We propose a novel approach to modeling advertising dynamics for a firm operating over distributed market domain based on controlled partial differential equations of diffusion type. Using our model, we consider a general type of finite-horizon profit maximization problem in a monopoly setting. By reformulating this profit maximization problem as an optimal control problem in infinite dimensions, we derive sufficient conditions for the existence of its optimal solutions under general profit functions, as well as state and control constraints, and provide general characterization of the optimal solutions. Sharper, feedback-form, characterizations of the optimal solutions are obtained for two variants of the general problem.
We consider a class of dynamic advertising problems under uncertainty in the presence of carryover and distributed forgetting effects, generalizing a classical model of Nerlove and Arrow. In particular, we allow the dynamics of the product goodwill to depend on its past values, as well as previous advertising levels. Building on previous work of two of the authors, the optimal advertising model is formulated as an infinite dimensional stochastic control problem. We obtain (partial) regularity as well as approximation results for the corresponding value function. Under specific structural assum…
We consider a class of optimal control problems of stochastic delay differential equations (SDDE) that arise in connection with optimal advertising under uncertainty for the introduction of a new product to the market, generalizing classical work of Nerlove and Arrow (1962). In particular, we deal with controlled SDDE where the delay enters both the state and the control. Following ideas of Vinter and Kwong (1981) (which however hold only in the deterministic case), we reformulate the problem as an infinite dimensional stochastic control problem to which we associate, through the dynamic progr…
With the emergence of new online channels and information technology, digital advertising tends to substitute more and more to traditional advertising by offering the opportunity to companies to target the consumers/users that are really interested by their products or services. We introduce a novel framework for the study of optimal bidding strategies associated to different types of advertising, namely, commercial advertising for triggering purchases or subscriptions, and social marketing for alerting population about unhealthy behaviours (anti-drug, vaccination, road-safety campaigns). Our …
Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and applications to optimal advertising and optimal investment problems
This paper deals with a class of optimal control problems which arises in advertising models with Volterra Ornstein-Uhlenbeck process representing the product goodwill. Such choice of the model can be regarded as a stochastic modification of the classical Nerlove-Arrow model that allows to incorporate both presence of uncertainty and empirically observed memory effects such as carryover or distributed forgetting. We present an approach to solve such optimal control problems based on an infinite dimensional lift which allows us to recover Markov properties by formulating a optimization problem …
Stochastic control problems related to optimal advertising under uncertainty are considered. In particular, we determine the optimal strategies for the problem of maximizing the utility of goodwill at launch time and minimizing the disutility of a stream of advertising costs that extends until the launch time for some classes of stochastic perturbations of the classical Nerlove-Arrow dynamics. We also consider some generalizations such as problems with constrained budget and with discretionary launching.
Advertising is a crucial component of marketing and an important way for companies to raise awareness of goods and services in the marketplace. Advertising campaigns are designed to convey a marketing image or message to an audience of potential consumers and television commercials can be an effective way of transmitting these messages to a large audience. In order to meet the requirements for a typical advertising order, television content providers must provide advertisers with a predetermined number of "impressions" in the target demographic. However, because the number of impressions for a…
Spread, then Target, and Advertise in Waves: Optimal Budget Allocation Across Advertising Channels
Yale University（耶鲁大学）；University of Pennsylvania（宾夕法尼亚大学）；Cornell University（康奈尔大学）；University of California, Davis（加州大学戴维斯分校）；Army Research Laboratory（陆军研究实验室）
We propose a new optimal model of product goodwill in a segmented market where the state variable is described by a partial differential equation of the Lotka--Sharp--McKendrick type. In order to maximize the sum of discounted profits over a finite time horizon, we control the advertising efforts which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of consumer recommendations in a segmented market. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we construct a…
A Profit-Maximizing Strategy for Advertising on the e-Commerce Platforms
South China University of Technology（华南理工大学）；Guangdong University of Foreign Studies（广东外语外贸大学）；ShenZhen TingZhu Health Management Co.,Ltd.（深圳市汀竹健康管理有限公司）
We consider the diffusive limit of a typical pure-jump Markovian control problem as the intensity of the driving Poisson process tends to infinity. We show that the convergence speed is provided by the Hölder constant of the Hessian of the limit problem, and explain how correction terms can be constructed. This provides an alternative efficient method for the numerical approximation of the optimal control of a pure-jump problem in situations with very high intensity of jump. We illustrate this approach in the context of a display advertising auction problem.
