Compacting multistage stochastic programming models through a new implicit extensive framework
Introduction
Stochastic optimization (SO) is a representation of practical problems, where uncertainty plays a major role. These problems can be approximated by equivalent convex problems, which should be feasible, solvable, dualizable and stable [57], [58]. These equivalent representations are not only known as deterministic equivalent but also denominated as extensive form. The size of these models might become large, and computationally expensive (or even intractable), to represent all characteristic of realistic environments properly. Thus, many research streams have risen to cope with such complexity. For instance, the algebraic modeling languages (AML), which have been applied to produce more compact representations with the aim of mitigating this prohibitive demand for computational resources. Several optimization modeling languages have already been proposed: GAMMS [6], AMPL [26], MODLER [32], AIMMS [5], and other examples are found on Kallrath [40]. The benefits of AMLs were outlined as a lower overall cost alternative to the matrix generators and as a convenient option in terms of formulations’ understandability, maintainability and verifiability [24], [43]. Even though, the early versions of these AMLs did not include support for more complex data structures, they have been improved and extended, enhancing their capability to describe a larger range of models, for instance, piecewise-linear functions and networks [25]. Entriken [21] and Colombo et al. [8] present other extensions.
Regarding stochastic optimization, the description provided by AMLs must specify the optimization model according to the event tree, i.e. both structures must have the same features (cardinality, scenario links, number of stages, number of scenarios). This congruence between the simulated scenario tree and algebraic optimization model brings some challenges. For example, the constraints that establish the link between the parent–child pair of each node are particularly difficult to be generated from an AML. This procedure is known by the nonanticipativity constraints, which prevents any decision making from taking advantage of information that will only become available in the future. Hence, scenarios that share a common history up to a point must have identical solutions until their information paths diverge. The correct information disclosure is what makes the stochastic optimization approach so realistic and yields computational challenges [27].
The unfolding of information for scenario trees is assigned by a tree-structure indexing system in AMLs. These tree-structures are defined by nonanticipativity constraints which can be explicitly or implicitly formulated in multistage stochastic optimization models. On the one hand, data from a nonanticipativity scenario portion is replicated, taking them as independent scenarios and explicit constraints that bind these redundant data, overloading the size of any multistage stochastic program [56]. There are some language extensions whose syntax are mostly based on the explicit formulation of the nonanticipativity, for example, the SPiNE [54]. On the other hand, Gassmann and Ireland [30], [31] realized that stochastic optimization modeling could greatly benefit from the implicit extensive form, which enables the declaration of the scenarios and the nonanticipative constraints based on only one dataset, thereby only one set of decision variables is defined. Thus, it reduces the model because only the essential random decision variables have to be described and the model becomes as concise as possible [29]. Based on this methodology, Valente et al. [53] created a notation that makes the filtration process the central syntactic construction of multistage stochastic recourse problems. In a comparable way, Calfa [7] used external matrices to inform the nonanticipativity condition implicitly and measured the memory demand decrease.
Unlike Valente et al. [53] and Colombo et al. [8] who make an extension on an AML language and Fourer and Lopes [28] who create a modeling tool, the contribution of this study is the description of a novel framework. This novel framework includes all steps of stochastic optimization (i.e., stochastic process simulation, optimization model and their interface) using the implicit extensive form for compact formulation. This architecture is based on ideas from Domenica et al. [17] who argue mainly for the scenario generation (simulation), and thoughts from Calfa [7] who mostly discuss stochastic optimization models. The stochastic processes are simulated, and an unique database is produced, only containing data for the distinguishable paths. It is modeled under the scenario generation perspective through a multi-way tree, and is implemented with an equivalent binary tree with dynamic memory allocation following the left-child-right-sibling strategy. An AML is employed to formulate the stochastic optimization model with information from a filtration process to compose bundles, following some ideas from Valente et al. [53]. The concept of bundles is extended for corresponding nonanticipativity constraints implicitly and differs from Valente [52] by referencing the relationship between a node and their children as suggested by [9]. This approach yields an implicit deterministic equivalent structure, which is generic enough to be applied on other SP models, for instance, the ones with recourse, chance constraints, or even integrated chance constraints, proposed by Charnes and Cooper [10].
The framework follows the implicit extensive form guidelines to allow the formalization of shorter stochastic programs: the simulation of stochastic processes is delineated for generating simpler data structures saving computational memory. Moreover, the concept of bundles is extended to provide an algebraic definition of nonanticipative constraints which makes possible their implementation without any AML extension. The proposed approach is tested for an asset-liability management (ALM) multistage stochastic program with joint chance constraints finding the optimal solutions for big model instances without the support of any relaxation or decomposition procedure.
The remainder of this paper is organized as follows. In Section 2, a theoretic multistage stochastic model with joint chance constraint and the implicit formulation of nonanticipativity constraints is formulated. In Section 3, the application of the filtration of a space probability and bundles to generate the implicitly extensive form of a multistage stochastic program is detailed. Furthermore, in Section 3, the scenario generation of an event tree driven by the filtration and bundles concepts is described. This framework is applied to an asset-liability management problem (ALM) in Section 4. Empirical results are discussed in Section 5. The final remarks are presented in Section 6. In Appendix A, a numerical example of bundle generation is provided. In Appendix B, the use of an AML for encoding the implicit deterministic equivalent version of the multistage stochastic program is also presented.
Section snippets
Multistage stochastic model with joint chance constraint
A stochastic multistage optimization problem presents a set for some dimension in which are decision variables . The parameter is a continuous random variable (RV) defined in the probability space . For instance, this RV could be asset returns, demand for products or weather conditions discretized throughout multiple time stages. Further, stages are considered as time periods and decisions as asset allocations, without causing the loss of generality. Thus, the portfolio
Proposed modeling framework
In this section, the framework is described as it is depicted in Fig. 1. It is reliant on the filtration process, , that originates a conditioned stochastic process such as informing how data will be disclosed in the scenarios of an event tree. Initially, the fitting of the filtration process and bundles is explained in Section 3.1. Next, in Section 3.2, an algorithm for bundle generation is presented. These definitions are used to compute multistage stochastic programs
Practical application – ALM
The scenario approach for multistage stochastic optimization has been used for a wide range of applications, e.g., thermal power system management [33], hydropower operations [59], portfolio optimization [16], freight transportation [45], but the computational challenges imposed by the number of required variables to assign these scenarios is limiting the applicability of this technique. In this session, the asset-liability management (ALM), a classical financial problem, is formulated using
Simulations and empirical results
In this section, the computational results obtained by optimizing the ALM problem described in Section 4 solving (27), (28), (29), (30) and (32), (33), (35) are presented. Using the implicit extensive form through the filtration-oriented approach with bundles and the implicit scenario generation with natural correspondence (Section 3), it was possible to perform the scenario generation and optimization of scenario trees with over 160,000 scenarios, without ex-ante relaxation or decomposition
Conclusions
The size of multistage stochastic programs can grow very quickly. Consequently, a large scenario tree becomes hard to compute. A framework which enables the solving of larger instances of multistage stochastic programs, without the support of any relaxation or decomposition on the initial program, is proposed. The scenario generation is carried out by a compact memory representation based on left-child, right-sibling description, relying on Knuth transformation encoded in C/C++. The
Author contributions
Alan Delgado de Oliveira: Conceptualization, methodology, software, data curation, writing – original draft preparation, investigation. Tiago Pascoal Filomena: Investigation, supervision, writing – reviewing and editing, validation.
Conflict of interest
The authors declare that there is no conflict of interest.
Acknowledgements
The authors want to thank the two anonymous referees and the editor for their valuable contributions. This work was funded by CAPES and CNPQ (Grant 302777/2017-2), two of the Brazilian research agencies.
Alan Delgado de Oliveira holds a Bachelor's degree in Computer science (2011), a Master's degree in Management Science (major in Quantitative Methods) (2013) and a Ph.D. in Management Science (major in Operations research) (2018) from the Federal University of Rio Grande do Sul. Lecturer of the Business School of Federal University of Rio Grande do Sul since 2019. He has taught courses on Decision science, Machine Learning and Deep Learning. His research interests are focused in stochastic
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Alan Delgado de Oliveira holds a Bachelor's degree in Computer science (2011), a Master's degree in Management Science (major in Quantitative Methods) (2013) and a Ph.D. in Management Science (major in Operations research) (2018) from the Federal University of Rio Grande do Sul. Lecturer of the Business School of Federal University of Rio Grande do Sul since 2019. He has taught courses on Decision science, Machine Learning and Deep Learning. His research interests are focused in stochastic programming research, non-linear programming, simulation, machine learning and deep learning.
Tiago Filomena is an Associate Professor of Operations Research and Finance at Federal University of Rio Grande do Sul, Brazil. He has a Ph.D. in Engineering (major in Operations Research & Financial Engineering) from The George Washington University, Washington, DC; with both a M.Sc. and B.Sc. in Engineering from Federal University of Rio Grande do Sul. His research interests include stochastic programming, non-linear programming, simulation, stochastic differential equations and data mining.