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Showing papers on "Stochastic programming published in 2014"


Book
25 Sep 2014

1,100 citations


Journal ArticleDOI
TL;DR: An overview of developments in robust optimization since 2007 is provided to give a representative picture of the research topics most explored in recent years, highlight common themes in the investigations of independent research teams and highlight the contributions of rising as well as established researchers both to the theory of robust optimization and its practice.

742 citations


Journal ArticleDOI
TL;DR: The proposed SMPCL approach outperforms conventional model predictive control and shows performance close to MPC with full knowledge of future driver power request in standard and real-world driving cycles.
Abstract: This paper develops an approach for driver-aware vehicle control based on stochastic model predictive control with learning (SMPCL). The framework combines the on-board learning of a Markov chain that represents the driver behavior, a scenario-based approach for stochastic optimization, and quadratic programming. By using quadratic programming, SMPCL can handle, in general, larger state dimension models than stochastic dynamic programming, and can reconfigure in real-time for accommodating changes in driver behavior. The SMPCL approach is demonstrated in the energy management of a series hybrid electrical vehicle, aimed at improving fuel efficiency while enforcing constraints on battery state of charge and power. The SMPCL controller allocates the power from the battery and the engine to meet the driver power request. A Markov chain that models the power request dynamics is learned in real-time to improve the prediction capabilities of model predictive control (MPC). Because of exploiting the learned pattern of the driver behavior, the proposed approach outperforms conventional model predictive control and shows performance close to MPC with full knowledge of future driver power request in standard and real-world driving cycles.

375 citations


Book
08 Jul 2014
TL;DR: The authors introduce new material to reflect recent developments in stochastic programming, including an analytical description of the tangent and normal cones of chance constrained sets and in-depth analysis of dynamic risk measures and concepts of time consistency.
Abstract: Optimization problems involving stochastic models occur in almost all areas of science and engineering, such as telecommunications, medicine, and finance. Their existence compels a need for rigorous ways of formulating, analyzing, and solving such problems. This book focuses on optimization problems involving uncertain parameters and covers the theoretical foundations and recent advances in areas where stochastic models are available. In Lectures on Stochastic Programming: Modeling and Theory, Second Edition, the authors introduce new material to reflect recent developments in stochastic programming, including: an analytical description of the tangent and normal cones of chance constrained sets; analysis of optimality conditions applied to nonconvex problems; a discussion of the stochastic dual dynamic programming method; an extended discussion of law invariant coherent risk measures and their Kusuoka representations; and in-depth analysis of dynamic risk measures and concepts of time consistency, including several new results. Audience: This book is intended for researchers working on theory and applications of optimization. It also is suitable as a text for advanced graduate courses in optimization.

366 citations


Book
31 Jan 2014
TL;DR: The main idea presented here is that it is possible to decompose a complex decision making problem into a sequence of elementary decisions, where each decision of the sequence is solved using a (stochastic) multi-armed bandit (simple mathematical model for decision making in stochastic environments).
Abstract: This work covers several aspects of the optimism in the face of uncertainty principle applied to large scale optimization problems under finite numerical budget. The initial motivation for the research reported here originated from the empirical success of the so-called Monte-Carlo Tree Search method popularized in computer-go and further extended to many other games as well as optimization and planning problems. Our objective is to contribute to the development of theoretical foundations of the field by characterizing the complexity of the underlying optimization problems and designing efficient algorithms with performance guarantees. The main idea presented here is that it is possible to decompose a complex decision making problem (such as an optimization problem in a large search space) into a sequence of elementary decisions, where each decision of the sequence is solved using a (stochastic) multi-armed bandit (simple mathematical model for decision making in stochastic environments). This so-called hierarchical bandit approach (where the reward observed by a bandit in the hierarchy is itself the return of another bandit at a deeper level) possesses the nice feature of starting the exploration by a quasi-uniform sampling of the space and then focusing progressively on the most promising area, at different scales, according to the evaluations observed so far, and eventually performing a local search around the global optima of the function. The performance of the method is assessed in terms of the optimality of the returned solution as a function of the number of function evaluations. Our main contribution to the field of function optimization is a class of hierarchical optimistic algorithms designed for general search spaces (such as metric spaces, trees, graphs, Euclidean spaces, ...) with different algorithmic instantiations depending on whether the evaluations are noisy or noiseless and whether some measure of the ''smoothness'' of the function is known or unknown. The performance of the algorithms depend on the local behavior of the function around its global optima expressed in terms of the quantity of near-optimal states measured with some metric. If this local smoothness of the function is known then one can design very efficient optimization algorithms (with convergence rate independent of the space dimension), and when it is not known, we can build adaptive techniques that can, in some cases, perform almost as well as when it is known.

