Horizon Problem Research Articles

Solving infinite horizon optimization problems through analysis of a one-dimensional global optimization problem

Infinite horizon optimization (IHO) problems present a number of challenges for their solution, most notably, the inclusion of an infinite data set. This hurdle is often circumvented by approximating its solution by solving increasingly longer finite horizon truncations of the original infinite horizon problem. In this paper, we adopt a novel transformation that reduces the infinite dimensional IHO problem into an equivalent one dimensional optimization problem, i.e., minimizing a Holder continuous objective function with known parameters over a closed and bounded interval of the real line. We exploit the characteristics of the transformed problem in one dimension and introduce an algorithm with a graphical implementation for solving the underlying infinite dimensional optimization problem.

Variable speed of light cosmology, primordial fluctuations and gravitational waves

A variable speed of light (VSL) cosmology is described in which the causal mechanism of generating primordial perturbations is achieved by varying the speed of light in a primordial epoch. This yields an alternative to inflation for explaining the formation of the cosmic microwave background (CMB) and the large scale structure (LSS) of the universe. The initial value horizon and flatness problems in cosmology are solved. The model predicts primordial scalar and tensor fluctuation spectral indices $n_s=0.96$ and $n_t=- 0.04$, respectively. We make use of the $\delta{\cal N}$ formalism to identify signatures of primordial nonlinear fluctuations, and this allows the VSL model to be distinguished from inflationary models. In particular, we find that the parameter $f_{\rm NL}=5$ in the variable speed of light cosmology. The value of the parameter $g_{\rm NL}$ evolves during the primordial era and shows a running behavior.

Opportunistic Finite Horizon Multicasting of Erasure-Coded Data

We propose an algorithm for opportunistic multicasting in wireless networks. Whereas prior multicast rate adaptation schemes primarily optimize long-term throughput, we investigate the finite horizon problem where a fixed number of packets has to be transmitted to a set of wireless receivers in the shortest amount of time-a common problem, e.g., for software updates or video multicast. In the finite horizon problem, the optimum rate critically depends on the recent reception history of the receivers and requires a fine balance between maximizing overall throughput and equalizing individual receiver throughput. We formulate a dynamic programming algorithm that optimally solves this problem. We then develop two low complexity heuristics that perform close to the optimal solution and are suitable for practical online scheduling. We further analyze the performance of our algorithms by means of simulation. They substantially outperform existing solutions based on throughput maximization or favoring the user with the worst channel, and we obtain a 30 percent performance improvement over the former and a 120 percent improvement over the latter in scenarios with Rayleigh fading. We further analyze the performance of the schemes under imperfect state information and observe an even higher improvement over the benchmark schemes.

Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property

We consider a vibrating string that is fixed at one end with Neumann control action at the other end. We investigate the optimal control problem of steering this system from given initial data to rest, in time T, by minimizing an objective functional that is the convex sum of the L2-norm of the control and of a boundary Neumann tracking term.We provide an explicit solution of this optimal control problem, showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at the end of the time interval, and in-between it decays exponentially. We show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term. This implies that, if the time T is large, then the optimal trajectory approximately consists of three arcs, where the first and the third short-time arcs are transient arcs, and in the middle arc the optimal control and the corresponding state are exponentially close to 0. This is an example of a turnpike phenomenon for a problem of optimal boundary control. If T=+∞ (infinite time horizon problem), then only the exponentially decaying component of the control remains, and the norms of the optimal control action and of the optimal state decay exponentially in time. In contrast to this situation, if the weight of the tracking term is zero and only the control cost is minimized, then the optimal control is distributed uniformly along the whole interval [0,T] and coincides with the control given by the Hilbert Uniqueness Method.In addition, we establish a similarity theorem stating that, for every T>0, there exists an appropriate weight λ<1 for which the optimal solutions of the corresponding finite horizon optimal control problem and of the infinite horizon optimal control problem coincide along the first part of the time interval [0,2]. We also discuss the turnpike phenomenon from the perspective of a general framework with a strongly continuous semi-group.

