Multistage Adaptive Optimization Research Articles

Hybrid strategies using linear and piecewise-linear decision rules for multistage adaptive linear optimization

Decision rules offer a rich and tractable framework for solving certain classes of multistage adaptive optimization problems. Recent literature has shown the promise of using linear and nonlinear decision rules in which wait-and-see decisions are represented as functions, whose parameters are decision variables to be optimized, of the underlying uncertain parameters. Despite this growing success, solving real-world stochastic optimization problems can become computationally prohibitive when using nonlinear decision rules, and in some cases, linear ones. Consequently, decision rules that offer a competitive trade-off between solution quality and computational time become more attractive. Whereas the extant research has always used homogeneous (i.e., either linear or piecewise-linear) decision rules, the major contribution of this paper is a computational exploration of hybrid decision rules combining the benefits of the two classes of decision rules. We first verify empirically that having higher uncertainty resolution or more linear pieces in early stages is more significant than having it in late stages in terms of solution quality. Then, we compare non-increasing and non-decreasing (i.e., higher uncertainty resolution in early and late stages, respectively) hybrid decision rules in a computational study to illustrate the trade-off between solution quality and computational cost. We also demonstrate a case where, unexpectedly, a linear decision rule is superior to a more complex piecewise-linear decision rule within a simulator. This observation bolsters the need to assess the quality of decision rules obtained from a look-ahead model within a simulator rather than just using the optimal look-ahead objective function value.

Multistage adaptive optimization using hybrid scenario and decision rule formulation

AbstractScenario‐based stochastic programming and linear decision rule (LDR)‐based robust optimization are prevalent methods for solving multistage adaptive optimization (MSAP) problems. In practical applications such as capacity expansion planning of chemical processes, often multiple sources of uncertainty affect the problem which introduces challenges to traditional stochastic optimization methods. While a large number of uncertain parameters exist in the problem, using scenario‐based method results in very large problem size and the solution becomes computationally expensive. In addition, when the constraints include multiplication of uncertain parameters and adaptive variables, the constraints are not linear with respect to uncertain parameters when the LDR method is used. In order to address these challenges, we propose two different hybrid methods where scenario and decision rule methods are combined to solve the MSAP problem. The article demonstrates the computational performance of the proposed hybrid methods using two chemical process planning examples.

Multistage Adaptive Optimization for Steelmaking and Continuous Casting Scheduling under Processing Time Uncertainty

The scheduling of steelmaking and continuous casting process is an important exercise for the steel industry. Uncertain processing time can significantly affect the schedule performance. To tackle this uncertainty, a multistage adaptive optimization method is proposed in this work. The proposed method can effectively handle a large number of uncertain parameters and generate an adjustable scheduling policy for online use. The method is flexible in choosing a robust objective or an expectation objective. Results demonstrate that the stochastic adaptive solution is less conservative and can be employed for optimal scheduling of the steelmaking process.

Binary decision rules for multistage adaptive mixed-integer optimization

Decision rules provide a flexible toolbox for solving computationally demanding, multistage adaptive optimization problems. There is a plethora of real-valued decision rules that are highly scalable and achieve good quality solutions. On the other hand, existing binary decision rule structures tend to produce good quality solutions at the expense of limited scalability and are typically confined to worst-case optimization problems. To address these issues, we first propose a linearly parameterised binary decision rule structure and derive the exact reformulation of the decision rule problem. In the cases where the resulting optimization problem grows exponentially with respect to the problem data, we provide a systematic methodology that trades-off scalability and optimality, resulting to practical binary decision rules. We also apply the proposed binary decision rules to the class of problems with random-recourse and show that they share similar complexity as the fixed-recourse problems. Our numerical results demonstrate the effectiveness of the proposed binary decision rules and show that they are (i) highly scalable and (ii) provide high quality solutions.

Design of Near Optimal Decision Rules in Multistage Adaptive Mixed-Integer Optimization

In recent years, decision rules have been established as the preferred solution method for addressing computationally demanding, multistage adaptive optimization problems. Despite their success, existing decision rules (a) are typically constrained by their a priori design and (b) do not incorporate in their modeling adaptive binary decisions. To address these problems, we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions, respectively. We then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed-integer optimization. We demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems. We provide global lower bounds and show that our approach is (i) practically tractable and (ii) provides high quality solutions that outperform alternative methods.

A Hierarchy of Near-Optimal Policies for Multistage Adaptive Optimization

In this paper, we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty, applicable to multistage robust optimization and stochastic programming. We introduce a hierarchy of near-optimal polynomial disturbance-feedback control policies, and show how these can be computed by solving a single semidefinite programming problem. The approach yields a hierarchy parameterized by a single variable (the degree of the polynomial policies), which controls the trade-off between the optimality gap and the computational requirements. We evaluate our framework in the context of three classical applications-two in inventory management, and one in robust regulation of an active suspension system-in which very strong numerical performance is exhibited, at relatively modest computational expense.