This work proposes a method for solving linear stochastic optimal control (SOC) problems using sum of squares and semidefinite programming. Previous work had used polynomial optimization to approximate the value function, requiring a high polynomial degree to capture local phenomena. To improve the scalability of the method to problems of interest, a domain decomposition scheme is presented. By using local approximations, lower degree polynomials become sufficient, and both local and global properties of the value function are captured. The domain of the problem is split into a non-overlapping partition, with added constraints ensuring $C^1$ continuity. The Alternating Direction Method of Multipliers (ADMM) is used to optimize over each domain in parallel and ensure convergence on the boundaries of the partitions. This results in improved conditioning of the problem and allows for much larger and more complex problems to be addressed with improved performance.
Towards stochastic realization theory for Generalized Linear Switched Systems with inputs: decomposition into stochastic and deterministic components and existence and uniqueness of innovation form
We consider multistage stochastic optimization problems involving multiple units. Each unit is a (small) control system. Static constraints couple units at each stage. We present a mix of spatial and temporal decompositions to tackle such large scale problems. More precisely, we obtain theoretical bounds and policies by means of two methods, depending whether the coupling constraints are handled by prices or by resources. We study both centralized and decentralized information structures. We report the results of numerical experiments on the management of urban microgrids. It appears that deco…
We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. We prove the almost sure convergence of these decomposition methods when the relatively complete recourse assumption holds. We also prove the almost sure convergence of these algorithms when applied to risk-averse mult…
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 strategies to select the most relevant cuts of the approximate recourse functions. We prove the convergence of the method in a finite number of iterations and use it to solve six portfolio problems with direct transaction costs under return uncertainty and six inventory management problems under demand uncertainty. On all problem instances CuSMuDA is much quicker than MuDA: between 5.1 and 1…
Second order conic programming (SOCP) has been used to model various applications in power systems, such as operation and expansion planning. In this paper, we present a two-stage stochastic mixed integer SOCP (MISOCP) model for the distribution system expansion planning problem that considers uncertainty and also captures the nonlinear AC power flow. To avoid costly investment plans due to some extreme scenarios, we further present a chance-constrained variant that could lead to cost-effective solutions. To address the computational challenge, we extend the basic Benders decomposition method …
The distributed operating room (OR) scheduling problem aims to find an assignment of surgeries to ORs across collaborating hospitals that share their waiting lists and ORs. We propose a stochastic extension of this problem where surgery durations are considered to be uncertain. In order to obtain solutions for the challenging stochastic model, we use sample average approximation, and develop two enhanced decomposition frameworks that use logic-based Benders (LBBD) optimality cuts and binary decision diagram based Benders cuts. Specifically, to the best of our knowledge, deriving LBBD optimalit…
We develop a decomposition algorithm for distributionally-robust two-stage stochastic mixed-integer convex cone programs, and its important special case of distributionally-robust two-stage stochastic mixed-integer second order cone programs. This generalizes the algorithm proposed by Sen and Sherali~[Mathematical Programming 106(2): 203-223, 2006]. We show that the proposed algorithm is finitely convergent if the second-stage problems are solved to optimality at incumbent first stage solutions, and solution to an optimization problem to identify worst-case probability distribution is availabl…
In this paper, we have studied a decomposition method for solving a class of nonconvex two-stage stochastic programs, where both the objective and constraints of the second-stage problem are nonlinearly parameterized by the first-stage variable. Due to the failure of the Clarke regularity of the resulting nonconvex recourse function, classical decomposition approaches such as Benders decomposition and (augmented) Lagrangian-based algorithms cannot be directly generalized to solve such models. By exploring an implicitly convex-concave structure of the recourse function, we introduce a novel dec…
In this paper, we present a sequential sampling-based algorithm for the two-stage distributionally robust linear programming (2-DRLP) models. The 2-DRLP models are defined over a general class of ambiguity sets with discrete or continuous probability distributions. The algorithm is a distributionally robust version of the well-known stochastic decomposition algorithm of Higle and Sen (Math. of OR 16(3), 650-669, 1991) for a two-stage stochastic linear program. We refer to the algorithm as the distributionally robust stochastic decomposition (DRSD) method. The key features of the algorithm incl…
A Stochastic Benders Decomposition Scheme for Large-Scale Stochastic Network Design
Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院及运筹学中心）；Department of Analytics, Marketing and Operations, Imperial College Business School（帝国理工商学院分析、营销与运营系）；Management Science and Operations, London Business School（伦敦商学院管理科学与运营系）；Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运筹学中心）
We apply logic-based Benders decomposition (LBBD) to two-stage stochastic planning and scheduling problems in which the second-stage is a scheduling task. We solve the master problem with mixed integer/linear programming and the subproblem with constraint programming. As Benders cuts, we use simple nogood cuts as well as analytical logic-based cuts we develop for this application. We find that LBBD is computationally superior to the integer L-shaped method, with a branch-and-check variant of LBBD faster by several orders of magnitude, allowing significantly larger instances to be solved. This …
We combine stochastic control methods, white noise analysis and Hida-Malliavin calculus applied to the Donsker delta functional to obtain new representations of semimartingale decompositions under enlargement of filtrations. The results are illustrated by explicit examples.
This paper proposes a data-driven version of the Benders decomposition algorithm applied to the stochastic unit commitment (SUC) problem. The proposed methodology aims at finding a trade-off between the size of the Benders master problem and the number of iterations until convergence. Using clustering techniques, we exploit the information contained in the Lagrange multipliers of the Benders subproblems in order to aggregate the optimality cuts, without compromising the critical information that is passed to the master problem. In addition, we develop an outer parallelization scheme that finds…
In energy management, it is common that strategic investment decisions (storage capacity, production units) are made at a slow time scale, whereas operational decisions (storage, production) are made at a fast time scale: for such problems, the total number of decision stages may be huge. In this paper, we consider multistage stochastic optimization problems with two time-scales, and we propose a time block decomposition scheme to address them numerically. More precisely, our approach relies on two assumptions. On the one hand, we suppose slow time scale stagewise independence of the noise pro…
Stochastic Alternating Least Squares (SALS) is a method that approximates the canonical decomposition of averages of sampled random tensors. Its simplicity and efficient memory usage make SALS an ideal tool for decomposing tensors in an online setting. We show, under mild regularization and readily verifiable assumptions on the boundedness of the data, that the SALS algorithm is globally convergent. Numerical experiments validate our theoretical findings and demonstrate the algorithm's performance and complexity.
A decomposition algorithm for solving large-scale security-constrained AC optimal power flow problems is presented. The formulation considered is the one used in the ARPA-E Grid Optimization (GO) Competition, Challenge 1, held from November 2018 through October 2019. The techniques found to be most effective in terms of performance in the challenge are presented, including strategies for contingency selection, fast contingency evaluation, handling complementarity constraints, avoiding issues related to degeneracy, and exploiting parallelism. The results of numerical experiments are provided to…
One of the challenges of cloud computing is to optimally and efficiently assign virtual ma- chines to physical machines. The aim of telecommunication operators is to minimize the map- ping cost while respecting constraints regarding location, assignment and capacity. In this paper we first propose an exact formulation leading to a 0-1 bilinear constrained problem. Then we introduce a variety of linear cuts by exploiting the problem structure and present a Lagrange decomposition based B&B algorithm to obtain optimal solutions efficiently. Numerically, we show that our valid inequalities close o…
