1,809 research outputs found
On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects
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
assumptions we study the effects of delays on the value function and optimal
strategy. In the absence of carryover effects, since the value function and the
optimal advertising policy can be characterized in terms of the solution of the
associated HJB equation, we obtain sharper characterizations of the optimal
policy.Comment: numerical example added; minor revision
Four-point functions of all-different-weight chiral primary operators in the supergravity approximation
Recently a Mellin-space formula was conjectured for the form of correlation
functions of BPS operators in planar SYM in the strong 't
Hooft coupling limit. In this work we report on the computation of two
previously unknown four-point functions of operators with weights and , from the effective type-IIB supergravity
action using AdS/CFT. These correlators are novel: they are the first
correlators with all-different weights and in particular
is the first next-next-next-to-extremal correlator to ever have been computed.
We also present simplifications of the known algorithm, without which these
computations could not have been executed without considerable computer power.
The main simplifications we found are present in the computation of the
exchange Lagrangian and in the computation of tensors. After bringing our
results in the appropriate form we successfully corroborate the recently
conjectured formula.Comment: 20+23 pages, 3 figures; v2: published versio
Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation
We present the computation of all the correlators of 1/2-BPS operators in
SYM with weights up to 8 as well as some very high-weight
correlation functions from the effective supergravity action. The computation
is done by implementing the recently developed simplified algorithm in
combination with the harmonic polynomial formalism. We provide a database of
these results attached to this publication and additionally check for almost
all of the functions in this database that they agree with the conjecture on
their Mellin-space form.Comment: 6 pages, database included; v2: database extended, appendix adde
Towards 4-point correlation functions of any 1/2-BPS operators from supergravity
The quartic effective action for Kaluza-Klein modes that arises upon
compactification of type IIB supergravity on the five-sphere S^5 is a starting
point for computing the four-point correlation functions of arbitrary weight
1/2-BPS operators in N=4 super Yang-Mills theory in the supergravity
approximation. The apparent structure of this action is rather involved, in
particular it contains quartic terms with four derivatives which cannot be
removed by field redefinitions. By exhibiting intricate identities between
certain integrals involving spherical harmonics of S^5 we show that the net
contribution of these four-derivative terms to the effective action vanishes.
Our result is in agreement with and provides further support to the recent
conjecture on the Mellin space representation of the four-point correlation
function of any 1/2-BPS operators in the supergravity approximation.Comment: 12 page
Optimal Distributed Dynamic Advertising
We propose a novel approach to modeling advertising dynamics for a firm operating over a distributed market domain based on controlled partial differential equations of the 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 a general characterization of the optimal solutions. Sharper, feedback-form characterizations of the optimal solutions are obtained for two variants of the general problem
Going Bunkers: The Joint Route Selection and Refueling Problem
Managing shipping vessel profitability is a central problem in marine transportation. We consider two commonly used types of vessels—“liners” (ships whose routes are fixed in advance) and “trampers” (ships for which future route components are selected based on available shipping jobs)—and formulate a vessel profit maximization problem as a stochastic dynamic program. For liner vessels, the profit maximization reduces to the problem of minimizing refueling costs over a given route subject to random fuel prices and limited vessel fuel capacity. Under mild assumptions about the stochastic dynamics of fuel prices at different ports, we provide a characterization of the structural properties of the optimal liner refueling policies. For trampers, the vessel profit maximization combines refueling decisions and route selection, which adds a combinatorial aspect to the problem. We characterize the optimal policy in special cases where prices are constant through time and do not differ across ports and prices are constant through time and differ across ports. The structure of the optimal policy in such special cases yields insights on the complexity of the problem and also guides the construction of heuristics for the general problem setting
Optimal Product Launch Times in a Duopoly: Balancing Life-Cycle Revenues With Product Cost
We present a model describing the demand dynamics of two new products competing for a limited target market. The demand trajectories of the two products are driven by a market saturation effect and an imitation effect reflecting the product experience of previous adopters. In this general setting, we provide analytical results for the sales trajectories and life-cycle sales of the competing products. We use these results to study the impact of launch time on overall life-cycle sales. We consider the perspective of one of the competing products and model the trade-off between the lost revenues resulting from a delayed launch and the lower unit-production costs. We find that the profit-maximizing launch time exhibits a counterintuitive behavior. In particular, we show that a firm facing a launch time delay from a competing product might benefit from accelerating its own product launch, as opposed to using the softened competitive situation to further improve its cost position. We identify conditions under which a marginal cost-benefit analysis leads to suboptimal launch-time decisions. Finally, we analyze the Nash equilibrium in launch-time decisions of the two competing products
Robust explicit model predictive control for hybrid linear systems with parameter uncertainties
Explicit model-predictive control (MPC) is a widely used control design
method that employs optimization tools to find control policies offline;
commonly it is posed as a semi-definite program (SDP) or as a mixed-integer SDP
in the case of hybrid systems. However, mixed-integer SDPs are computationally
expensive, motivating alternative formulations, such as zonotope-based MPC
(zonotopes are a special type of symmetric polytopes). In this paper, we
propose a robust explicit MPC method applicable to hybrid systems. More
precisely, we extend existing zonotope-based MPC methods to account for
multiplicative parametric uncertainty. Additionally, we propose a convex
zonotope order reduction method that takes advantage of the iterative structure
of the zonotope propagation problem to promote diagonal blocks in the zonotope
generators and lower the number of decision variables. Finally, we developed a
quasi-time-free policy choice algorithm, allowing the system to start from any
point on the trajectory and avoid chattering associated with discrete switching
of linear control policies based on the current state's membership in
state-space regions. Last but not least, we verify the validity of the proposed
methods on two experimental setups, varying physical parameters between
experiments
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