On Hamiltonian as Limiting Gradient in Infinite Horizon Problem

Necessary conditions of optimality in the form of the Pontryagin maximum principle are derived for the Bolza-type discounted problem with free right end. The optimality is understood in the sense of the uniformly overtaking optimality. Such process is assumed to exist, and the corresponding payoff of the optimal process (expressed in the form of improper integral) is assumed to converge in the Riemann sense. No other assumptions on the asymptotic behaviour of trajectories or adjoint variables are required. In this paper, we prove that there exists a corresponding limiting solution of the Pontryagin maximum principle that satisfies the Michel transversality condition; in particular, the stationarity condition of the maximized Hamiltonian and the fact that the maximized Hamiltonian vanishes at infinity are proved. The connection of this condition with the limiting subdifferentials of payoff function along the optimal process at infinity is showed. The case of payoff without discount multiplier is also considered.

Non-parametric reconstruction of an inflaton potential from Einstein–Cartan–Sciama–Kibble gravity with particle production

The coupling between spin and torsion in the Einstein-Cartan-Sciama-Kibble theory of gravity generates gravitational repulsion at very high densities, which prevents a singularity in a black hole and may create there a new universe. We show that quantum particle production in such a universe near the last bounce, which represents the Big Bang gives the dynamics that solves the horizon, flatness, and homogeneity problems in cosmology. For a particular range of the particle production coefficient, we obtain a nearly constant Hubble parameter that gives an exponential expansion of the universe with more than 60 $e$-folds, which lasts about $\sim 10^{-42}$ s. This scenario can thus explain cosmic inflation without requiring a fundamental scalar field and reheating. From the obtained time dependence of the scale factor, we follow the prescription of Ellis and Madsen to reconstruct in a non-parametric way a scalar field potential which gives the same dynamics of the early universe. This potential gives the slow-roll parameters of cosmic inflation, from which we calculate the tensor-to-scalar ratio, the scalar spectral index of density perturbations, and its running as functions of the production coefficient. We find that these quantities do not significantly depend on the scale factor at the Big Bounce. Our predictions for these quantities are consistent with the Planck 2015 observations.

The investment horizon problem: a possible resolution

In the conventional model of investments, the optimal fractions in the risky assets do not depend on the time horizon. This is against empirical evidence, and against the typical recommendations of portfolio managers. We demonstrate that if the intertemporal coefficient of relative risk aversion is allowed to depend on time, or the age of the investor, the investment horizon problem can be resolved. Accordingly, the only standard assumption in applied economics/finance that we relax in order to obtain our conclusion, is the state and time separability of the intertemporal felicity index in the investor’s utility function. We include a random, idiosyncratic horizon of the consumer, which is essential when there is no intermediate consumption, we consider bequest, and also demonstrate that preferences aggregate.

Constraints on just enough inflation preceded by a thermal era

If the inflationary era is preceded by a radiation dominated era in which the inflaton too was in thermal equilibrium at some very early time then the CMB data places an upper bound on the comoving temperature of the (decoupled) inflaton quanta. In addition, if one considers models of "just enough" inflation, where the number of e-foldings of inflation is just enough to solve the horizon and flatness problems, then we get a lower bound on the Hubble parameter during inflation, $H_{\rm inf}$, which is in severe conflict with the upper bound from tensor perturbations. Alternatively, imposing the upper bound on $H_{\rm inf}$ implies that such scenarios are compatible with the data only if the number of relativistic degrees of freedom in the thermal bath in the pre-inflationary Universe is extremely large (greater than $10^9$ or $10^{11})$. We are not aware of scenarios in which this can be satisfied.

The horizon problem in Bordeaux wine cooperatives

This paper explores if and how the horizon problem affects wine cooperatives according to the downstream strategy they are following: remaining in their traditional form, i.e., as bulk wine providers to negociants; forming a union with other cooperatives; or vertical integration. We identify three variables, each of which should have a different influence on the price paid to producers when the horizon problem prevails or not: cooperative margin, asset obsolescence and leverage. We observe that 'traditional' cooperatives prioritise the payment to producers against the renewal of assets, while cooperatives in union seem to anticipate the need for investment by a decrease of the price paid to producers when the capital obsolescence reaches a certain level, confirming our main hypothesis. However, the other results have led us to reconsider our assumptions and to question the access to bank credits for such wine cooperatives.

The horizon problem in Bordeaux wine cooperatives

This paper explores if and how the horizon problem affects wine cooperatives according to the downstream strategy they are following: remaining in their traditional form, i.e., as bulk wine providers to negociants; forming a union with other cooperatives; or vertical integration. We identify three variables, each of which should have a different influence on the price paid to producers when the horizon problem prevails or not: cooperative margin, asset obsolescence and leverage. We observe that 'traditional' cooperatives prioritise the payment to producers against the renewal of assets, while cooperatives in union seem to anticipate the need for investment by a decrease of the price paid to producers when the capital obsolescence reaches a certain level, confirming our main hypothesis. However, the other results have led us to reconsider our assumptions and to question the access to bank credits for such wine cooperatives.

Quantum Bubble Polarization in Curved Spacetime

The conventional singular hot big bang scenario is questioned. A new model which does not include an initial singularity g00 = ∞ at t = 0, neither a brief period of exponential expansion a(t)~eHt is considered. The main parameters T and ρ are kept finite. The beginning of the universe we occupy is revisited in the framework of quantum field theory in curved spacetime. However, a straightforward alternative mechanism for not only solving the most fundamental problems in modern cosmology-flatness problem, horizon problem and magnetic monopole problem, but even suppressing their number is provided. In the particular paper, we discuss the essential role quantum entanglement plays in the structure of the spacetime and the apparent contradiction between quantum mechanics and general relativity in terms of classical field theory in 3 + 1 dimensions.

CEOs’ Career Development Path and Horizon Problem at their Last Career Stage

This paper takes into account CEOs’ career development paths to advance the knowledge about their long-term investment decisions at their last career stage. CEOs’ career development, especially whe...

Do Managers Act Opportunistically Towards the End of Their Career?

Ethical considerations engulf managerial decisions and compensation. As managers approach the end of their careers, their personal career horizons become shorter and they might start to behave opportunistically by taking a more risk-averse and short-term orientation. Long-term risky investments, such as research and development, suffer the most from this problem as their payoff is expected long after CEO retires. To mitigate such behavior, most executive compensation contracts include long-term performance incentives. On the other hand, we hypothesize that long-term debt-like compensation in the form of pension entitlements can make the career horizon problem more severe. In this study, we examine the impact of managerial opportunism, influenced by pension compensation, on the research and development investments. We find that on average CEOs do not curtail research and development as their career horizons become shorter. But, the defined-benefit pension component of executive compensation lead CEOs, who are closer to retirement, to decrease R&D investments. Our results imply that executive compensation contracts need to be appropriately adjusted when managers approach retirement.

Finite time linear quadratic based optimal control of BLDC motor employing distributed parameters modeling

An optimal control theory for linear quadratic finite time horizon problem is presented and combined with distributed parameters model of the BLDC (BrushLess Direct-Current) motor. Method appropriateness for minimization of the phase current control error and energy delivered to the drive is proven. The paper focuses on finding the best weighting configuration of the objective function. Presented control strategy is performed and presented employing the numerical computations.

Real-Time Energy Storage Management With Renewable Integration: Finite-Time Horizon Approach

We consider the design of cost-effective management of energy storage with renewable integration for load supply. We take a finite time horizon approach and formulate the control optimization problem aimed at minimizing the system cost over a fixed time period. Recognizing the unpredictable and nonstationary stochastic nature of system dynamics, we assume unknown arbitrary dynamics of renewable generation, load, and electricity pricing in formulating our problem. Furthermore, we incorporate detailed battery operation cost into the system cost. Different from the infinite time horizon problems in existing works, the coupling of control decisions over time, due to finite battery capacity, is more challenging to manage. We develop a special technique to tackle the technical challenges in solving the problem. Through problem modification and transformation, we are able to apply Lyapunov optimization to design a real-time control algorithm that relies only on the current system dynamics. The proposed control solution has a closed-form expression and thus is simple to implement. Through analysis, the proposed algorithm is shown to have a bounded performance gap to the optimal noncausal $T$ -slot lookahead control policy. Simulation studies show the effectiveness of our proposed algorithm as compared with two alternative real-time and noncausal algorithms.

Applications of light speed expansion and gravitational self-energy density in black hole cosmology

From the beginning of Planck scale to the scale of the current Hubble radius: 1) Considering the relation, subjects of black holes and cosmology, both can be integrated into evolving black hole cosmology and cosmic horizon problem can be relinquished. 2) Considering ‘continuous light speed expansion’ of the cosmic black hole horizon, attributed results of cosmic inflation can be re-addressed completely. If ‘nature’ of the universe is to expand with light speed, then there is no need to think about the existence of currently believed ‘Lambda term’. In addition, ‘light speed expanding cosmic space’ can be called as ‘flat space’. 3) Considering the ratio of gravitational self-energy density and thermal energy density to be  (where  is the Planck scale temperature, and is cosmic temperature at any time). Quantum gravity can be implemented in low energy scale current cosmological observations. Considering the above concepts, currently believed dark matter energy density and visible matter energy density both can be accurately fitted with the ratio of current gravitational self-energy density and current thermal energy density. To proceed further, the authors would like to highlight the following three points: 1) Deep-space red shift non-linearity can be expected to be connected with cosmological gravitational and relativistic effects and cannot be considered as a major criterion of cosmic evolution. 2) Until one finds solid applications of super luminal speeds and super luminal expansions in other areas of physics like astrophysics and nuclear astrophysics, currently believed ‘cosmic inflation’ cannot be considered as a real physical model and alternative proposals of inflation can be given a chance in exploring the evolving history of the universe. 3) Implementing Planck scale in current paradigm of cosmological observations and standard cosmology is very challenging and is inevitable.