278 citations


Journal ArticleDOI
TL;DR: An overview of the use of Monte Carlo sampling-based methods for stochastic optimization problems with sampling is given, with the goal of introducing the topic to students and researchers and providing a practical guide for someone who needs to solve a stochastically optimization problem with sampling.

256 citations


Journal ArticleDOI
TL;DR: This paper extends the concept of minmax robustness to multi-objective optimization and calls this extension robust efficiency for uncertain multi- objective optimization problems, and uses ingredients from robust (single objective) and (deterministic) multi-Objective optimization to gain insight into the new area of robust multi- Objective optimization.

236 citations


Journal ArticleDOI
TL;DR: This work presents a novel accelerated primal-dual (APD) method for solving a class of deterministic and stochastic saddle point problems (SPPs) and demonstrates an optimal rate of convergence not only in terms of its dependence on the number of the iteration, but also on a variety of problem parameters.
Abstract: We present a novel accelerated primal-dual (APD) method for solving a class of deterministic and stochastic saddle point problems (SPPs). The basic idea of this algorithm is to incorporate a multistep acceleration scheme into the primal-dual method without smoothing the objective function. For deterministic SPP, the APD method achieves the same optimal rate of convergence as Nesterov's smoothing technique. Our stochastic APD method exhibits an optimal rate of convergence for stochastic SPP not only in terms of its dependence on the number of the iteration, but also on a variety of problem parameters. To the best of our knowledge, this is the first time that such an optimal algorithm has been developed for stochastic SPP in the literature. Furthermore, for both deterministic and stochastic SPP, the developed APD algorithms can deal with the situation when the feasible region is unbounded, as long as a saddle point exists. In the unbounded case, we incorporate the modified termination criterion introduced b...

232 citations


Journal ArticleDOI
TL;DR: In this article, a chance-constrained stochastic programming formulation with economic and reliability metrics is presented for the day-ahead scheduling, where reserve requirements and line flow limits are formulated as chance constraints in which power system reliability requirements are to be satisfied with a presumed level of high probability.
Abstract: This paper proposes a day-ahead stochastic scheduling model in electricity markets. The model considers hourly forecast errors of system loads and variable renewable sources as well as random outages of power system components. A chance-constrained stochastic programming formulation with economic and reliability metrics is presented for the day-ahead scheduling. Reserve requirements and line flow limits are formulated as chance constraints in which power system reliability requirements are to be satisfied with a presumed level of high probability. The chance-constrained stochastic programming formulation is converted into a linear deterministic problem and a decomposition-based method is utilized to solve the day-ahead scheduling problem. Numerical tests are performed and the results are analyzed for a modified 31-bus system and an IEEE 118-bus system. The results show the viability of the proposed formulation for the day-ahead stochastic scheduling. Comparative evaluations of the proposed chance-constrained method and the Monte Carlo simulation (MCS) method are presented in the paper.

222 citations


Journal ArticleDOI
TL;DR: A stochastic programming framework to choose optimal energy and reserve bids for the storage units that takes into account the fluctuating nature of the market prices due to the randomness in the renewable power generation availability is formulated.
Abstract: In this paper, we consider a scenario where a group of investor-owned independently-operated storage units seek to offer energy and reserve in the day-ahead market and energy in the hour-ahead market. We are particularly interested in the case where a significant portion of the power generated in the grid is from wind and other intermittent renewable energy resources. In this regard, we formulate a stochastic programming framework to choose optimal energy and reserve bids for the storage units that takes into account the fluctuating nature of the market prices due to the randomness in the renewable power generation availability. We show that the formulated stochastic program can be converted to a convex optimization problem to be solved efficiently. Our simulation results also show that our design can assure profitability of the private investment on storage units. We also investigate the impact of various design parameters, such as the size and location of the storage unit on increasing the profit.

