Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion

We consider the incorporation of a time-consistent coherent risk measure into a multi-stage stochastic programming model, so that the model can be solved using a SDDP-type algorithm. We describe the implementation of this algorithm, and study the solutions it gives for an application of hydro-thermal scheduling in the New Zealand electricity system. The performance of policies using this risk measure at different levels of risk aversion is compared with the risk-neutral policy.     

Risk-averse stochastic optimal control: An efficiently computable statistical upper bound

In this paper, we discuss an application of the Stochastic Dual Dynamic Programming (SDDP) type algorithm to nested risk-averse formulations of Stochastic Optimal Control (SOC) problems. We propose a construction of a statistical upper bound for the optimal value of risk-averse SOC problems. This outlines an approach to a solution of a long standing problem in that area of research. The bound holds for a large class of convex and monotone conditional risk mappings. Finally, we show the validity of the statistical upper bound to solve a real-life stochastic hydro-thermal planning problem.     

On complexity of multistage stochastic programs

In this paper we derive estimates of the sample sizes required to solve a multistage stochastic programming problem with a given accuracy by the (conditional sampling) sample average approximation method. The presented analysis is self-contained and is based on a relatively elementary, one-dimensional, Cramér's Large Deviations Theorem.     

Approximating infinite horizon stochastic optimal control in discrete time with constraints

Traditional approaches to solving stochastic optimal control problems involve dynamic programming, and solving certain optimality equations. When recast as stochastic programming problems, structural aspects such as convexity are retained, and numerical solution procedures based on decomposition and duality may be exploited. This paper explores a class of stationary, infinite-horizon stochastic optimization problems with discounted cost criterion. Constraints on both states and controls are permitted, and modeled in the objective function by allowing it to take infinite values. Approximating techniques are developed using variational analysis, and intuitive lower bounds are obtained via averaging the future. These bounds could be used in a finite-time horizon stochastic programming setting to find solutions numerically. Research supported in part by a grant of the National Science Foundation. AMS Classification 46N10, 49N15, 65K10, 90C15, 90C46     

On complexity of multistage stochastic programs under heavy tailed distributions

In this paper, the complexity of sample average approximation (SAA) of multistage stochastic programs under heavy tailed distributions is investigated. Specifically, we estimate confidence levels when the accuracy parameter and sample size are given under independently and identically distributed (iid) and non-iid conditional samples, respectively. Different from the existing works, we emphasize the impact of heavy tailed distributions, non-iid conditional sampling and stages dependence of the random process in multistage stochastic programs.     

Central limit theorem and chaoticity

This paper studies the relations between stochastic properties and chaotic properties of a dynamical system satisfying the central limit theorem. For such a system, it is proved that every nonempty open set in its defining space contains a point with positive lower density of its return time set and that the system is syndetically sensitive, provided that it is strongly topologically ergodic. Moreover, it is shown that the system admits many chaotic properties if its domain is restricted to a tree.     

Continuous-time zero-sum stochastic game with stopping and control

We consider a zero-sum stochastic game for continuous-time Markov chain with countable state space and unbounded transition and pay-off rates. The additional feature of the game is that the controllers together with taking actions are also allowed to stop the process. Under suitable hypothesis we show that the game has a value and it is the unique solution of certain dynamic programming inequalities with bilateral constraints. In the process we also prescribe a saddle point equilibrium.     

HARA frontiers of optimal portfolios in stochastic markets

In this paper, we consider the optimal portfolio selection problem in continuous-time settings where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has the structure of the HARA family and the market states change according to a Markov process. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. This includes the distributions of the random asset returns as well as the utility function. We analyzed Black–Scholes type continuous-time models where the market parameters are driven by Markov processes. The Markov process that affects the state of the market is independent of the underlying Brownian motion that drives the stock prices. The problem of maximizing the expected utility of the terminal wealth is investigated and solved by stochastic optimal control methods for exponential, logarithmic and power utility functions. We found explicit solutions for optimal policy and the associated value functions. We also constructed the optimal wealth process explicitly and discussed some of its properties. In particular, it is shown that the optimal policy provides linear frontiers.     

一类带有MaxEMin评判的补偿型随机规划算法

补偿型随机规划一般假定随机变量的概率分布具有完备信息, 但实际情况往往只能获得部分信息. 针对离散概率具有一类线性部分信息条件而建立了带有MaxEMin评判的两阶段随机规划模型, 借助二次规划和对偶分解方法得到了可行性切割和最优切割, 给出了基于L-型的求解算法, 并证明了算法的收敛性. 通过数值实验表明了算法的有效性.     

Convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints

Sample average approximation (SAA) method has recently been applied to solve stochastic programs with second order stochastic dominance (SSD) constraints. In particular, Hu et al. (Math Program 133:171–201, 2012) presented a detailed convergence analysis of $\epsilon $ -optimal values and $\epsilon $ -optimal solutions of sample average approximated stochastic programs with polyhedral SSD constraints. In this paper, we complement the existing research by presenting convergence analysis of stationary points when SAA is applied to a class of stochastic minimization problems with SSD constraints. Specifically, under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts. Moreover, by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations, we derive exponential rate of convergence of stationary points.     

A preliminary set of applications leading to stochastic semidefinite programs and chance-constrained semidefinite programs

Ariyawansa and Zhu have recently introduced (two-stage) stochastic semidefinite programs (with recourse) (SSDPs) [1] and chance-constrained semidefinite programs (CCSDPs) [2] as paradigms for dealing with uncertainty in applications leading to semidefinite programs. Semidefinite programs have been the subject of intense research during the past 15 years, and one of the reasons for this research activity is the novelty and variety of applications of semidefinite programs. This research activity has produced, among other things, efficient interior point algorithms for semidefinite programs. Semidefinite programs however are defined using deterministic data while uncertainty is naturally present in applications. The definitions of SSDPs and CCSDPs in [1] and [2] were formulated with the expectation that they would enhance optimization modeling in applications that lead to semidefinite programs by providing ways to handle uncertainty in data. In this paper, we present results of our attempts to create SSDP and CCSDP models in four such applications. Our results are promising and we hope that the applications presented in this paper would encourage researchers to consider SSDP and CCSDP as new paradigms for stochastic optimization when they formulate optimization models.     

An optimal solution for the stochastic version of the Wagner–Whitin dynamic lot-size model

We present an algorithm for determining the optimal solution over the entire planning horizon for the dynamic lot-size model where demand is stochastic and non-stationary. The optimal solution to the deterministic problem is the well-known Wagner–Whitin algorithm. The present work contributes principally to knowledge building and provides a tool for researchers. One potentially useful contribution to practice is the solution to an important special case, where demand follows normal distributions. Other contributions to practice will likely flow from the development of improved heuristics and the improved basis to evaluate heuristic performance.     

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

Dynamic programming for multidimensional stochastic control problems

In this paper we study a general multidimensional diffusion-type stochastic control problem. Our model contains the usual regular control problem, singular control problem and impulse control problem as special cases. Using a unified treatment of dynamic programming, we show that the value function of the problem is a viscosity solution of certain Hamilton-Jacobi-Bellman (HJB) quasivariational inequality. The uniqueness of such a quasi-variational inequality is proved. Supported in part by USA Office of Naval Research grant #N00014-96-1-0262. Supported in part by the NSFC Grant #79790130, the National Distinguished Youth Science Foundation of China Grant #19725106 and the Chinese Education Ministry Science Foundation.     

Variance reduction in sample approximations of stochastic programs

This paper studies the use of randomized Quasi-Monte Carlo methods (RQMC) in sample approximations of stochastic programs. In numerical integration, RQMC methods often substantially reduce the variance of sample approximations compared to Monte Carlo (MC). It seems thus natural to use RQMC methods in sample approximations of stochastic programs. It is shown, that RQMC methods produce epi-convergent approximations of the original problem. RQMC and MC methods are compared numerically in five different portfolio management models. In the tests, RQMC methods outperform MC sampling substantially reducing the sample variance and bias of optimal values in all the considered problems.     

Central limit theorem and stable laws for intermittent maps

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form x+x¹⁺, for (0, 1). In particular, for >1/2, we show that the Birkhoff sums of a Hölder observable f converge to a normal law or a stable law, depending on whether f(0)=0 or f(0)0. The proof uses spectral techniques introduced by Sarig, and Wieners Lemma in non-commutative Banach algebras.Mathematics Subject Classification (2000):37A30, 37A50, 37C30, 37E05, 47A56, 60F05     

Large deviation limit for discrete-time, totally observed stochastic control problems with multiplicative cost

We show that a large class of discrete-time dynamic games can be obtained as a limit of stochastic control problems with multiplicative cost. Our approach consists in analyzing the large deviation properties of the Markov kernels associated with the stochastic dynamics, and allows us to give a unitary treatment of several nonlinear models.     

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.     

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.