A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks

We study optimal control problems in infinite horizon whxen the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (corresponding to a toy traffic model). We adapt the results in Soner (SIAM J Control Optim 24(6):1110–1122, 1986) to prove the regularity of the value function and the dynamic programming principle. Extending the networks and Krylov’s “shaking the coefficients” method, we prove that the value function can be seen as the solution to a linearized optimization problem set on a convenient set of probability measures. The approach relies entirely on viscosity arguments. As a by-product, the dual formulation guarantees that the value function is the pointwise supremum over regular subsolutions of the associated Hamilton–Jacobi integrodifferential system. This ensures that the value function satisfies Perron’s preconization for the (unique) candidate to viscosity solution.

Linear Programming and the Control of Diffusion Processes

Recent work by Han and Van Roy [Han J, Van Roy B (2011) Control of diffusions via linear programming. Infanger G, ed. Stochastic Programming: The State of the Art, in Honor of George B. Dantzig (Springer, New York), 329–354] introduced a linear programming technique to compute good suboptimal solutions to high-dimensional control problems in a diffusion-based setting. Their problem formulation worked with finite horizon problems where the horizon, T, is an exponentially distributed random variable. We extend their approach to finite horizon problems with a fixed horizon T. We also apply these techniques to dynamic portfolio optimization problems and then simulate the resulting policies to obtain lower bounds on the optimal value functions. We also use these policies in conjunction with convex duality methods designed for portfolio optimization problems to construct upper bounds on the optimal value functions. In our numerical experiments we find that the primal and dual bounds are very close, and so we conclude, for these problems at least, that the linear programming approach performs very well.

Loop quantum cosmology: The horizon problem and the probability of inflation

Anomaly-free perturbations of loop quantum cosmology reveal a deformed space-time structure, in which the signature changes when the energy density isρ = ρc/2. Furthermore, in loop quantum cosmology, one can obtain an effective causal structure only for a low density region (ρ≤ρc/2), which gives a natural initial condition to consider the horizon problem. Choosing the initial value atρ(0) =ρc/2 in this paper, we investigate the horizon problem and the probability of inflation in the fr amework of loop quantum cosmology. Two models are considered: the quadratic inflation and the natural infla tion. We use the Liouville measure to calculate the probability of inflation which solves the horizon problem, a nd find that, for the quadratic inflation model, the probability is very close to unity, while for the natural infl ation model, the probability is about 35%.

GOVERNANÇA COOPERATIVA: O PROBLEMA DO HORIZONTE EM COOPERATIVAS AGROPECUÁRIAS DO RIO GRANDE DO SUL

The study aimed to analyze the horizon problem in agricultural cooperatives in Rio Grande do Sul, as well as the strategies employed by the cooperatives to minimize them. This problem arises due to the temporal perspective of the cooperated be different of the cooperative organization, because the cooperative members tend to prioritize the projects and the investments that aim to benefit their economic activity in the short term. Methodologically, we chose the mixed method in the execution of search, ie, we were used qualitative and quantitative information. Thus, we sampled 10% of agricultural cooperatives linked to the Organization of Cooperatives of the State of Rio Grande do Sul (OCERGS). It was found that the cooperatives that have the grains, as the main business activity, presented more intensity of the horizon problem. Most of the variables used to capture this problem showed elevated levels of agreement. It was also observed that the financial condition of the cooperative affects the incidence of the horizon problem. Among the main strategies to minimize this problem are: the distribution of surpluses, the programs of relationships and incentives with the membership and the return of social capital to the cooperated in the cooperative after large period. This study highlights the need to adopt new organizational models for overcoming of the horizon problem, but to achieve this strategy it is necessary to update current legislation taking into account the new environment of agribusiness.