Monotonic bounds in multistage mixed-integer stochastic programming

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.     

Sublinear upper bounds for stochastic programs with recourse

Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.This research has been partially supported by the National Science Foundation. The first author's work was also supported in part by Office of Naval Research Grant N00014-86-K-0628 and by the National Research Council under a Research Associateship at the Naval Postgraduate School, Monterey, California.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Implementing bounds-based approximations in convex-concave two-stage stochastic programming

This paper is concerned with implementational issues and computational testing of bounds-based approximations for solving two-stage stochastic programs with fixed recourse. The implemented bounds are those derived by the authors previously, using first and cross moment information of the random parameters and a convex-concave saddle property of the recourse function. The paper first examines these bounds with regard to their tightness, monotonic behavior, convergence properties, and computationally exploitable decomposition structures. Subsequently, the bounds are implemented under various partitioning/refining strategies for the successive approximation. The detailed numerical experiments demonstrate the effectiveness in solving large scenario-based two-stage stochastic optimization problems throughsuccessive scenario clusters induced by refining the approximations.     

Asymptotic behavior of truncated stochastic approximation procedures

We study asymptotic behavior of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix-valued random step-size sequence, and a dynamically changing random regression function. In particular, we show that under quitemild conditions, stochastic approximation procedures are asymptotically linear in the statistical sense, that is, they can be represented as weighted sums of random variables. Therefore, a suitable formof the central limit theoremcan be applied to derive asymptotic distribution of the corresponding processes. The theory is illustrated by various examples and special cases.     

Barycentric scenario trees in convex multistage stochastic programming

This work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support. Based on earlier results, sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem. Error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation. Convergence of the approximate solutions is proven under the stated assumptions. Preliminary computational results are outlined. This work has been supported by Schweizerischen Nationalfonds Grant Nr. 21-39 575.93.     

An optimal Gauss–Markov approximation for a process with stochastic drift and applications

We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein–Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by exploiting the aforementioned results.     

Computable error bounds of multidimensional Euler inversion and their financial applications

Transform inversion is an efficient approximation procedure in operations research, yet the inversion results are sometimes unstable which calls for comprehensive error analysis. This article proposes a multidimensional Euler inversion (MEI) algorithm with computable error bounds. We design mild sufficient conditions that validate the inversion formula, and provide closed-form upper bounds of the inversion errors. Numerical experiments are conducted to compute the joint probability of default and barrier option prices under complicated stochastic models, and output the associated error bounds.     

Optimal designs for weighted approximation and integration of stochastic processes on [0,∞)

We study minimal errors and optimal designs for weighted L₂-approximation and weighted integration of Gaussian stochastic processes X defined on the half-line [0,∞). Under some regularity conditions, we obtain sharp bounds for the minimal errors for approximation and upper bounds for the minimal errors for integration. The upper bounds are proven constructively for approximation and non-constructively for integration. For integration of the r-fold integrated Brownian motion, the upper bound is proven constructively and we have a matching lower bound.     

Restricted recourse strategies for bounding the expected network recourse function

This paper presents bounds for the expected recourse function for stochastic programs with network recourse. Cyclic recourse, a concept introduced by Wallace [18], allows the approximation of the recourse problem by restricting the optimal flows on a set of cycles and by augmenting the original network to induce separability. We introduce a new procedure that uses again a set of cycles but does not approximate the problem; instead it solves it heuristically without altering the original network or requiring separability. The method produces tighter bounds and is computationally feasible for large networks. Numerical experiments with selected networks illustrate the effectiveness of the approach.     

Convergent Bounds for Stochastic Programs with Expected Value Constraints

This article describes a bounding approximation scheme for convex multistage stochastic programs (MSP) that constrain the conditional expectation of some decision-dependent random variables. Expected value constraints of this type are useful for modelling a decision maker’s risk preferences, but they may also arise as artifacts of stage-aggregation. We develop two finite-dimensional approximate problems that provide bounds on the (infinite-dimensional) original problem, and we show that the gap between the bounds can be made smaller than any prescribed tolerance. Moreover, the solutions of the approximate MSPs give rise to a feasible policy for the original MSP, and this policy’s optimality gap is shown to be smaller than the difference of the bounds. The considered problem class comprises models with integrated chance constraints and conditional value-at-risk constraints. No relatively complete recourse is assumed.     

