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We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget , that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: (1) adversarial online convex optimization and (2) the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well-performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the “price of non-stationarity,” which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one.

Optimal choice of actions is a fundamental problem relevant to fields as diverse as neuroscience, psychology, economics, computer science, and control engineering. Despite this broad relevance the abstract setting is similar: we have an agent choosing actions over time, an uncertain dynamical system whose state is affected by those actions, and a performance criterion that the agent seeks to optimize. Solving problems of this kind remains hard, in part, because of overly generic formulations. Here, we propose a more structured formulation that greatly simplifies the construction of optimal control laws in both discrete and continuous domains. An exhaustive search over actions is avoided and the problem becomes linear. This yields algorithms that outperform Dynamic Programming and Reinforcement Learning, and thereby solve traditional problems more efficiently. Our framework also enables computations that were not possible before: composing optimal control laws by mixing primitives, applying deterministic methods to stochastic systems, quantifying the benefits of error tolerance, and inferring goals from behavioral data via convex optimization. Development of a general class of easily solvable problems tends to accelerate progress—as linear systems theory has done, for example. Our framework may have similar impact in fields where optimal choice of actions is relevant.

This paper offers an improved memory-type ratio estimator in stratified random sampling under linear and non-linear cost functions. The issue is given as all integer non-linear programming problems (AINLPPs). The sampling properties mainly the bias and the mean squared error of the introduced estimator are derived up to the first order of approximation. The optimum value of the characterizing scalar is obtained by the Lagrange method of maxima–minima. The least value of the MSE of the suggested estimator is also obtained for this optimum value of the charactering constant. The suggested estimator is compared both theoretically and empirically with the competing estimators. Under this setup, the optimum allocation with mean square error of the suggested estimator is attained, and the estimator is compared to other comparable estimators. The AINLPP is solved using the genetic programming approach, which is applied to both actual and simulated data sets from a bivariate normal distribution.

Permanent magnet synchronous motors (PMSM) have become prevalent in industry and play an essential role in managing industrial processes, automation systems, and renewable energy sources due to their superior efficiency, torque, and power density. However, because it operates like a non‐linear system with quick dynamics, variable parameters during operation, and unknown disturbances, PMSM presents challenges for machine control. Non‐linear controls are required to account for the non‐linearities of the permanent magnet synchronous machine. Recently, predictive control techniques for non‐linear multi‐variable systems have gained popularity. In this work, a novel approach to robust non‐linear generalized predictive control (RNGPC) has been developed for PMSM, with the aim of tracking the reference speed while maintaining minimum reactive power, robustness to external disturbances, and parameter uncertainties. A new finite horizon cost function is integrated, with an integral action introduced in the control law. The main advantage of this technique is that it does not require the measurement and observation of external disturbance as well as parametric uncertainties. The control strategy method has been tested in the MATLAB/Simulink environment with various operating conditions. The results showed good robustness against parameter changes and ensured fast convergence. In this work, a new approach of robust non‐linear generalized predictive controller (RNGPC) has been developed for permanent magnet synchronous motors (PMSM). The control objective is tracking the reference speed while maintaining minimum reactive power and robustness to external disturbances and parameter uncertainties.

Production function techniques often impose functional form and other restrictions that limit their applicability. One common limitation in popular production function techniques is the requirement that all inputs and outputs must be positive numbers. There is a need to develop a production function analysis technique that is less restrictive in the assumptions it makes, and inputs it can process. This paper proposes such a general technique by linking fields of neural networks and econometrics. Specifically, two radial basis function (RBF) neural networks are proposed for stochastic production and cost frontier analyses. The functional forms of production and cost functions are considered unknown except that they are multivariate. Using simulated and real-world datasets, experiments are performed, and results are provided. The results illustrate that the proposed technique has broad applicability and performs equal to or better than the traditional stochastic frontier analysis technique.

