Progressive Hedging Research Articles

N-1 Reliable Unit Commitment via Progressive Hedging

AbstractProgressive hedging is a scenario-based decomposition method that can be applied to solve the stochastic unit commitment problem. Progressive hedging is not guaranteed to converge to the global optimal solution since unit commitment involves integer variables. Slow convergence rates and cyclic behaviors have been previously observed in practice. Hedging is conventionally performed on the unit commitment status variables. In this paper, several hedging methods are proposed to improve progressive hedging for the N-1 (single contingency) stochastic unit commitment problem. In particular, hedging on the start-up and shutdown variables is proposed and tested in this paper. The performance of the progressive hedging is rather sensitive to the choice in the penalty factors. Thus, several strategies for choosing the penalty factors are evaluated for the cases when using the unit commitment status variables versus using the start-up and shutdown binary variables as the hedging mechanisms. Finally, a hybrid...

Adaptive monitoring of the progressive hedging penalty for reservoir systems management

Reservoir systems operations problems are in essence stochastic because of the uncertain nature of natural inflows. This leads to very large stochastic models that may not be easy to handle numerically. In this paper, we revisit the decomposition method developed by Rockafellar and Wets (Math Oper Res 119–147, 1991) by proposing new heuristics to initialize and dynamically adjust the penalty parameter of the augmented Lagrangian function on which this method is based. The heuristics are tested on multi-reservoir problems generated randomly and compared with the traditional strategy of setting the penalty parameter to a fixed value.

Scenario aggregation for supply chain quantity-flexibility contract

In this research, we apply a scenario aggregation approach to solving the supply chain contract model formulated by two-stage stochastic programming problem. The supply chain contract can achieve the coordination between the buyer and the supplier. We formulate the stochastic programming model for a quantity-flexibility contract. The scenario aggregation method called the progressive hedging method is used to solve this problem. Experimental results show the convergence behaviour of the algorithm and the sensitivity of parameters.

Incorporating Demand Response with Load Shifting into Stochastic Unit Commitment

High costs for fossil fuels and increasing installations of intermittent energy sources are imposing major challenges on power grid management. Uncertainty in generation and demand for electric energy require flexible generation capacity and stochastic optimization of generation schedules. Emerging smart grid technology is one mean to increase efficiency in power generation and to mitigate effects of increasing uncertainty. In this paper, we analyze the potential of demand side resources (DSRs) that can be dispatched to reduce load at peak times. We present a stochastic dynamic programming model for the unit commitment problem in a day ahead market and include dispatch decisions for DSRs. Unlike previous research, we model the stochastic load shifting effect to previous and subsequent periods that must be taken into account when making dispatch decisions. We also present a solution algorithm that combines an approximate dynamic programming algorithm with stochastic progressive hedging to solve the problem. We prove convergence results for our algorithms and derive a lower bound on the optimal solution. Using data from the California Independent System Operator region, we show that our approach can solve real world instances in reasonable time. We conduct an extensive numerical study using real world data to show that substantial savings can be achieved by utilizing DSRs.

The sample average approximation method for empty container repositioning with uncertainties

One of the challenges faced by liner operators today is to effectively operate empty containers in order to meet demand and to reduce inefficiency in an uncertain environment. To incorporate uncertainties in the operations model, we formulate a two-stage stochastic programming model with random demand, supply, ship weight capacity, and ship space capacity. The objective of this model is to minimize the expected operational cost for Empty Container Repositioning (ECR). To solve the stochastic programs with a prohibitively large number of scenarios, the Sample Average Approximation (SAA) method is applied to approximate the expected cost function. To solve the SAA problem, we consider applying the scenario aggregation by combining the approximate solution of the individual scenario problem. Two heuristic algorithms based on the progressive hedging strategy are applied to solve the SAA problem. Numerical experiments are provided to show the good performance of the scenario-based method for the ECR problem with uncertainties.

PySP: modeling and solving stochastic programs in Python

Although stochastic programming is a powerful tool for modeling decision-making under uncertainty, various impediments have historically prevented its wide-spread use. One factor involves the ability of non-specialists to easily express stochastic programming problems as extensions of their deterministic counterparts, which are typically formulated first. A second factor relates to the difficulty of solving stochastic programming models, particularly in the mixed-integer, non-linear, and/or multi-stage cases. Intricate, configurable, and parallel decomposition strategies are frequently required to achieve tractable run-times on large-scale problems. We simultaneously address both of these factors in our PySP software package, which is part of the Coopr open-source Python repository for optimization; the latter is distributed as part of IBM’s COIN-OR repository. To formulate a stochastic program in PySP, the user specifies both the deterministic base model (supporting linear, non-linear, and mixed-integer components) and the scenario tree model (defining the problem stages and the nature of uncertain parameters) in the Pyomo open-source algebraic modeling language. Given these two models, PySP provides two paths for solution of the corresponding stochastic program. The first alternative involves passing an extensive form to a standard deterministic solver. For more complex stochastic programs, we provide an implementation of Rockafellar and Wets’ Progressive Hedging algorithm. Our particular focus is on the use of Progressive Hedging as an effective heuristic for obtaining approximate solutions to multi-stage stochastic programs. By leveraging the combination of a high-level programming language (Python) and the embedding of the base deterministic model in that language (Pyomo), we are able to provide completely generic and highly configurable solver implementations. PySP has been used by a number of research groups, including our own, to rapidly prototype and solve difficult stochastic programming problems.