How does a stochastic optimization/approximation algorithm adapt to a randomly evolving optimum/root with jump Markov sample paths

Stochastic optimization/approximation algorithms are widely used to recursively estimate the optimum of a suitable function or its root under noisy observations when this optimum or root is a constant or evolves randomly according to slowly time-varying continuous sample paths. In comparison, this paper analyzes the asymptotic properties of stochastic optimization/approximation algorithms for recursively estimating the optimum or root when it evolves rapidly with nonsmooth (jump-changing) sample paths. The resulting problem falls into the category of regime-switching stochastic approximation algorithms with two-time scales. Motivated by emerging applications in wireless communications, and system identification, we analyze asymptotic behavior of such algorithms. Our analysis assumes that the noisy observations contain a (nonsmooth) jump process modeled by a discrete-time Markov chain whose transition frequency varies much faster than the adaptation rate of the stochastic optimization algorithm. Using stochastic averaging, we prove convergence of the algorithm. Rate of convergence of the algorithm is obtained via bounds on the estimation errors and diffusion approximations. Remarks on improving the convergence rates through iterate averaging, and limit mean dynamics represented by differential inclusions are also presented. The research of G. Yin was supported in part by the National Science Foundation under DMS-0603287, in part by the National Security Agency under MSPF-068-029, and in part by the National Natural Science Foundation of China under #60574069. The research of C. Ion was supported in part by the Wayne State University Rumble Fellowship. The research of V. Krishnamurthy was supported in part by NSERC (Canada).     

Convergence of stochastic search algorithms to finite size pareto set approximations

In this work we investigate the convergence of stochastic search algorithms toward the Pareto set of continuous multi-objective optimization problems. The focus is on obtaining a finite approximation that should capture the entire solution set in a suitable sense, which will be defined using the concept of ε-dominance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will be determined entirely by the archiving strategy. We propose and analyse two different archiving strategies which lead to a different limit behavior of the algorithms, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting Pareto set approximation.      

Truncated stochastic approximation with moving bounds: convergence

In this paper we consider a wide class of truncated stochastic approximation procedures. These procedures have three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function. We establish convergence and consider several examples to illustrate the results.     

On the information-based complexity of stochastic programming

Existing complexity results in stochastic linear programming using the Turing model depend only on problem dimensionality. We apply techniques from the information-based complexity literature to show that the smoothness of the recourse function is just as important. We derive approximation error bounds for the recourse function of two-stage stochastic linear programs and show that their worst case is exponential and depends on the solution tolerance, the dimensionality of the uncertain parameters and the smoothness of the recourse function.     

Relaxations and approximations of chance constraints under finite distributions

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

Strong and weak orders in averaging for SPDEs

We show an averaging result for a system of stochastic evolution equations of parabolic type with slow and fast time scales. We derive explicit bounds for the approximation error with respect to the small parameter defining the fast time scale. We prove that the slow component of the solution of the system converges towards the solution of the averaged equation with an order of convergence 1/2 in a strong sense–approximation of trajectories–and 1 in a weak sense–approximation of laws. These orders turn out to be the same as for the SDE case.     

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0, 1[^d. The error of an algorithm is defined in L₂-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of nonuniform time discretizations is crucial.     

Spectral measure and approximation of homogenized coefficients

This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients. In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimension d?≥ 2, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.     

Error Estimates for a Stochastic Impulse Control Problem

We obtain error bounds for monotone approximation schemes of a stochastic impulse control problem. This is an extension of the theory for error estimates for the Hamilton-Jacobi-Bellman equation. We obtain almost the same estimate on the rate of convergence as in the equation without impulsions [2], [3].