The feedback-based wavefront shaping emerges as a promising method for deep tissue microscopy, energy control in bio-incubation, and re-configurable structural illuminations. The cost function plays a crucial role in the feedback-based wavefront optimization for focusing light through scattering media. However, popularly used cost functions, such as intensity ( ) and peak-to-background ratio (PBR) struggle to achieve precise intensity control and uniformity across the focus spot. We have proposed an -norm-based quadratic cost function (QCF) for establishing both intensity and position correlations between image pixels, which helps to advance the focusing light through scattering media, such as biological tissue and ground glass diffusers. The proposed cost function has been integrated into the genetic algorithm, establishing pixel-to-pixel correlations that enable precise and controlled contrast optimization, while maintaining uniformity across the focus spot and effectively suppressing the background intensity. We have conducted both simulations and experiments using the proposed QCF, comparing its performance with the commonly used and PBR-based cost functions. The results evidently indicate that the QCF achieves superior performance in terms of precise intensity control, uniformity, and background intensity suppression. By contrast, both the and PBR cost functions exhibit uncontrolled intensity gain compared with the proposed QCF. The proposed QCF is most suitable for applications requiring precise intensity control at the focus spot, better uniformity, and effective background intensity suppression. This method holds significant promise for applications where intensity control is critical, such as photolithography, photothermal treatments, dosimetry, and energy modulation within and outside bio-incubation systems.

This article concerns a Cournot duopoly game with homogeneous expectations. The cost functions of the two players are assumed to be asymmetric to capture possible asymmetries in firms’ technologies or firms’ input costs. Large values of the speed of adjustment of the players destabilize the Nash Equilibrium (N.E.) and cause the appearance of a chaotic trajectory in the Discrete Dynamical System (D.D.S.). The scope of this article is to control the chaotic dynamics that appear outside the stability field, assuming asymmetric cost functions of the two players. Specifically, one player uses linear costs, while the other uses nonlinear costs (quadratic or cubic). The cubic cost functions are widely used in the Economic Dispatch Problem. The delayed feedback control method is applied by introducing a new control parameter at the D.D.S. It is shown that larger values of the control parameter keep the N.E. locally asymptotically stable even for higher values of the speed of adjustment.

Drivers are aware that energy consumption is a key factor in the cost of travel and that it affects their route choice on a road network. This paper presents a framework for modelling the estimation of energy consumption together with a link cost function that connects with the demand–supply interaction on a road network. Demand–supply interaction models, used in static and dynamic traffic assignment, have a significant limitation: they cannot simulate a generalised perceived cost in aggregate form without considering energy consumption as a component of the cost in a complete cost function. This paper presents a framework that explicitly takes into account the cost of energy consumption inside a consolidated traffic assignment model. The framework explicitly models the circular dependency between energy consumption and traffic conditions on the link. The model is specified and supported by a test numerical application in a test system, which validates the proposed framework.

Model Predictive Control (MPC) algorithms typically use the classical L2 cost function, which minimises squared differences of predicted control errors. Such an approach has good numerical properties, but the L1 norm that measures absolute values of the control errors gives better control quality. If a nonlinear model is used for prediction, the L1 norm leads to a difficult, nonlinear, possibly non-differentiable cost function. A computationally efficient alternative is discussed in this work. The solution used consists of two concepts: (a) a neural approximator is used in place of the non-differentiable absolute value function; (b) an advanced trajectory linearisation is performed on-line. As a result, an easy-to-solve quadratic optimisation task is obtained in place of the nonlinear one. Advantages of the presented solution are discussed for a simulated neutralisation benchmark. It is shown that the obtained trajectories are very similar, practically the same, as those possible in the reference scheme with nonlinear optimisation. Furthermore, the L1 norm even gives better performance than the classical L2 one in terms of the classical control performance indicator that measures squared control errors.

Change point detection becomes increasingly important because it can support data analysis by providing labels to the data in an unsupervised manner. In the context of process data analytics, change points in the time series of process variables may have an important indication about the process operation. For example, in a batch process, the change points can correspond to the operations and phases defined by the batch recipe. Hence identifying change points can assist labelling the time series data. Various unsupervised algorithms have been developed for change point detection, including the optimisation approach which minimises a cost function with certain penalties to search for the change points. The Bayesian approach is another, which uses Bayesian statistics to calculate the posterior probability of a specific sample being a change point. The paper investigates how the two approaches for change point detection can be applied to process data analytics. In addition, a new type of cost function using Tikhonov regularisation is proposed for the optimisation approach to reduce irrelevant change points caused by randomness in the data. The novelty lies in using regularisation-based cost functions to handle ill-posed problems of noisy data. The results demonstrate that change point detection is useful for process data analytics because change points can produce data segments corresponding to different operating modes or varying conditions, which will be useful for other machine learning tasks.