When stochastic control problems do not possess separability and/or monotonicity, the dynamic programming pioneered by Bellman in 1950s fails to work as a time-decomposition solution method. Such cases have posted a great challenge to the control society in both theoretical foundation and solution methodologies for many years. With the help of the progressive hedging algorithm proposed by Rockafellar and Wets in 1991, we develop a novel scenario-decomposition solution framework for stochastic control problems which could be nonseparable and/or non-monotonic, thus extending the reach of stochas…
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 dimensi…
Microsoft Research New England（微软新英格兰研究院）；University of California Irvine（加州大学欧文分校）；University of California Berkeley（加州大学伯克利分校）；Cornell University（康奈尔大学）
We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allows for efficient optimization. Using this property we show that stochastic gradient descent converges to a local minimum in a polynomial number of iterations. To the best of our knowledge this is the first work that gives global convergence guarantees for stochastic gradient desc…
RTE（法国输电系统运营商RTE）；Université Paris 13（巴黎第十三大学）；Polytechnique Montréal（蒙特利尔高等综合理工学院）；University of Edinburgh（爱丁堡大学）
Semidefinite Programming (SDP) provides tight lower bounds for Optimal Power Flow problems. However, solving large-scale SDP problems requires exploiting sparsity. In this paper, we experiment several clique decomposition algorithms that lead to different reformulations and we show that the resolution is highly sensitive to the clique decomposition procedure. Our main contribution is to demonstrate that minimizing the number of additional edges in the chordal extension is not always appropriate to get a good clique decomposition.
In this paper, we consider a discrete-time stochastic Stackelberg game with a single leader and multiple followers. Both the followers and the leader together have conditionally independent private types, conditioned on action and previous state, that evolve as controlled Markov processes. The objective is to compute the stochastic Stackelberg equilibrium of the game where the leader commits to a dynamic strategy. Each follower's strategy is the best response to the leader's strategies and other followers' strategies while the each leader's strategy is optimum given the followers play the best…
In this paper, we study backward doubly stochastic recursive optimal control problem where the cost function is described by the solution of a backward doubly stochastic differential equation. We give the dynamical programming principle for this kind of optimal control problem and show that the value function is the unique Sobolev weak solution for the corresponding stochastic Hamilton-Jacobi-Bellman equation.
In this paper, we study the stochastic optimal control problem for the PReP vaccine in the stochastic model for HIV/AIDS with PReP. By using the stochastic maximum principle, we derive the stochastic optimal control of PReP for the unconstrained control problem, as well as for two different types of budget constrains. We illustrate the results by a numerical example. We first study the PReP stochastic differential equation dynamics with a constant, deterministic PReP treatment rate. Then, we compare this to the stochastic optimal control in the unconstrained case.
Zhejiang University（浙江大学）；Singapore University of Technology and Design（新加坡科技设计大学）
We consider a two-stage stochastic optimization problem, in which a long-term optimization variable is coupled with a set of short-term optimization variables in both objective and constraint functions. Despite that two-stage stochastic optimization plays a critical role in various engineering and scientific applications, there still lack efficient algorithms, especially when the long-term and short-term variables are coupled in the constraints. To overcome the challenge caused by tightly coupled stochastic constraints, we first establish a two-stage primal-dual decomposition (PDD) method to d…
We study Benamou's domain decomposition algorithm for optimal transport in the entropy regularized setting. The key observation is that the regularized variant converges to the globally optimal solution under very mild assumptions. We prove linear convergence of the algorithm with respect to the Kullback--Leibler divergence and illustrate the (potentially very slow) rates with numerical examples. On problems with sufficient geometric structure (such as Wasserstein distances between images) we expect much faster convergence. We then discuss important aspects of a computationally efficient imple…
This work recasts time-dependent optimal control problems governed by partial differential equations in a Dynamic Mode Decomposition with control framework. Indeed, since the numerical solution of such problems requires a lot of computational effort, we rely on this specific data-driven technique, using both solution and desired state measurements to extract the underlying system dynamics. Thus, after the Dynamic Mode Decomposition operators construction, we reconstruct and perform future predictions for all the variables of interest at a lower computational cost with respect to the standard s…
