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We present a global optimization algorithm for two-stage stochastic nonlinear programs (NLPs). The algorithm uses a tailored reduced-space spatial branch and bound (BB) strategy to exploit the nearly decomposable structure of the problem. At each node in the BB scheme, a lower bound is constructed by relaxing the so-called non-anticipativity constraints and an upper bound is constructed by fixing the first-stage variables to the current candidate solution. A key advantage of this approach is that both lower and upper bounds can be computed by solving individual scenario subproblems. Another key property of this approach is that it only needs to perform branching on the first-stage variables to guarantee convergence (branching on the second-stage variables is performed implicitly during the computation of lower and upper bounds). Notably, convergence results for this scheme also hold for two-stage stochastic MINLPs with mixed-integer first-stage variables and continuous recourse variables. We present a serial implementation of the algorithm in Julia, that we call SNGO. The implementation is interfaced to the structured modeling language Plasmo.jl, which facilitates benchmarking and model processing. Our implementation incorporates typical features that help accelerate the BB search such as LP-based lower bounding techniques, local search-based upper bounding techniques, and relaxation-based bounds tightening techniques. These strategies require the solution of extensive forms of the stochastic program but can potentially be solved using structured interior-point solvers (when the problem is an NLP). Numerical experiments are performed for a controller tuning formulation, a parameter estimation formulation for microbial growth models, and a stochastic test set from GLOBALlib. We compare the computational results against SCIP and demonstrate that the proposed approach achieves significant speedups. [ABSTRACT FROM AUTHOR]

We study the problem of designing systems in order to minimize cost while meeting a given flexibility target. Flexibility is attained by enforcing a joint chance constraint, which ensures that the system will exhibit feasible operation with a given target probability level. Unfortunately, joint chance constraints are complicated mathematical objects that often need to be reformulated using mixed-integer programming (MIP) techniques. In this work, we cast the design problem as a conflict resolution problem that seeks to minimize cost while maximizing flexibility. We propose a purely continuous relaxation of this problem that provides a significantly more scalable approach relative to MIP methods and show that the formulation delivers solutions that closely approximate the Pareto set of the original joint chance-constrained problem. [ABSTRACT FROM AUTHOR]

We present an asynchronous bundle-trust-region algorithm within the context of Lagrangian dual decomposition for stochastic mixed-integer programs. The approach solves the Lagrangian master problem by using a bundle method with a trust-region constraint. This scheme enables asynchronous computations and can thus help mitigate severe load imbalance issues (associated with the solution of scenario subproblems) and improve parallel efficiency. We provide a convergence analysis and an implementation of the proposed scheme. We also present extensive numerical results on eighty instances of a large-scale stochastic unit commitment problem, and demonstrate that the proposed approach provides significant reductions in solution time and achieves strong scaling. [ABSTRACT FROM AUTHOR]

We present an asynchronous bundle-trust-region algorithm within the context of Lagrangian dual decomposition for stochastic mixed-integer programs. The approach solves the Lagrangian master problem by using a bundle method with a trust-region constraint. This scheme enables asynchronous computations and can thus help mitigate severe load imbalance issues (associated with the solution of scenario subproblems) and improve parallel efficiency. We provide a convergence analysis and an implementation of the proposed scheme. We also present extensive numerical results on eighty instances of a large-scale stochastic unit commitment problem, and demonstrate that the proposed approach provides significant reductions in solution time and achieves strong scaling. [ABSTRACT FROM AUTHOR]

Highlights • Stochastic programming and statistical extrapolation to mitigate long-term extreme loads. • Formulations can be cast as large-scale nonlinear programming problems. • Approach can identify controller settings in a more systematic manner. Abstract We propose stochastic programming formulations to enforce mechanical load requirements in wind turbine controller design procedures. The formulations use statistical extrapolation techniques to construct a probabilistic (chance) constraint that controls the long-term probability of exceeding an extreme load threshold (as described by the IEC-61400 standard). This approach is based on the observation that extreme loads follow a generalized extreme value distribution, which enables an explicit algebraic representation of the probabilistic constraint. We illustrate how to use the formulations to find design parameters for pitch angle and torque controllers that maximize power output while constraining long-term extreme loads. We also use the formulation to explore the ability of a hypothetical model predictive controller to mitigate extreme loads. The proposed formulations can be cast as large-scale (but structured) nonlinear programming problems that contain up to 7.5 million variables and constraints. We show that these problems can be solved in less than 1.3 h on a multi-core computer with existing optimization tools. [ABSTRACT FROM AUTHOR]

We argue that stochastic programming provides a powerful framework to tune and analyze the performance limits of controllers. In particular, stochastic programming formulations can be used to identify controller settings that remain robust across diverse scenarios (disturbances, set‐points, and modeling errors) observed in real‐time operations. We also discuss how to use historical data and sampling techniques to construct operational scenarios and inference analysis techniques to provide statistical guarantees on limiting controller performance. Under the proposed framework, it is also possible to use risk metrics to handle extreme (rare) events and stochastic dominance concepts to conduct systematic benchmarking studies. We provide numerical studies to illustrate the concepts and to demonstrate that modern modeling and local/global optimization tools can tackle large‐scale applications. The proposed work also opens the door to data‐based controller tuning strategies that can be implemented in real‐time operations. © 2017 American Institute of Chemical Engineers AIChE J, 64: 2997–3010, 2018 [ABSTRACT FROM AUTHOR]