In this paper, we consider the problem of distributed optimal control of linear dynamical systems with a quadratic cost criterion. We study the case of output feedback control for two interconnected dynamical systems, and show that the linear optimal solution can be obtained from a combination of two uncoupled Riccati equations and two coupled Riccati equations.
In this paper, we consider distributed optimization design for resource allocation problems over weight-balanced graphs. With the help of singular perturbation analysis, we propose a simple sub-optimal continuous-time optimization algorithm. Moreover, we prove the existence and uniqueness of the algorithm equilibrium, and then show the convergence with an exponential rate. Finally, we verify the sub-optimality of the algorithm, which can approach the optimal solution as an adjustable parameter tends to zero.
We consider a heat conduction problem $S$ with mixed boundary conditions in a n-dimensional domain $Ω$ with regular boundary $Γ$ and a family of problems $S_α$, where the parameter $α>0$ is the heat transfer coefficient on the portion of the boundary $Γ_{1}$. In relation to these state systems, we formulate simultaneous \emph{distributed-boundary} optimal control problems on the internal energy $g$ and the heat flux $q$ on the complementary portion of the boundary $Γ_{2}$. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint …
Distributed Algorithm for the Global Optimal Controller of Nonlinear Multi-Agent Systems
School of Control Science and Engineering, Shandong University（山东大学控制科学与工程学院）；School of Computer Science and Technology, Shandong University（山东大学计算机科学与技术学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
The solution to the infinite horizon optimal control problem for linear distributed time-delay systems is presented. The proposal is based on the use of the Cauchy solution for distributed time-delay systems. In contrast with previous results (for punctual time delay systems), the form of the functional and its properties are formally justified. An important property is demonstrated: the Bellman functional admits upper and lower bounds, thus it is proved to be positive definite. Additionally, some experimental results on a temperature process are presented.
A Pontryagin maximum principle for an optimal control problem in three dimensional linearized compressible viscous flows is established using the Ekeland variational principle. The controls are distributed over a bounded domain, while the state variables are subject to a set of constraints and governed by the linearized compressible Navier-Stokes equations. The maximum principle is of integral-type and obtained for minimizers of a tracking-type integral cost functional.
We consider a steady-state heat conduction problem in a multidimensional bounded domain Omega for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion Gamma_1 of the boundary and a constant heat flux q in the remaining portion Gamma_2 of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary Gamma_1 with heat transfer coefficient alpha and external temperature b. We obtain explicitly, for a rectangular domain in R^2, an annulus in R^2 and a s…
This paper proposes coordination laws for optimal energy generation and distribution in energy network, which is composed of physical flow layer and cyber communication layer. The physical energy flows through the physical layer; but all the energies are coordinated to generate and flow by distributed coordination algorithms on the basis of communication information. First, distributed energy generation and energy distribution laws are proposed in a decoupled way without considering the interactive characteristics between the energy generation and energy distribution. Then, a joint coordinatio…
We consider a control problem constrained by the unsteady stochastic Stokes equations with nonhomogeneous boundary conditions in connected and bounded domains. In this paper, controls are defined inside the domain as well as on the boundary. Using a stochastic maximum principle, we derive necessary and sufficient optimality conditions such that explicit formulas for the optimal controls are derived. As a consequence, we are able to control the stochastic Stokes equations using distributed controls as well as boundary controls in a desired way.
In this manuscript we consider a class optimal control problem for stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a $C^{1,α}$-partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising).
Abstract Objective: The objectives of this paper are to 1) construct a new network model compatible with distributed computation, 2) construct the full optimal power flow (OPF) in a distributed fashion so that an effective, non-inferior solution can be found, and 3) develop a scalable algorithm that guarantees the convergence to a local minimum.
Here an original idea is suggested to prove the existence of optimal control for some types of non- linear problems. The obtained results can be considered as individual existence theorems (in some sense).
The rapidly increasing penetration of inverter-based resources into a power transmission network requires more sophisticated voltage control strategies considering their inherent output variabilities. In addition, faults and load variations affect the voltage profile over the power network. This paper proposes a Primal Dual Gradient Dynamics based optimal distributed voltage control approach that optimizes outputs of distributed reactive power sources to maintain an acceptable voltage profile while preserving operational limits. Case studies of this new approach on IEEE test systems have verif…
A discretization of an optimal control problem of a stochastic parabolic equation driven by multiplicative noise is analyzed. The state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. The convergence rate $ O(τ^{1/2} + h^2) $ is rigorously derived.
