential systems with semi stable equilibria is analyzed in fact a way of circumventing the very severe unconditional stability barrier 3 is proposed the usage of time meshes with step size ratios under some constant greater than one as usual in standard numerical codes is considered in order to recover stability for many implicit methods the main result roughly says that any strongly a stable runge kutta method is stable for this kind of differential systems on this sort of meshes

stochastic decomposition an algorithm for two stage linear programs with recourse we present a cutting plane algorithm for two stage stochastic linear programs with recourse motivated by benders decomposition our method uses randomly generated observations of random variables to construct statistical estimates of supports of the objective function in general the resulting piecewise linear approximations do not agree with the objective function in finite time however certain subsequences of the estimated supports are shown to accumulate at supports of the objective function with probability one from this we establish the convergence of the algorithm under relatively mild assumptions

an algorithm for multistage dynamic networks with random arc capacities with an application to dynamic fleet management we consider the class of multistage dynamic networks with random arc capacities a framework that is well suited to model dynamic fleet management problems we propose a successive convex approximation approach that produces an approximation to the expected recourse function which captures the future effects of current decisions under uncertainty this method decomposes the network in each stage into tree subproblems whose expected recourse functions are easy to obtain we also compare this method with two alternative methods on a set of dynamic fleet management problems the numerical results show that this method is superior to the two alternative methods

carleman estimates for stochastic parabolic equations with neumann boundary conditions and applications abstract by a dual method two carleman estimates for forward and backward stochastic parabolic equations with neumann boundary conditions are established then they are used to study a null controllability problem and a state observation problem for some stochastic forward parabolic equations with neumann boundary conditions

solving lagrangian variational inequalities with applications to stochastic programming lagrangian variational inequalities feature both primal and dual elements in expressing first order conditions for optimality in a wide variety of settings where multipliers in a very general sense need to be brought in their stochastic version relates to problems of stochastic programming and covers not only classical formats with inequality constraints but also composite models with nonsmooth objectives the progressive hedging algorithm as a means of solving stochastic programming problems has however focused so far only on optimality conditions that correspond to variational inequalities in primal variables alone here that limitation is removed by appealing to a recent extension of progressive hedging to multistage stochastic variational inequalities in general

an augmented lagrangian method for distributed optimization we propose a novel distributed method for convex optimization problems with a certain separability structure the method is based on the augmented lagrangian framework we analyze its convergence and provide an application to two network models as well as to a two stage stochastic optimization problem the proposed method compares favorably to two augmented lagrangian decomposition methods known in the literature as well as to decomposition methods based on the ordinary lagrangian function

a linearization method for nonsmooth stochastic programming problems for a stochastic programming problem with a nonsmooth objective a recursive stochastic subgradient method is proposed in which successive directions are obtained from quadratic programming subproblems the subproblems result from linearization of original constrains and from approximation of the objective by a quadratic function involving stochastic subgradient estimates constructed in the course of computation a special lyapunov function technique is used to show that the method is convergent with probability 1 to stationary points and recovers the subgradient and lagrange multipliers that appear in necessary conditions of optimality

a weighted identity for stochastic partial differential operators and its applications abstract in this paper a weighted identity for some stochastic partial differential operators with complex principal parts is established this identity presents a unified approach in establishing carleman type estimates for some deterministic stochastic partial differential equations based on this identity one can deduce some known global carleman estimates for stochastic parabolic equations stochastic schrodinger equations stochastic transport equations and their deterministic counterparts meanwhile as its applications we derive two different carleman estimates for linear forward stochastic complex ginzburg landau equations they can be used to study the controllability observability and inverse problems for some stochastic complex ginzburg landau equations respectively

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

shape a stochastic hybrid approximation procedure for two stage stochastic programs we consider the problem of approximating the expected recourse function for two stage stochastic programs our problem is motivated by applications that have special structure such as an underlying network that allows reasonable approximations to the expected recourse function to be developed in this paper we show how these approximations can be improved by combining them with sample gradient information from the true recourse function for the case of strictly convex nonlinear approximations we prove convergence for this hybrid approximation the method is attractive for practical reasons because it retains the structure of the approximation

analysis of sample path optimization sample path optimization is a method for optimizing limit functions occurring in stochastic modeling problems such as steady state functions in discrete event dynamic systems it is closely related to retrospective optimization techniques and to m estimation the method has been computationally tested elsewhere on problems arising in production and in project planning with apparent success in this paper we provide a mathematical justification for sample path optimization by showing that under certain assumptions which hold for the problems just mentioned the method will almost surely find a point that is in a specified sense sufficiently close to the set of optimizers of the limit function