We present a stochastic programming framework for a multiple timescale economic dispatch problem to address integration of renewable energy resources into power systems. This framework allows certain slow-response energy resources to be controlled at an hourly timescale, while fast-response resources, including renewable resources, and related network decisions can be controlled at a sub-hourly timescale. To this end, we study two models motivated by actual scheduling practices of system operators. Using an external simulator as driver for sub-hourly wind generation, we optimize these economic dispatch models using stochastic decomposition, a sample-based approach for stochastic programming. Computational experiments, conducted on the IEEE-RTS96 system and the Illinois system, reveal that optimization with sub-hourly dispatch not only results in lower expected operational costs, but also predicts these costs with far greater accuracy than with models allowing only hourly dispatch. Our results also demonstrate that when compared with standard approaches using the extensive formulation of stochastic programming, the sequential sampling approach of stochastic decomposition provides better predictions with much less computational time. [ABSTRACT FROM AUTHOR]

• Present modeling abstraction that we call random field optimization. • Abstraction captures uncertainty that lives on continuous space-time domains. • Demonstrate the applicability using case studies. [Display omitted] We present a new modeling paradigm for optimization that we call random field optimization. Random fields are a powerful modeling abstraction that aims to capture the behavior of random variables that live on infinite-dimensional spaces (e.g., space and time) such as stochastic processes (e.g., time series, Gaussian processes, and Markov processes), random matrices, and random spatial fields. This paradigm involves sophisticated mathematical objects (e.g., stochastic differential equations and space-time kernel functions) and has been widely used in neuroscience, geoscience, physics, civil engineering, and computer graphics. Despite of this, however, random fields have seen limited use in optimization; specifically, existing optimization paradigms that involve uncertainty (e.g., stochastic programming and robust optimization) mostly focus on the use of finite random variables. This trend is rapidly changing with the advent of statistical optimization (e.g., Bayesian optimization) and multi-scale optimization (e.g., integration of molecular sciences and process engineering). Our work extends a recently-proposed abstraction for infinite-dimensional optimization problems by capturing more general uncertainty representations. Moreover, we discuss solution paradigms for this new class of problems based on finite transformations and sampling, and identify open questions and challenges. [ABSTRACT FROM AUTHOR]

• Present dual dynamic programming framework for mixed-integer MPC • Approach uses general state-spaces representations • Show that the approach is scalable and outperforms state-of-the-art solvers We study logistical investment flexibility provided by modular processing technologies for mitigating risk. Specifically, we propose a multi-stage stochastic programming formulation that determines optimal capacity expansion plans that mitigate demand uncertainty. The formulation accounts for multi-product dependencies between small/large units and for trade-offs between expected profit and risk. The formulation uses a cumulative risk measure to avoid time-consistency issues of traditional, per-stage risk-minimization formulations, and we argue that this approach is more compatible with typical investment metrics such as the net present value. Case studies of different complexity are presented to illustrate the developments. Our studies reveal that the Pareto frontier of a flexible setting (allowing for deployment of small units) dominates the Pareto frontier of an inflexible setting (allowing only for deployment of large units). Notably, this dominance is prevalent despite benefits arising from economies of scale of large processing units. [ABSTRACT FROM AUTHOR]

• Propose ADMM to precondition the iterative Krylov solver GMRES. • Approach overcomes scalability limits of Schur complement decomposition. • Use approach to solve problems arising in power networks with millions of variables. We study the solution of block-structured linear algebra systems arising in optimization by using iterative solution techniques. These systems are the core computational bottleneck of many problems of interest such as parameter estimation, optimal control, network optimization, and stochastic programming. Our approach uses a Krylov solver (GMRES) that is preconditioned with an alternating method of multipliers (ADMM). We show that this ADMM-GMRES approach overcomes well-known scalability issues of Schur complement decomposition in problems that exhibit a high degree of coupling. The effectiveness of the approach is demonstrated using linear systems that arise in stochastic optimal power flow problems and that contain up to 2 million total variables and 4000 coupling variables. We find that ADMM-GMRES is nearly an order of magnitude faster than Schur complement decomposition. Moreover, we demonstrate that the approach is robust to the selection of the augmented Lagrangian penalty parameter, which is a key advantage over the direct use of ADMM. [ABSTRACT FROM AUTHOR]

We present a scalable computing framework for the solution stochastic multiobjective optimization problems. The proposed framework uses a nested conditional value-at-risk (nCVaR) metric to find compromise solutions among conflicting random objectives. We prove that the associated nCVaR minimization problem can be cast as a standard stochastic programming problem with expected value (linking) constraints. We also show that these problems can be implemented in a modular and compact manner using PLASMO (a Julia -based structured modeling framework) and can be solved efficiently using PIPS-NLP (a parallel nonlinear solver). We apply the framework to a CHP design study in which we seek to find compromise solutions that trade-off cost, water, and emissions in the face of uncertainty in electricity and water demands. [ABSTRACT FROM AUTHOR]

Highlights: [•] Addresses the problem water shortages and effect weather conditions on plant performance. [•] Presents the problem as a stochastic programming problem. [•] Uses an efficient new algorithm BONUS to solve the problem. [•] Provides real time optimization solutions for various scenarios. [ABSTRACT FROM AUTHOR]