Class Of Dynamic Optimization Problems Research Articles

Survey of Dynamic Resource-Constrained Reward Collection Problems: Unified Model and Analysis

Dynamic resource allocation problems arise under a variety of settings. In “Survey of Dynamic Resource-Constrained Reward Collection Problems: Unified Model and Analysis,” Balseiro, Besbes, and Pizarro introduce a unifying model for a large class of dynamic optimization problems dubbed dynamic resource-constrained reward collection (DRC2) problems. Surveying the literature, they show that this class encompasses a variety of disparate and classical problems typically studied separately, such as dynamic pricing with capacity constraints, dynamic bidding with budgets, network revenue management, online matching, or order fulfillment. Furthermore, they establish that the DRC2 class is amenable to analysis by characterizing the performance of a central, certainty-equivalent heuristic. Notably, they provide a novel unifying analysis that isolates the drivers of performance, recovers as corollaries some existing specialized results, generalizes other existing results by weakening the assumptions required, and yields new results in specialized settings for which no such characterization was available.

Survey of Dynamic Resource Constrained Reward Collection Problems: Unified Model and Analysis

Dynamic resource allocation problems arise under a variety of settings and have been studied across disciplines such as Operations Research and Computer Science. The present paper introduces a unifying model for a very large class of dynamic optimization problems, that we call dynamic resource constrained reward collection (DRC2). We show that this class encompasses a variety of disparate and classical dynamic optimization problems such as dynamic pricing with capacity constraints, dynamic bidding with budgets, network revenue management, online matching, or order fulfillment, to name a few. Furthermore, we establish that the class of DRC2 problems, while highly general, is amenable to analysis. In particular, we characterize the performance of the fluid certainty equivalent control heuristic for this class. Notably, this very general result recovers as corollaries some existing specialized results, generalizes other existing results by weakening the assumptions required, but also yields new results in specialized settings for which no such characterization was available. As such, the DRC2 class isolates some common features of a broad class of problems, and offers a new object of analysis.

Solving online dynamic time-linkage problems under unreliable prediction

Dynamic time-linkage optimization problems (DTPs) are a special class of dynamic optimization problems (DOPs) with the feature of time-linkage. Time-linkage means that the decisions taken now could influence the problem states in future. Although DTPs are common in practice, attention from the field of evolutionary optimization is little. To date, the prediction method is the major approach to solve DTPs in the field of evolutionary optimization. However, in existing studies, the method of how to deal with the situation where the prediction is unreliable has not been studied yet for the complete Black-Box Optimization (BBO) case. In this paper, the prediction approach EA+predictor, proposed by Bosman, is improved to handle such situation. A stochastic-ranking selection scheme based on the prediction accuracy is designed to improve EA+predictor under unreliable prediction, where the prediction accuracy is based on the rank of the individuals but not the fitness. Experimental results show that, compared with the original prediction approach, the performance of the improved algorithm is competitive.

Dynamic consistency for stochastic optimal control problems

For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step t 0, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time steps t 0,t 1,…,T; at the next time step t 1, he is able to formulate a new optimization problem starting at time t 1 that yields a new sequence of optimal decision rules. This process can be continued until the final time T is reached. A family of optimization problems formulated in this way is said to be dynamically consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of dynamic consistency, well-known in the field of economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (Ann. Oper. Res. 152(1):5–22, 2007) and studied in the stochastic programming framework by Shapiro (Oper. Res. Lett. 37(3):143–147, 2009) and for Markov Decision Processes (MDP) by Ruszczynski (Math. Program. 125(2):235–261, 2010). We here link this notion with the concept of “state variable” in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen.

Conditional Choice Probability Estimation of Dynamic Discrete Choice Models With Unobserved Heterogeneity

We adapt the expectation–maximization algorithm to incorporate unobserved heterogeneity into conditional choice probability (CCP) estimators of dynamic discrete choice problems. The unobserved heterogeneity can be time-invariant or follow a Markov chain. By developing a class of problems where the difference in future value terms depends on a few conditional choice probabilities, we extend the class of dynamic optimization problems where CCP estimators provide a computationally cheap alternative to full solution methods. Monte Carlo results confirm that our algorithms perform quite well, both in terms of computational time and in the precision of the parameter estimates.

Estimation of Structural Optimization Models: A Note on Identification

Bajari, Benkard and Levin (2007) propose an estimation methodology for a broad class of dynamic optimization problems. To carry out their procedure, one needs to select a set of alternative policy functions and compare the implied expected payoffs with that from the data. We show that this can generally lead to objective functions that are not capable of consistently estimating an identified model.

DOTcvpSB, a software toolbox for dynamic optimization in systems biology

BackgroundMathematical optimization aims to make a system or design as effective or functional as possible, computing the quality of the different alternatives using a mathematical model. Most models in systems biology have a dynamic nature, usually described by sets of differential equations. Dynamic optimization addresses this class of systems, seeking the computation of the optimal time-varying conditions (control variables) to minimize or maximize a certain performance index. Dynamic optimization can solve many important problems in systems biology, including optimal control for obtaining a desired biological performance, the analysis of network designs and computer aided design of biological units.ResultsHere, we present a software toolbox, DOTcvpSB, which uses a rich ensemble of state-of-the-art numerical methods for solving continuous and mixed-integer dynamic optimization (MIDO) problems. The toolbox has been written in MATLAB and provides an easy and user friendly environment, including a graphical user interface, while ensuring a good numerical performance. Problems are easily stated thanks to the compact input definition. The toolbox also offers the possibility of importing SBML models, thus enabling it as a powerful optimization companion to modelling packages in systems biology. It serves as a means of handling generic black-box models as well.ConclusionHere we illustrate the capabilities and performance of DOTcvpSB by solving several challenging optimization problems related with bioreactor optimization, optimal drug infusion to a patient and the minimization of intracellular oscillations. The results illustrate how the suite of solvers available allows the efficient solution of a wide class of dynamic optimization problems, including challenging multimodal ones. The toolbox is freely available for academic use.

Stochastic Depletion Problems: Effective Myopic Policies for a Class of Dynamic Optimization Problems

This paper presents a general class of dynamic stochastic optimization problems we refer to as stochastic depletion problems. A number of challenging dynamic optimization problems of practical interest are stochastic depletion problems. Optimal solutions for such problems are difficult to obtain, both from a pragmatic computational perspective as well as from a theoretical perspective. As such, simple heuristics are desirable. We isolate two simple properties that, if satisfied by a problem within this class, guarantee that a myopic policy incurs a performance loss of at most 50% relative to the optimal adaptive control policy for that problem. We are able to verify that these two properties are satisfied for several interesting families of stochastic depletion problems and, as a consequence, we identify computationally efficient approximations to optimal control policies for a number of interesting dynamic stochastic optimization problems.

The economics of the thermohaline circulation—A problem with multiple thresholds of unknown locations

A potentially serious environmental threat facing humanity is the possibility of a collapse of the thermohaline circulation. Resulting climate changes, including absolute cooling near Greenland and northwest Europe, could be abrupt. Collapse of the thermohaline circulation may be triggered if the temperature or the rate of temperature change exceeds certain thresholds. The locations of these thresholds are unknown. Economic regulation of this problem requires solution methods for a class of dynamic optimization problems with multiple thresholds located in n-dimensional space. This class of problems has hitherto not been discussed in the literature. We present a model for the economic regulation of CO 2 emissions in the presence of threshold-triggered risk of a collapsing thermohaline circulation and derive optimality conditions for regulation.

Coordinate Transformations and Derivation of Open-Loop Nash Equilibria

Leitmann (Ref. 1) introduced coordinate transformations to derive global optima of a class of dynamic optimization problems. We present applications of this method to derive open-loop Nash equilibria for finite-time horizon differential games. The method of coordinate transformations is especially useful in cases where the original game does not satisfy the global curvature conditions normally imposed in sufficient optimality conditions.

Dynamic optimization with state variable path constraints

A new method for solving dynamic optimization problems that contain path constraints on the state variables is described. We establish the equivalence between the inequality path-constrained dynamic optimization problem and a hybrid discrete/continuous dynamic optimization problem that contains switching phenomena. The control parameterization method for solving dynamic optimization problems, which transforms the dynamic optimization problem into a finite-dimensional nonlinear program (NLP), is combined with an algorithm for constrained dynamic simulation so that any admissible combination of the control parameters produces an initial value problem that is feasible with respect to the path constraints. We show that the dynamic model, which is in general described by a system of differential-algebraic equations (DAEs), can become high-index during the state-constrained portions of the trajectory. During these constrained portions of the trajectory, a subset of the control variables are allowed to be determined by the solution of the high-index DAE. The algorithm proceeds by detecting activation and deactivation of the constraints during the solution of the initial value problem, and solving the resulting high-index DAEs and their related sensitivity systems using the method of dummy derivatives. This method is applicable to a large class of dynamic optimization problems.

A modified quasilinearization algorithm for the computation of optimal singular control

A quasilinearization algorithm is modified to handle a wider class of dynamic optimization problems with singular arcs. That is, the performance index and the system equations may be non-linear in the control variable. Four examples were solved to demonstrate the effectiveness of the modified algorithm. It is shown that the proposed algorithm could offer a larger region of convergence and faster convergence in addition to its ability to handle a wider class of problem.