A widely used heuristic for solving stochastic optimization problems is to use a deterministic rolling horizon procedure, which has been modified to handle uncertainty (e.g. buffer stocks, schedule slack). This approach has been criticized for its use of a deterministic approximation of a stochastic problem, which is the major motivation for stochastic programming. We recast this debate by identifying both deterministic and stochastic approaches as policies for solving a stochastic base model, which may be a simulator or the real world. Stochastic lookahead models (stochastic programming) require a range of approximations to keep the problem tractable. By contrast, so-called deterministic models are actually parametrically modified cost function approximations which use parametric adjustments to the objective function and/or the constraints. These parameters are then optimized in a stochastic base model which does not require making any of the types of simplifications required by stochastic programming. We formalize this strategy and describe a gradient-based stochastic search strategy to optimize the parameters.
Rolling forecasts have been almost overlooked in the renewable energy storage literature. In this paper, we provide a new approach for handling uncertainty not just in the accuracy of a forecast, but in the evolution of forecasts over time. Our approach shifts the focus from modeling the uncertainty in a lookahead model to accurate simulations in a stochastic base model. We develop a robust policy for making energy storage decisions by creating a parametrically modified lookahead model, where the parameters are tuned in the stochastic base model. Since computing unbiased stochastic gradients w…
The most common approaches for solving stochastic resource allocation problems in the research literature is to either use value functions ("dynamic programming") or scenario trees ("stochastic programming") to approximate the impact of a decision now on the future. By contrast, common industry practice is to use a deterministic approximation of the future which is easier to understand and solve, but which is criticized for ignoring uncertainty. We show that a parameterized version of a deterministic lookahead can be an effective way of handling uncertainty, while enjoying the computational si…
In this paper we develop a novel, discrete-time optimal control framework for mechanical systems with uncertain model parameters. We consider finite-horizon problems where the performance index depends on the statistical moments of the stochastic system. Our approach constitutes an extension of the original Differential Dynamic Programming method and handles uncertainty through generalized Polynomial Chaos (gPC) theory. The developed iterative scheme is capable of controlling the probabilistic evolution of the dynamic system and can be used in planning and control. Moreover, its scalable and f…
We propose an online parametric estimation method of stochastic differential equations with discrete observations and misspecified modelling based on online gradient descent. Our study provides uniform upper bounds for the risks of the estimators over a family of stochastic differential equations. The derivation of the bounds involves three underlying theoretical results: the analysis of the stochastic mirror descent algorithm based on dependent and biased subgradients, the simultaneous exponential ergodicity of classes of diffusion processes, and the proposal of loss functions whose approxima…
We consider two-stage recourse models in which the second-stage problem has integer decision variables and uncertainty in the second-stage cost vector, technology matrix, and the right-hand side vector. Such mixed-integer recourse models are typically non-convex and thus hard to solve. There exist convex approximations of these models with accompanying error bounds. However, it is unclear how these error bounds depend on the distributions of the second-stage cost vector $q$. In fact, the only error bound that is known hinges on the assumption that $q$ has a finite support. In this paper, we de…
Aerodynamic optimization is ubiquitous in the design of most engineering systems interacting with fluids. A common approach is to optimize a performance function defined by a choice of an aerodynamic model, e.g., turbulence RANS model, and at nominal operating conditions. Practical experience indicates that such a deterministic approach may result in considerably sub-optimal designs when the adopted aerodynamic model does not lead to accurate flow predictions or when the actual operating conditions differ from those considered in the design. One approach to address this shortcoming is to consi…
In this paper we address the problem of designing receding horizon control algorithms for linear discrete-time systems with parametric uncertainty. We do not consider presence of stochastic forcing or process noise in the system. It is assumed that the parametric uncertainty is probabilistic in nature with known probability density functions. We use generalized polynomial chaos theory to design the proposed stochastic receding horizon control algorithms. In this framework, the stochastic problem is converted to a deterministic problem in higher dimensional space. The performance of the propose…
Department of Operations Research and Financial Engineering（运筹学与金融工程系）；University of Waterloo（滑铁卢大学）
The most common approaches for solving multistage stochastic programming problems in the research literature have been to either use value functions ("dynamic programming") or scenario trees ("stochastic programming") to approximate the impact of a decision now on the future. By contrast, common industry practice is to use a deterministic approximation of the future which is easier to understand and solve, but which is criticized for ignoring uncertainty. We show that a parameterized version of a deterministic optimization model can be an effective way of handling uncertainty without the compl…
Kyungpook National University（庆北国立大学）；KINDI Centre for Computing Research, Qatar University（卡塔尔大学KINDI计算研究中心）；Nanyang Technological University（南洋理工大学）
Non-parametric tests can determine the better of two stochastic optimization algorithms when benchmarking results are ordinal, like the final fitness values of multiple trials. For many benchmarks, however, a trial can also terminate once it reaches a pre-specified target value. When only some trials reach the target value, two variables characterize a trial's outcome: the time it takes to reach the target value (or not) and its final fitness value. This paper describes a simple way to impose linear order on this two-variable trial data set so that traditional non-parametric methods can determ…
From economics point of view, we investigate a new optimal control problem driven by a stochastic differential equation with a multi-time states cost functional. By constructing a series of first-order adjoint equations, we establish the stochastic maximum principle and sufficient optimality conditions for this new optimal control problem. A constraints problem also be studied. In the end, we develop a near optimal control problem for a general cost functional.