exact controllability for stochastic transport equations this paper studies the exact controllability for stochastic transport equations by two controls one is a boundary control imposed on the drift term and the other is an internal control imposed on the diffusion term by means of the duality argument this controllability problem can be reduced to an observability problem for backward stochastic transport equations and the desired observability estimate is obtained by a new global carleman estimate also we present some results about the lack of exact controllability which show that the action of two controls is necessary to some extent this indicates that the controllability problems for stochastic pdes differ from their deterministic counterpart

applications of the method of partial inverses to convex programming decomposition a primal dual decomposition method is presented to solve the separable convex programming problem convergence to a solution and lagrange multiplier vector occurs from an arbitrary starting point the method is equivalent to the proximal point algorithm applied to a certain maximal monotone multifunction in the nonseparable case it specializes to a known method the proximal method of multipliers conditions are provided which guarantee linear convergence

global carleman estimates for the linear stochastic kuramoto sivashinsky equations and their applications abstract in this paper we establish two new global carleman estimates for the linear stochastic kuramoto sivashinsky equations the first one is for the backward linear stochastic kuramoto sivashinsky equation based on this estimate and the duality argument we obtain the null controllability of the forward linear stochastic kuramoto sivashinsky equation by three controls one is an internal control in the diffusion term and the others are boundary controls the second one is for the forward linear stochastic kuramoto sivashinsky equation according to this estimate we obtain a unique continuation property for the forward linear stochastic kuramoto sivashinsky equation

differential systems with semi stable equilibria and numerical methods in this paper the long time stability of numerical methods for a class of nonlinear autonomous differential systems possessing a semi stable equilibrium is considered the local dynamics of the class under the h assumptions is reviewed some interesting differential systems of stiff type are seen to belong to this class it is also shown that the implicit euler method is the only one of practical interest which is unconditionally stable

a decomposition procedure based on approximate newton directions the efficient solution of large scale linear and nonlinear optimization problems may require exploiting any special structure in them in an efficient manner we describe and analyze some cases in which this special structure can be used with very little cost to obtain search directions from decomposed subproblems we also study how to correct these directions using decomposable preconditioned conjugate gradient methods to ensure local convergence in all cases the choice of appropriate preconditioners results in a natural manner from the structure in the problem finally we conduct computational experiments to compare the resulting procedures with direct methods

learning algorithms for separable approximations of discrete stochastic optimization problems we propose the use of sequences of separable piecewise linear approximations for solving nondifferentiable stochastic optimization problems the approximations are constructed adaptively using a combination of stochastic subgradient information and possibly sample information on the objective function itself we prove the convergence of several versions of such methods when the objective function is separable and has integer break points and we illustrate their behavior on numerical examples we then demonstrate the performance on nonseparable problems that arise in the context of two stage stochastic programming problems and demonstrate that these techniques provide near optimal solutions with a very fast rate of convergence compared with other solution techniques

an efficient family of strongly a stable runge kutta collocation methods for stiff systems and daes part i stability and order results for each integer s 3 a new uniparametric family of stiffly accurate strongly a stable s stage runge kutta methods is obtained these are collocation methods with a first internal stage of explicit type the methods are based on interpolatory quadrature rules with precision degree equal to 2s 4 and all of them have two prefixed nodes c 1 0 and c s 1 the amount of implicitness of our s stage method is similar to that involved with the s stage lobattoiiia method or with the s 1 stage radauiia method the new family of runge kutta methods proves to be of interest for the numerical integration of stiff systems and differential algebraic equations in fact on several stiff test problems taken from the current literature two methods selected in our 4 stage family seem to be slightly more efficient than the 3 stage radauiia method and also more robust than the 4 stage lobattoiiia method

decomposition coordination algorithms in stochastic optimization this paper considers an extension to the situation of stochastic programming of the auxiliary problem principle formerly introduced in a deterministic setting to serve as a general framework for decomposition coordination optimization algorithms the idea is based upon that of the stochastic gradient that is independent noise realizations are considered successively along the iterations as a consequence deterministic subproblems are solved at each iteration whereas iterations fulfill the two tasks of coordination and stochastic approximation at the same time coupling cost function expectation of some performance index and deterministic coupling constraints are considered price dual decomposition encompassing extensions of the uzawa and arrow hurwicz algorithms to this stochastic case are studied as well as resource allocation primal decomposition

on the convergence of runge kutta methods for stiff non linear differential equations this paper studies the convergence properties of general runge kutta methods when applied to the numerical solution of a special class of stiff non linear initial value problems it is proved that under weaker assumptions on the coefficients of a runge kutta method than in the standard theory of b convergence it is possible to ensure the convergence of the method for stiff non linear systems belonging to the above mentioned class thus it is shown that some methods which are not algebraically stable like the lobatto iiia or a stable sirk methods are convergent for the class of stiff problems under consideration finally some results on the existence and uniqueness of the runge kutta solution are also presented

