Subregular Recourse in Nonlinear Multistage Stochastic Optimization

Adaptive Partition-based SDDP Algorithms for Multistage Stochastic Linear Programming

In this paper, we extend the adaptive partition-based approach for solving two-stage stochastic programs with fixed recourse to the multistage stochastic programming setting. The proposed algorithms integrate the adaptive partition-based strategy with a popular approach for solving multistage stochastic programs, the stochastic dual dynamic programming, via different tree-traversal strategies in order to enhance its computational efficiency. Our numerical experiments on a hydro-thermal power generation planning problem show the effectiveness of the proposed algorithms.

随机二次动态规划

Stochastic Quadratic Dynamic Programming

本文提出了一种名为SQDP的算法，用于求解具有强凸回溯函数的多阶段随机优化问题，通过将仿射切线替换为二次切线扩展了经典SDDP方法，并证明了算法的收敛性。

Single cut and multicut SDDP with cut selection for multistage stochastic linear programs: convergence proof and numerical experiments

We introduce a variant of Multicut Decomposition Algorithms (MuDA), called CuSMuDA (Cut Selection for Multicut Decomposition Algorithms), for solving multistage stochastic linear programs that incorporates a class of cut selection strategies to choose the most relevant cuts of the approximate recourse functions. This class contains Level 1 and Limited Memory Level 1 cut selection strategies, initially introduced for respectively Stochastic Dual Dynamic Programming (SDDP) and Dual Dynamic Programming (DDP). We prove the almost sure convergence of the method in a finite number of iterations and …

Stochastic Dual Dynamic Programming for Multistage Stochastic Mixed-Integer Nonlinear Optimization

In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorit…

A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization

We introduce a unified framework for the study of multilevel mixed integer linear optimization problems and multistage stochastic mixed integer linear optimization problems with recourse. The framework highlights the common mathematical structure of the two problems and allows for the development of a common algorithmic framework. Focusing on the two-stage case, we investigate, in particular, the nature of the value function of the second-stage problem, highlighting its connection to dual functions and the theory of duality for mixed integer linear optimization problems, and summarize differen…

A general model-and-run solver for multistage robust discrete linear optimization

The necessity to deal with uncertain data is a major challenge in decision making. Robust optimization emerged as one of the predominant paradigms to produce solutions that hedge against uncertainty. In order to obtain an even more realistic description of the underlying problem where the decision maker can react to newly disclosed information, multistage models can be used. However, due to their computational difficulty, multistage problems beyond two stages have received less attention and are often only addressed using approximation rather than optimization schemes. Even less attention is p…

多阶段适应鲁棒优化的变换-近端束算法及其在约束鲁棒最优控制中的应用

A Transformation-Proximal Bundle Algorithm for Multistage Adaptive Robust Optimization and Application to Constrained Robust Optimal Control

本文提出了一种新的变换-近端束算法用于多阶段适应鲁棒优化问题，通过将补救决策分为状态和控制决策，仅对状态决策应用仿射控制策略，允许控制决策完全适应，从而将原问题转化为等价的两阶段适应鲁棒优化问题。该变换为任何两阶段ARO解决方案算法提供了通用框架，开辟了多种多阶段ARO算法的新途径。近端束方法被开发用于所得到的两阶段问题，并进行了收敛性分析。在库存控制应用中，仿射扰动反馈控制策略的平均差距为34.88%，而所提出的算法生成的平均差距仅为1.68%。

Duality and sensitivity analysis of multistage linear stochastic programs

We investigate the dual of a Multistage Stochastic Linear Program (MSLP) to study two questions for this class of problems. The first of these questions is the study of the optimal value of the problem as a function of the involved parameters. For this sensitivity analysis problem, we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Since these formulas involve optimal dual solutions, we need an algorithm that computes such solutions to use them. In this context, the second question we address is…

Stochastic Lipschitz Dynamic Programming

We propose a new algorithm for solving multistage stochastic mixed integer linear programming (MILP) problems with complete continuous recourse. In a similar way to cutting plane methods, we construct nonlinear Lipschitz cuts to build lower approximations for the non-convex cost to go functions. An example of such a class of cuts are those derived using Augmented Lagrangian Duality for MILPs. The family of Lipschitz cuts we use is MILP representable, so that the introduction of these cuts does not change the class of the original stochastic optimization problem. We illustrate the application o…

用于确定性与随机强凸优化问题的带二次割的束方法

Bundle methods with quadratic cuts for deterministic and stochastic strongly convex optimization problems

本文提出基于二次割的QCC、QB及DASC方法，用于求解确定性与随机强凸优化问题，并证明其复杂度与收敛性，实验表明DASC在大强凸常数下显著快于SDDP。