209 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the operation of a GB integrated gas and electricity network considering the uncertainty in wind power forecast using three operational planning methods: deterministic, two-stage stochastic programming, and multistage Stochastic Programming.
Abstract: In many power systems, in particular in Great Britain (GB), significant wind generation is anticipated and gas-fired generation will continue to play an important role. Gas-fired generating units act as a link between the gas and electricity networks. The variability of wind power is, therefore, transferred to the gas network by influencing the gas demand for electricity generation. Operation of a GB integrated gas and electricity network considering the uncertainty in wind power forecast was investigated using three operational planning methods: deterministic, two-stage stochastic programming, and multistage stochastic programming. These methods were benchmarked against a perfect foresight model which has no uncertainty associated with the wind power forecast. In all the methods, thermal generators were controlled through an integrated unit commitment and economic dispatch algorithm that used mixed integer programming. The nonlinear characteristics of the gas network, including the gas flow along pipes and the operation of compressors, were taken into account and the resultant nonlinear problem was solved using successive linear programming. The operational strategies determined by the stochastic programming methods showed reductions of the operational costs compared to the solution of the deterministic method by almost 1%. Also, using the stochastic programming methods to schedule the thermal units was shown to make a better use of pumped storage plants to mitigate the variability and uncertainty of the net demand.

Journal ArticleDOI
TL;DR: In this article, a bi-level, three-stage Stochastic Mathematical Program with Equilibrium Constraints (SMPEC) is proposed for quantifying and optimizing travel time resilience in roadway networks under non-recurring natural or human-induced disaster events.
Abstract: A bi-level, three-stage Stochastic Mathematical Program with Equilibrium Constraints (SMPEC) is proposed for quantifying and optimizing travel time resilience in roadway networks under non-recurring natural or human-induced disaster events. At the upper-level, a sequence of optimal preparedness and response decisions is taken over pre-event mitigation and preparedness and post-event response stages of the disaster management life cycle. Assuming semi-adaptive user behavior exists shortly after the disaster and after the implementation of immediate response actions, the lower-level problem is formulated as a Partial User Equilibrium, where only affected users are likely to rethink their routing decisions. An exact Progressive Hedging Algorithm is presented for solution of a single-level equivalent, linear approximation of the SMPEC. A recently proposed technique from the literature that uses Schur’s decomposition with SOS1 variables in creating a linear equivalent to complementarity constraints is employed. Similarly, recent advances in piecewise linearization are exploited in addressing nonseparable link travel time functions. The formulation and solution methodology are demonstrated on an illustrative example.

Journal ArticleDOI
TL;DR: A design and planning approach is proposed for addressing general multi-period, multi-product closed-loop supply chains (CLSCs), structured as a 10-layer network, with uncertain levels in the amount of raw material supplies and customer demands.

Journal ArticleDOI
TL;DR: This work studies two optimization criteria for the transmission expansion planning problem under the robust optimization paradigm, where the maximum cost and maximum regret of the expansion plan over all uncertainties are minimized, respectively.
Abstract: Due to the long planning horizon, transmission expansion planning is typically subjected to a lot of uncertainties including load growth, renewable energy penetration, policy changes, etc. In addition, deregulation of the power industry and pressure from climate change introduced new sources of uncertainties on the generation side of the system. Generation expansion and retirement become highly uncertain as well. Some of the uncertainties do not have probability distributions, making it difficult to use stochastic programming. Techniques like robust optimization that do not require a probability distribution became desirable. To address these challenges, we study two optimization criteria for the transmission expansion planning problem under the robust optimization paradigm, where the maximum cost and maximum regret of the expansion plan over all uncertainties are minimized, respectively. With these models, our objective is to make planning decisions that are robust against all scenarios. We use a two-layer algorithm to solve the resulting tri-level optimization problems. Then, in our case studies, we compare the performance of the minimax cost approach and the minimax regret approach under different characterizations of uncertainties.

Journal ArticleDOI
TL;DR: In this article, a three-stage mixed-integer stochastic programming model for disaster response planning is presented, considering the opening of local distribution facilities, initial allocation of supplies, and last mile distribution of aid.
Abstract: This paper presents a three-stage mixed-integer stochastic programming model for disaster response planning, considering the opening of local distribution facilities, initial allocation of supplies, and last mile distribution of aid. The vehicles available for transportation, the state of the infrastructure and the demand of the potential beneficiaries are considered as stochastic elements. Extensive computational testing performed on realistic instances shows that the solutions produced by the stochastic programming model are significantly better than those produced by a deterministic expected value approach.

