Regularized Decomposition of High-Dimensional Multistage Stochastic Programs with Markov Uncertainty

We develop a quadratic regularization approach for the solution of high-dimensional multistage stochastic optimization problems characterized by a potentially large number of time periods/stages (e.g. hundreds), a high-dimensional resource state variable, and a Markov information process. The resulting algorithms are shown to converge to an optimal policy after a finite number of iterations under mild technical assumptions. Computational experiments are conducted using the setting of optimizing energy storage over a large transmission grid, which motivates both the spatial and temporal dimensions of our problem. Our numerical results indicate that the proposed methods exhibit significantly faster convergence than their classical counterparts, with greater gains observed for higher-dimensional problems

Solution methods and bounds for two-stage risk-neutral and multistage risk-averse stochastic mixed-integer programs with applications in energy and manufacturing

This dissertation presents an integrated method for solving stochastic mixed-integer programs, develops a lower bounding approach for multistage risk-averse stochastic mixed-integer programs, and proposes an optimization formulation for mixed-model assembly line sequencing (MMALS) problems. It is well known that a stochastic mixed-integer program is difficult to solve due to its non-convexity and stochastic factors. The scenario decomposition algorithms display computational advantage when dealing with a large number of possible realizations of uncertainties, but each has its own advantages and disadvantages. This dissertation presents a solution method for solving large-scale stochastic mixed-integer programs that integrates two scenario-decomposition algorithms: Progressive Hedging (PH) and Dual Decomposition (DD). In this integrated method, fast progress in early iterations of PH speeds up the convergence of DD to an exact solution. In many applications, the decision makers are risk-averse and are more concerned with large losses in the worst scenarios than with average performance. The PH algorithm can serve as a time-efficient heuristic for risk-averse stochastic mixed-integer programs with many scenarios, but the scenario reformulation for time consistent multistage risk-averse models does not exist. This dissertation develops a scenario-decomposed version of time consistent multistage risk-averse programs, and proposes a lower bounding approach that can assess the quality of PH solutions and thus identify whether the PH algorithm is able to find near-optimal solutions within a reasonable amount of time. The existing optimization formulations for MMALS problems do not consider many real-world uncertainty factors such as timely part delivery and material quality. In addition, real-time sequencing decisions are required to deal with inevitable disruptions. This dissertation formulates a multistage stochastic optimization problem with part availability uncertainty. A risk-averse model is further developed to guarantee customers’ satisfaction regarding on-time performance. Computational studies show that the integration of PH helps DD to reduce the run-time significantly, and the lower bounding approach can obtain convergent and tight lower bounds to help PH evaluate quality of solutions. The PH algorithm and the lower bounding approach also help the proposed MMALS formulation to make real-time sequencing decisions

Numerical Methods for Convex Multistage Stochastic Optimization

Optimization problems involving sequential decisions in a stochastic environment were studied in Stochastic Programming (SP), Stochastic Optimal Control (SOC) and Markov Decision Processes (MDP). In this paper we mainly concentrate on SP and SOC modelling approaches. In these frameworks there are natural situations when the considered problems are convex. Classical approach to sequential optimization is based on dynamic programming. It has the problem of the so-called ``Curse of Dimensionality", in that its computational complexity increases exponentially with increase of dimension of state variables. Recent progress in solving convex multistage stochastic problems is based on cutting planes approximations of the cost-to-go (value) functions of dynamic programming equations. Cutting planes type algorithms in dynamical settings is one of the main topics of this paper. We also discuss Stochastic Approximation type methods applied to multistage stochastic optimization problems. From the computational complexity point of view, these two types of methods seem to be complimentary to each other. Cutting plane type methods can handle multistage problems with a large number of stages, but a relatively smaller number of state (decision) variables. On the other hand, stochastic approximation type methods can only deal with a small number of stages, but a large number of decision variables

Joint dynamic probabilistic constraints with projected linear decision rules

We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically