Multidimensional Optimal Control Problems Research Articles

Joint dynamic pricing and marketing‐mix strategies for revenue management applications with stochastic demand

AbstractTo attract potential customers and to effectively sell their inventories over time, retailers often invest in different marketing levers and apply dynamic pricing strategies. However, composing a profitable marketing mix in dynamic settings is challenging. The high problem complexity increases further if one seeks to account for stochastic and inventory‐dependent demand, which is essential for various practical applications. In this paper, we examine such stochastic dynamic pricing and marketing models under a finite as well as an infinite time horizon. We assume that marketing dimensions, such as information advertising, design quality, and the like, are substitutes only if they are all positive; otherwise, the demand vanishes. For a reasonably general demand formulation with asymmetric multidimensions marketing, we show how to transform the multidimensional optimal control problem into a two‐dimensional one, simplifying the analytical complexity. Further, for well‐established cases with isoelastic and exponential demand, we present optimal closed‐form solutions allowing detailed sensitivity analyses. Using numerical examples, we illustrate and analyze how optimally controlled sales processes are influenced by different model parameters and infer managerial insights. Our results provide general qualitative and quantitative insights into the complex interplay of marketing‐mix controls over time.

A necessary and sufficient condition of optimality for a class of multidimensional control problems

AbstractIn this paper, we introduce a necessary and sufficient condition of optimality for a new class of multidimensional optimal control problems governed by path‐independent curvilinear integral functionals and mixed constraints involving first‐order partial differential equations (PDEs) of m‐flow type. Furthermore, as a consequence, we establish the equivalence between the class of (strongly) b‐invex functionals and the class of (strongly) b‐pseudoinvex functionals. The mathematical development in the paper is supported by illustrative examples, as well.

Approximation of the Multidimensional Optimal Control Problem for the Heat Equation (Applicable to Computational Fluid Dynamics (CFD))

This work is devoted to finding an estimate of the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. An important aspect of the work is not only the establishment of convergence of solutions of a sequence of discrete problems to the solution of the original differential problem, but the determination of the order of convergence, which plays a very important role in applications. The paper uses the discretization method of the differential problem and the method of integral estimates. The reduction of a differential multidimensional mixed problem to a difference one is based on the approximation of the desired solution and its derivatives by difference expressions, for which the error of such an approximation is known. The idea of using integral estimates is typical for such problems, but in the multidimensional case significant technical difficulties arise. To estimate errors, we used multidimensional analogues of the integration formula by parts, Friedrichs and Poincare inequalities. The technique used in this work can be applied under some additional assumptions, and for nonlinear multidimensional mixed problems of parabolic type. To find a numerical solution, the variable direction method is used for the difference problem of a parabolic type equation. The resulting algorithm is implemented using program code written in the Python 3.7 programming language.

On a modified optimal control problem with first-order PDE constraints and the associated saddle-point optimality criterion

In this paper, based on a new multidimensional optimal control problem, in short (OCP), we introduce a modified multidimensional optimal control problem (which is simpler or easier to solve than the original one) involving first-order partial differential equations (PDEs) and inequality-type constraints. Optimality conditions associated with this new optimal control problem are formulated and proved, as well. Furthermore, under some generalized convexity assumptions, we establish an equivalence between an optimal solution of (OCP) and a saddle-point associated with the Lagrange functional corresponding to the modified multidimensional optimal control problem. Also, in order to illustrate the main results and their effectiveness, some applications are presented.

Sufficient optimality conditions for multidimensional optimal control problems with mixed constraints governed by an elliptic PDE

In this article, the idea of a dual dynamic programming is applied to the optimal control problems with multiple integrals governed by a semi-linear elliptic PDE and mixed state-control constraints. The main result called a verification theorem provides the new sufficient conditions for optimality in terms of a solution to the dual equation of a multidimensional dynamic programming. The optimality conditions are also obtained by using the concept of an optimal dual feedback control. Besides seeking the exact minimizers of problems considered some kind of an approximation is given and the sufficient conditions for an approximated optimal pair are derived.

Economic dynamics of reservoir sedimentation management: Optimal control with singularly perturbed equations of motion

We analyze the economic dynamics of reservoir sedimentation management using the hydrosuction-dredging sediment-removal system. System dynamics depend on two interdependent hydraulic processes evolving at different rates. The accumulation of water impounded in the reservoir evolves on a ‘fast’ time scale, while the loss of water storage capacity to trapped sediments evolves on a ‘slow’ time scale. We formulate a multidimensional optimal control problem with singularly perturbed equations of motion to accommodate the disparate time scales. We apply singular perturbation methods to approximate (via polynomial series expansion) a ‘slow’ manifold reducing multi-dimensional solution space to the single-dimensional subspace confining long-term dynamics.