Journal ArticleDOI
TL;DR: In this paper, the authors present a new approach for exactly solving chance-constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints, using both decomposition and integer programming techniques to combine the results of these subproblems to yield strong valid inequalities.
Abstract: We present a new approach for exactly solving chance-constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods are only able to find provably good solutions in certain very special cases. Our approach uses both decomposition, to enable processing subproblems corresponding to one possible outcome at a time, and integer programming techniques, to combine the results of these subproblems to yield strong valid inequalities. Computational results on a chance-constrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixed-integer programming formulation and a simple decomposition approach that does not use strong valid inequalities. We also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels, as would be done when solving for the efficient frontier of risk and cost.

Journal ArticleDOI
TL;DR: In this paper, the authors discuss different approaches to construct uncertainty sets based on historical data, with the purpose of reducing the conservativeness while maintaining the same level of robustness of the solution.
Abstract: For robust unit commitment problems addressing load, renewable energy generation, and demand response uncertainties, constructing a proper uncertainty set plays an important role in determining the conservativeness of the model. In this letter, we discuss different approaches to construct uncertainty sets based on historical data, with the purpose of reducing the conservativeness while maintaining the same level of robustness of the solution.

Journal ArticleDOI
01 Jan 2014-Energy
TL;DR: In this article, a two-stage stochastic programming model is proposed to schedule energy and reserve provided by both of generating units and responsive loads in power systems with high penetration of wind power.

Journal ArticleDOI
TL;DR: In this paper, the authors introduce an efficient stochastic dynamic programming model to optimally charge an electric vehicle while accounting for the uncertainty inherent to its use, and show that the randomness intrinsic to driving needs has a substantial impact on the charging strategy to be implemented.

Journal ArticleDOI
TL;DR: The experimental analysis showed that the proposed GA with a new multi-parent crossover converges quickly to the optimal solution and thus exhibits a superior performance in comparison to other algorithms that also solved those problems.

Journal ArticleDOI
TL;DR: This paper comprehensively discusses about the effectiveness of incorporating a risk measure in a two-stage stochastic model and proves the capabilities and acceptability of the developed risk-averse approach and the affects of risk parameters in the model behavior.

Journal ArticleDOI
TL;DR: This work revisits the sparse linear algebra computations of the parallel solver PIPS with the goal of improving the shared-memory performance and decreasing the time to solution.
Abstract: We present a scalable approach and implementation for solving stochastic optimization problems on high-performance computers. In this work we revisit the sparse linear algebra computations of the parallel solver PIPS with the goal of improving the shared-memory performance and decreasing the time to solution. These computations consist of solving sparse linear systems with multiple sparse right-hand sides and are needed in our Schur-complement decomposition approach to compute the contribution of each scenario to the Schur matrix. Our novel approach uses an incomplete augmented factorization implemented within the PARDISO linear solver and an outer BiCGStab iteration to efficiently absorb pivot perturbations occurring during factorization. This approach is capable of both efficiently using the cores inside a computational node and exploiting sparsity of the right-hand sides. We report on the performance of the approach on high-performance computers when solving stochastic unit commitment problems of unpre...

Journal ArticleDOI
TL;DR: A stochastic dynamic programming model is developed that co-optimizes the use of energy storage for multiple applications, such as energy, capacity, and backup services, while accounting for market and system uncertainty.
Abstract: We develop a stochastic dynamic programming model that co-optimizes the use of energy storage for multiple applications, such as energy, capacity, and backup services, while accounting for market and system uncertainty. Using the example of a battery that has been installed in a home as a distributed storage device, we demonstrate the ability of the model to co-optimize services that ‘compete’ for the capacity of the battery. We also show that these multiple uses of a battery can provide substantive value.

Journal ArticleDOI
TL;DR: In this paper, a decentralized methodology to optimally schedule generating units while simultaneously determining the geographical allocation of the required reserve is proposed for an interconnected multi-area power system with cross-border trading in the presence of wind power uncertainty.
Abstract: This paper proposes a decentralized methodology to optimally schedule generating units while simultaneously determining the geographical allocation of the required reserve. We consider an interconnected multi-area power system with cross-border trading in the presence of wind power uncertainty. The multi-area market-clearing model is represented as a two-stage stochastic programming model. The proposed decentralized procedure relies on an augmented Lagrangian algorithm that requires no central operator intervention but just moderate interchanges of information among neighboring regions. The methodology proposed is illustrated using an example and a realistic case study.