information relaxations and duality in stochastic dynamic programs we describe a general technique for determining upper bounds on maximal values or lower bounds on minimal costs in stochastic dynamic programs in this approach we relax the nonanticipativity constraints that require decisions to depend only on the information available at the time a decision is made and impose a penalty that punishes violations of nonanticipativity in applications the hope is that this relaxed version of the problem will be simpler to solve than the original dynamic program the upper bounds provided by this dual approach complement lower bounds on values that may be found by simulating with heuristic policies we describe the theory underlying this dual approach and establish weak duality strong duality and complementary slackness results that are analogous to the duality results of linear programming we also study properties of good penalties finally we demonstrate the use of this dual approach in an adaptive inventory control problem with an unknown and changing demand distribution and in valuing options with stochastic volatilities and interest rates these are complex problems of significant practical interest that are quite difficult to solve to optimality in these examples our dual approach requires relatively little additional computation and leads to tight bounds on the optimal values

distributional efficiency in multiobjective stochastic linear programming several concepts of distributional efficiency are proposed for the multiobjective stochastic linear programming mslp problem in contexts where the probability distribution of random parameters is known and the decision maker dm has an unknown multi attribute utility function belonging to a given glass u we present a general efficient set the u admissible solutions and two subsets the u unanimous and u advocated solutions the latter being particularly relevant to the case of a single dm we show how advocated solutions can be generated and or tested when u is the class of non decreasing additive concave functions

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

a decomposition method and its application to convex programming a method is proposed for minimizing a function of the form q x d f1 x fm x where q is definite quadratic and fi are proper closed convex the key feature of the method is its capability of reducing the problem to a sequence of subproblems of the form min q x z fi x the method is an extension of the successive projection method given in han s p 1988 a successive projection method math programming 40 1 14 and has some similar features as the partial inverse method of spingarn spingarn j e 1983 partial inverse of a monotone operator appl math optim 10 247 265 in combination with the proximal point algorithm martinet b 1970 regularisation d inequations variationelles par approximations successives rev francaise inform rech oper 154 159 martinet b 1972 determination approchee d un point fixe d une application pseudo contractante c r acad sci paris 274 163 165 rockafellar r t 1976 monotone operators and the proximal point algorithm siam j control

first and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraints abstract the purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints the control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable hilbert space we employ some classical set valued analysis tools and theories of the transposition solution of vector valued backward stochastic evolution equations and the relaxed transposition solution of operator valued backward stochastic evolution equations to derive these optimality conditions the correction part of the second order adjoint equation which does not appear in the first order optimality condition plays a fundamental role in the second order optimality condition

stopping rules for a class of sampling based stochastic programming algorithms monte carlo sampling based algorithms hold much promise for solving stochastic programs with many scenarios a critical component of such algorithms is a stopping criterion to ensure the quality of the solution in this paper we develop a stopping rule theory for a class of algorithms that estimate bounds on the optimal objective function value by sampling we provide rules for selecting sample sizes and terminating the algorithm under which asymptotic validity of confidence intervals for the quality of the proposed solution can be verified empirical coverage results are given for a simple example

stochastic dual dynamic integer programming multistage stochastic integer programming msip combines the difficulty of uncertainty dynamics and non convexity and constitutes a class of extremely challenging problems a common formulation for these problems is a dynamic programming formulation involving nested cost to go functions in the linear setting the cost to go functions are convex polyhedral and decomposition algorithms such as nested benders decomposition and its stochastic variant stochastic dual dynamic programming sddp which proceed by iteratively approximating these functions by cuts or linear inequalities have been established as effective approaches however it is difficult to directly adapt these algorithms to msip due to the nonconvexity of integer programming value functions in this paper we propose an extension to sddp called stochastic dual dynamic integer programming sddip for solving msip problems with binary state variables the crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts termed lagrangian cuts derived from a lagrangian relaxation of a specific reformulation of the subproblems in each stage where local copies of state variables are introduced we show that the lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of sddip with probability one we show that under fairly reasonable assumptions an msip problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size thus our proposed sddip approach is applicable to very general classes of msip problems extensive computational experiments on three classes of real world problems namely electric generation expansion financial portfolio management and network revenue management show that the proposed methodology is very effective in solving large scale multistage stochastic integer optimization problems

asynchronous level bundle methods in this paper we consider nonsmooth convex optimization problems with additive structure featuring independent oracles black boxes working in parallel existing methods for solving these distributed problems in a general form are synchronous in the sense that they wait for the responses of all the oracles before performing a new iteration in this paper we propose level bundle methods handling asynchronous oracles these methods require original upper bounds using upper models or scarce coordinations to deal with asynchronicity we prove their convergence using variational analysis techniques and illustrate their practical performance on a lagrangian decomposition problem