Applying different decomposition schemes using the progressive hedging algorithm to the operation planning problem of a hydrothermal system

This paper addresses the solution of the Medium-Term Operation Planning (MTOP) problem. This operation scheduling problem aims to define the output of each power plant to minimize the expected production cost over a medium-term planning horizon. In hydrothermal systems, the MTOP is strongly influenced by the amount of water inflow to the reservoirs of hydroelectric plants, which is uncertain. Thus, the System Operator (SO) must consider these uncertainties in the problem resolution, which can be solved by means of Stochastic Programming (SP) techniques, such as the Progressive Hedging (PH) proposed in this paper. This paper presents suitable decomposition schemes to reduce the CPU time, such that it is possible to use a detailed model for the problem. Additionally, a parallel computational approach based on the PH algorithm is also implemented. To demonstrate the efficiency of the proposed schemes, a large hydrothermal system is investigated in the case studies.

Comparing stochastic optimization methods to solve the medium-term operation planning problem

The Medium-Term Operation Planning (MTOP) of hydrothermal systems aims to define the generation for each power plant, minimizing the expected operating cost over the planning horizon. Mathematically, this task can be characterized as a linear, stochastic, large-scale problem which requires the application of suitable optimization tools. To solve this problem, this paper proposes to use the Nested Decomposition, frequently used to solve similar problems (as in Brazilian case), and Progressive Hedging, an alternative method, which has interesting features that make it promising to address this problem. To make a comparative analysis between these two methods with respect to the quality of the solution and the computational burden, a benchmark is established, which is obtained by solving a single Linear Programming problem (the Deterministic Equivalent Problem). An application considering a hydrothermal system is carried out.

Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems

Numerous planning problems can be formulated as multi-stage stochastic programs and many possess key discrete (integer) decision variables in one or more of the stages. Progressive hedging (PH) is a scenario-based decomposition technique that can be leveraged to solve such problems. Originally devised for problems possessing only continuous variables, PH has been successfully applied as a heuristic to solve multi-stage stochastic programs with integer variables. However, a variety of critical issues arise in practice when implementing PH for the discrete case, especially in the context of very difficult or large-scale mixed-integer problems. Failure to address these issues properly results in either non-convergence of the heuristic or unacceptably long run-times. We investigate these issues and describe algorithmic innovations in the context of a broad class of scenario-based resource allocation problem in which decision variables represent resources available at a cost and constraints enforce the need for sufficient combinations of resources. The necessity and efficacy of our techniques is empirically assessed on a two-stage stochastic network flow problem with integer variables in both stages.

On the implementation of a log-barrier progressive hedging method for multistage stochastic programs

A progressive hedging method incorporated with self-concordant barrier for solving multistage stochastic programs is proposed recently by Zhao [G. Zhao, A Lagrangian dual method with self-concordant barrier for multistage stochastic convex nonlinear programming, Math. Program. 102 (2005) 1–24]. The method relaxes the nonanticipativity constraints by the Lagrangian dual approach and smoothes the Lagrangian dual function by self-concordant barrier functions. The convergence and polynomial-time complexity of the method have been established. Although the analysis is done on stochastic convex programming, the method can be applied to the nonconvex situation. We discuss some details on the implementation of this method in this paper, including when to terminate the solution of unconstrained subproblems with special structure and how to perform a line search procedure for a new dual estimate effectively. In particular, the method is used to solve some multistage stochastic nonlinear test problems. The collection of test problems also contains two practical examples from the literature. We report the results of our preliminary numerical experiments. As a comparison, we also solve all test problems by the well-known progressive hedging method.

Practical aspects in solving the medium-term operation planning problem of hydrothermal power systems by using the progressive hedging method

The medium-term operation planning (MTOP) of hydrothermal systems aims to define generation targets that minimize the expected operation cost over the planning period, considering a set of constraints with respect to the power system and the generating units. Mathematically, this problem can be characterized as a linear, stochastic and large scale, which requires the application of suitable optimization tools. The purpose of this paper is to apply the progressive hedging (PH) method to solve the MTOP problem. For that, two critical aspects of the PH are addressed: the use of warm start and the choice of a suitable penalty parameter. In addition, this paper compares the performance of the PH to other popular methods, the full high-scale linear programming problem, also known as deterministic equivalent, and the nested decomposition, when solving similar problems. As a result we show that the PH is a competitive method to solve the MTOP problem.

