Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higherdimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of nonlinear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear and nonlinear decision rules, and we assess their performance in the context of a dynamic production planning problem.
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 finite number of pieces and solve the problem 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.
In this paper, we develop a unified framework for studying constrained robust optimal control problems with adjustable uncertainty sets. In contrast to standard constrained robust optimal control problems with known uncertainty sets, we treat the uncertainty sets in our problems as additional decision variables. In particular, given a finite prediction horizon and a metric for adjusting the uncertainty sets, we address the question of determining the optimal size and shape of the uncertainty sets, while simultaneously ensuring the existence of a control policy that will keep the system within its constraints for all possible disturbance realizations inside the adjusted uncertainty set. Since our problem subsumes the classical constrained robust optimal control design problem, it is computationally intractable in general. We demonstrate that by restricting the families of admissible uncertainty sets and control policies, the problem can be formulated as a tractable convex optimization problem. We show that our framework captures several families of (convex) uncertainty sets of practical interest, and illustrate our approach on a demand response problem of providing control reserves for a power system.
Decision rules provide a flexible toolbox for solving computationally demanding, multistage adaptive optimization problems. There is a plethora of realvalued 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.
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