Journal ArticleDOI
TL;DR: In this paper, a stochastic programming-based tool is proposed to support adaptive transmission planning under market and regulatory uncertainties, where the objective is to minimize expected transmission and generation costs over the time horizon.
Abstract: We propose a stochastic programming-based tool to support adaptive transmission planning under market and regulatory uncertainties. We model investments in two stages, differentiating between commitments that must be made now and corrective actions that can be undertaken as new information becomes available. The objective is to minimize expected transmission and generation costs over the time horizon. Nonlinear constraints resulting from Kirchhoff's voltage law are included. We apply the tool to a 240-bus representation of the Western Electricity Coordinating Council and model uncertainty using three scenarios with distinct renewable electricity mandates, emissions policies, and fossil fuel prices. We conclude that the cost of ignoring uncertainty (the cost of using naive deterministic planning methods relative to explicitly modeling uncertainty) is of the same order of magnitude as the cost of first-stage transmission investments. Furthermore, we conclude that heuristic rules for constructing transmission plans based on scenario planning can be as suboptimal as deterministic plans.

Journal ArticleDOI
TL;DR: The proposed method is shown to outperform deterministic model predictive control in terms of average EV charging cost and an enhancement to the classical discrete stochastic dynamic programming method is proposed.
Abstract: This paper investigates the application of stochastic dynamic programming to the optimization of charging and frequency regulation capacity bids of an electric vehicle (EV) in a smart electric grid environment. We formulate a Markov decision problem to minimize an EV's expected cost over a fixed charging horizon. We account for both Markov random prices and a Markov random regulation signal. We also propose an enhancement to the classical discrete stochastic dynamic programming method. This enhancement allows optimization over a continuous space of decision variables via linear programming at each state. Simple stochastic process models are built from real data and used to simulate the implementation of the proposed method. The proposed method is shown to outperform deterministic model predictive control in terms of average EV charging cost.

Journal ArticleDOI
14 Aug 2014-Energy
TL;DR: In this article, the optimal operation of a VPP considering the risk factors affecting its daily operation profits is modelled in both day ahead and balancing markets as a two-stage stochastic mixed integer linear programming in order to maximize a GenCo (generation companies) expected profit.

Journal ArticleDOI
TL;DR: In this paper, a probabilistic mixed integer linear programming model for the design of a reverse logistic network is proposed, which is first converted into an equivalent deterministic model. And then, a priority based genetic algorithm is proposed to find reverse logistics network to satisfy the demand imposed by manufacturing centers and recycling centers with minimum total cost under uncertainty condition.

Journal ArticleDOI
TL;DR: This paper presents a novel framework for smart energy management based on the concept of quality-of-service in electricity (QoSE), and derives several “hard” performance bounds for the proposed algorithm, and evaluates its performance with trace-driven simulations.
Abstract: Microgrid (MG) is a promising component for future smart grid (SG) deployment. The balance of supply and demand of electric energy is one of the most important requirements of MG management. In this paper, we present a novel framework for smart energy management based on the concept of quality-of-service in electricity (QoSE). Specifically, the resident electricity demand is classified into basic usage and quality usage. The basic usage is always guaranteed by the MG, while the quality usage is controlled based on the MG state. The microgrid control center (MGCC) aims to minimize the MG operation cost and maintain the outage probability of quality usage, i.e., QoSE, below a target value, by scheduling electricity among renewable energy resources, energy storage systems, and macrogrid. The problem is formulated as a constrained stochastic programming problem. The Lyapunov optimization technique is then applied to derive an adaptive electricity scheduling algorithm by introducing the QoSE virtual queues and energy storage virtual queues. The proposed algorithm is an online algorithm. We derive several “hard” performance bounds for the proposed algorithm, and evaluate its performance with trace-driven simulations. The simulation results demonstrate the efficacy of the proposed electricity scheduling algorithm.

Journal ArticleDOI
TL;DR: This paper presents a two-stage stochastic programming model used to design and manage biodiesel supply chains and solves the problem using algorithms that combine Lagrangian relaxation and L-shaped solution methods, and develops a case study using data from the state of Mississippi.