Progressive Hedging Innovations for a Class of Stochastic Resource Allocation Problems

Progressive hedging (PH) is a scenario-based decomposition technique for solving stochastic programs. While PH has been successfully applied to a number of problems, a variety of issues arise when implementing PH in practice, especially when dealing with very difficult or large-scale mixed-integer problems. In particular, decisions must be made regarding the value of the penalty parameter p, criteria for termination, and techniques for accelerating convergence. We investigate these issues in the context of a class of scenario-based resource allocation problem in which decision variables represent resources available at a cost and constraints enforce needs for sufficient combinations of resources. We introduce techniques for selecting p in proportion to the unit-cost of the associated decision variable, a mathematically-based heuristic for setting the p parameter, novel techniques for convergence acceleration, and methods for detecting and recovering from oscillatory behavior. The efficacy of these techniques is empirically assessed on two test cases: a difficult laboratory stochastic network flow problem and a largescale, real-world, robust spare parts procurement planning problem. Both test problems have integer variables in both stages. Additionally, we demonstrate that variable-specific p values are more effective than traditional fixed p values, and that the PH algorithm can serve as an effective heuristic even when the mathematical conditions for convergence are not respected.

Using scenario trees and progressive hedging for stochastic inventory routing problems

The Stochastic Inventory Routing Problem is a challenging problem, combining inventory management and vehicle routing, as well as including stochastic customer demands. The problem can be described by a discounted, infinite horizon Markov Decision Problem, but it has been showed that this can be effectively approximated by solving a finite scenario tree based problem at each epoch. In this paper the use of the Progressive Hedging Algorithm for solving these scenario tree based problems is examined. The Progressive Hedging Algorithm can be suitable for large-scale problems, by giving an effective decomposition, but is not trivially implemented for non-convex problems. Attempting to improve the solution process, the standard algorithm is extended with locking mechanisms, dynamic multiple penalty parameters, and heuristic intermediate solutions. Extensive computational results are reported, giving further insights into the use of scenario trees as approximations of Markov Decision Problem formulations of the Stochastic Inventory Routing Problem.

Progressive hedging as a meta-heuristic applied to stochastic lot-sizing

In a great many situations, the data for optimization problems cannot be known with certainty and furthermore the decision process will take place in multiple time stages as the uncertainties are resolved. This gives rise to a need for stochastic programming (SP) methods that create solutions that are hedged against future uncertainty. The progressive hedging algorithm (PHA) of Rockafellar and Wets is a general method for SP. We cast the PHA in a meta-heuristic framework with the sub-problems generated for each scenario solved heuristically. Rather than using an approximate search algorithm for the exact problem as is typically the case in the meta-heuristic literature, we use an algorithm for sub-problems that is exact in its usual context but serves as a heuristic for our meta-heuristic. Computational results reported for stochastic lot-sizing problems demonstrate that the method is effective.

Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming

Many problems faced by decision makers are characterized by a multistage decision process with uncertainty about the future and some decisions constrained to take on values of either zero or one (for example, either open a facility at a location or do not open it). Although some mathematical theory exists concerning such problems, no general-purpose algorithms have been available to address them. In this article, we introduce the first implementation of general purpose methods for finding good solutions to multistage, stochastic mixed-integer (0, 1) programming problems. The solution method makes use of Rockafellar and Wets' progressive hedging algorithm that averages solutions rather than data. Solutions to the induced quadratic (0,1) mixed-integer subproblems are obtained using a tabu search algorithm. We introduce the notion of integer convergence for progressive hedging. Computational experiments verify that the method is effective. The software that we have developed reads standard (SMPS) data files.

A scenario aggregation algorithm for the solution of stochastic dynamic minimax problems

In this paper, we present a scenario aggregation algorithm for the solution of the dynamic minimax problem in stochastic programming. We consider the case where the joint probability distribution has a known finite support. The algorithm applies the Alternating Direction of Multipliers Method on a reformulation of the minimax problem using a double duality framework. The problem is solved by decomposition into scenario sub-problems, which are deterministic multi-period problems. Convergence properties are deduced from the Alternating Direction of Multipliers. The resulting algorithm can be seen as an extension of Rockafellar and Wets Progressive Hedging algorithm to the dynamic minimax context.

Scenario aggregation in single-resource production planning models with uncertain demand

Abstract In this paper we present an application of the scenario aggregation approach proposed by Rockafellar and Wets to a simple standard multi-product multi-period production planning problem with uncertain demand and setup cost modelled by logical zero-one variables. The uncertainty in demand is expressed by a number of demand scenarios. As compared with more traditional approaches that require distributional assumptions and/or estimates of parameters from historical demand data, the scenario approach offers greater flexibility and makes it possible to take subjective information into account. The scenario aggregation principle and the corresponding progressive hedging algorithm offer a theoretically sound basis for generating consistent solutions for production planning models with uncertain demand. Since the production planning problem studied in this paper is of mixed-integer type the original scenario aggregation approach cannot be applied directly. However, since the integer variables in the prod...