Parallel Algorithms for Stochastic Dynamic Programming with Continuous State and Control Variables

We compare two partitioning methods for solving a multi-dimensional optimal control problem in parallel using continuous state, continuous control, stochastic dynamic programming (SCP). The algorithm uses tensor products to interpolate the multi-dimensional, continuous variable state space. Unlike previous parallel SDP applications, an iterative optimization routine is used to find the optimal continuous control for each node point in the state space. This routine causes load balance and dependence problems not encountered in previous parallel SDP applications. Even with these additional computational complexities, superb efficiencies are achieved on a shared memory MIMD architecture. The two methods of solution differ in the following way: Method 1 uses the outer state loop index to send subsets of the state space (called parcels) to each concurrent process. The number of parcels sent as parallel tasks is the number of discretizations of one of the state variables. Method 2 allows the user to determine the number of parcels to be sent as parallel tasks. Method 2 splits the state space into a greater number of parcels, so that the processors are more likely to finish at the same time at a synchronization barrier. Computational results indicate that Method 2 has better load balance, with efficiencies up to 93.5%. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Mathematical Bioeconomics: The Optimal Management of Renewable Resources.

Introduction. 1. Elementary Dynamics of Exploited Populations. 1.1 The Logistic Growth Model. 1.2 Generalized Logistic Models: Depensation. 1.3 Summary and Critique. 2. Economic Models of Renewable-Resource Harvesting. 2.1 The Open-Access Fishery. 2.2 Economic Overfishing. 2.3 Biological Overfishing. 2.4 Optimal Fishery Management. 2.5 The Optimal Harvest Policy. 2.6 Examples Based on the Schaefer Model. 2.7 Linear Variational Problems. 2.8 The Possibility of Extinction. 2.9 Summary and Critique. 3. Capital-Theoretic Aspects of Resource Management. 3.1 Interest and Discount Rates. 3.2 Capital Theory and Renewable Resources. 3.3 Nonautonomous Models. 3.4 Applications to Policy Problems: Labor Mobility in the Fishery. 4. Optimal Control Theory. 4.1 One-Dimensional Control Problems. 4.2 A Nonlinear Fishery Model. 4.3 Economic Interpretation of the Maximum Principle. 4.4 Multidimensional Optimal Control Problem. 4.5 Optimal Investment in Renewable-Resource Harvesting. 5. Supply and Demand: Nonlinear Models. 5.1 The Elementary Theory of Supply and Demand. 5.2 Supply and Demand in Fisheries. 5.3 Nonlinear Cost Effects: Pulse Fishing. 5.4 Game-Theoretic Models. 5.5 Transboundary Fishery Resources: A Further Application of the Theory. 5.6 Summary and Critique. 6. Dynamical Systems. 6.1 Basic Theory. 6.2 Dynamical Systems in the Plane: Linear Theory. 6.3 Isoclines. 6.4 Nonlinear Plane-Autonomous Systems. 6.5 Limit Cycles. 6.6 Gause's Model of Interspecific Competition. 7. Discrete-Time and Metered Models. 7.1 A General Metered Stock-Recruitment Model. 7.2 The Beverton-Holt Stock-Recruitment Model. 7.3 Depensation Models. 7.4 Overcompensation. 7.5 A Simple Cohort Model. 7.6 The Production Function of a Fishery. 7.7 Optimal Harvest Policies. 7.8 The Discrete Maximum Principle. 7.9 Dynamic Programming. 8. The Theory of Resource Regulation. 8.1 A Behavioral Model. 8.2 Optimization Analysis. 8.3 Limited Entry. 8.4 Taxes and Allocated Transferable Quotas. 8.5 Total Catch Quotas. 8.6 Summary and Critique. 9. Growth and Aging. 9.1 Forestry Management: The Faustmann Model. 9.2 The Beverton-Holt Fisheries Model. 9.3 Dynamic Optimization in the Beverton-Holt Model. 9.4 The Case of Bounded F. 9.5 Multiple, Cohorts: Nonselective Gear. 9.6 Pulse Fishing. 9.7 Multiple Cohorts: Selective Gear. 9.8 Regulation. 9.9 Summary and Critique. 10. Multispecies Models. 10.1 Differential Productivity. 10.2 Harvesting Competing Populations. 10.3 Selective Harvesting. 10.4 A Diffusion Model: The Inshore-Offshore Fishery. 10.5 Summary and Critique. 11. Stochastic Resource Models. 11.1 Stochastic Dynamic Programming. 11.2 A Stochastic Forest Rotation Model. 11.3 Uncertainty and Learning. 11.4 Searching for Fish. 11.5 Summary and Critique. Supplementary Reading. References. Index.

Sufficiency Conditions for Weak Local Minima in Multidimensional Optimal Control Problems with Mixed Control-State Restrictions

In [13] a new sufficiency criterion for strong local minimality in multidimensional non-convex control problems with pure state constraint was developed. In this paper we use a similar method to obtain sufficient conditions for weak local minimality in multidimensional control problems with mixed state-control restrictions. The result is obtained by applying duality theory for control problems of Klötzler [11] as well as first and second order optimality conditions for optimization problems described by C^1 -functions having a locally Lipschitzian gradient mapping. The main theorem contains the result of Zeidan [17] for one-dimensional problems withoutstate restrictions.