We investigate the computation of the gradient of the value function in parametric convex optimization problems. We derive general expression for the gradient of the value function in terms of the cost function, constraints and Lagrange multipliers. In particular, we show that for the strictly convex parametric quadratic program the value function is continuously differentiable at every point in the interior of feasible space for which the Linear Independent Constraint Qualification holds.
Derivatives of Stochastic Gradient Descent in parametric optimization
Université Paul Sabatier（保罗·萨巴蒂埃大学）；Toulouse School of Economics（图卢兹经济学院）；Université Toulouse Capitole（图卢兹第一大学）；CNRS（法国国家科学研究中心）；Laboratoire J. A. Dieudonné（迪厄多内实验室）
We propose a tube-based guaranteed cost model predictive controller considering a homothetic formulation for constrained linear systems subject to multiplicative structured norm-bounded uncertainties. It provides an upper bound to the general min-max model predictive control. The invariance property of the proposed tube holds for any arbitrary scaling. It yields a second-order cone programming problem which is less computationally expensive than standard semi-definite programming problems. We also present a numerical example with a comparative study among the proposed approach, an open-loop gu…
Information Processing and Telecommunications Center（信息处理与电信中心）；Universidad Politécnica de Madrid（马德里理工大学）
Multiagent systems where agents interact among themselves and with a stochastic environment can be formalized as stochastic games. We study a subclass named Markov potential games (MPGs) that appear often in economic and engineering applications when the agents share a common resource. We consider MPGs with continuous state-action variables, coupled constraints and nonconvex rewards. Previous analysis followed a variational approach that is only valid for very simple cases (convex rewards, invertible dynamics, and no coupled constraints); or considered deterministic dynamics and provided open-…
In this paper we study the optimal stochastic control problem for stochastic differential systems reflected in a domain. The cost functional is a recursive one, which is defined via generalized backward stochastic differential equations developed by Pardoux and Zhang [20]. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. For this, we also prove some new estimates for stochastic differential systems reflected in a doma…
We explore generalizations of some integrated learning and optimization frameworks for data-driven contextual stochastic optimization that can adapt to heteroscedasticity. We identify conditions on the stochastic program, data generation process, and the prediction setup under which these generalizations possess asymptotic and finite sample guarantees for a class of stochastic programs, including two-stage stochastic mixed-integer programs with continuous recourse. We verify that our assumptions hold for popular parametric and nonparametric regression methods.
In this paper, we study Nash equilibrium payoffs for nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of a doubly controlled backward stochastic differential equation. Our results extend former ones by Buckdahn, Cardaliaguet and Rainer (2004) and are based on a backward stochastic differential equation approach.
Car-sharing issue is a popular research field in sharing economy. In this paper, we investigate the car-sharing relocation problem (CSRP) under uncertain demands. Normally, the real customer demands follow complicating probability distribution which cannot be described by parametric approaches. In order to overcome the problem, an innovative framework called Data-Driven Kernel Stochastic Programming (DDKSP) that integrates a non-parametric approach - kernel density estimation (KDE) and a two-stage stochastic programming (SP) model is proposed. Specifically, the probability distributions are de…
