Non-differentiable Model Research Articles

The solution of the economic dispatch problem via an efficient Teaching-Learning-Based Optimization method

This paper is concerned with the economic generation dispatch problem. It is a well-known fact that practical aspects of power plant equipment, as well as the objectives to be met, may result in a nonconvex, nondifferentiable model that poses difficulties to conventional mathematical programming methods. This paper proposes the use of metaheuristic Teaching-Learning-Based Optimization to overcome such difficulties. This metaheuristic is well known for requiring a few parameters and, most importantly, it does not require the tuning of problem-dependent parameters. The algorithm proposed in this work is parameter-free; that is, the few parameters required by the Teaching-Learning-Based Optimization method are set automatically based on the power system’s data. In addition, the handling of constraints, such as generators’ prohibited zones and the generator-load-loss power balance, is performed in a very efficient way. Simulation results are shown for power systems containing 3 to 40 generation units, and the results provided by the proposed method are shown and discussed based on comparisons with other metaheuristics and a mathematical programming technique.

An all-densities pedestrian simulator based on a dynamic evaluation of the interpersonal distances

In this paper we deal with pedestrian modeling, aiming at simulating crowd behavior in normal and emergency scenarios, including highly congested mass events. We are specifically concerned with a new agent-based, continuous-in-space, discrete-in-time, nondifferential model, where pedestrians have finite size and are compressible to a certain extent. The model also takes into account the pushing behavior appearing at extremely high densities. The main novelty is that pedestrians are not assumed to generate any kind of “field” which governs the dynamics of the others in the space around them. Instead, the behavior of each pedestrian solely relies on its knowledge of the environment and the evaluation of interpersonal distances between it and the others. The model is able to reproduce the concave/concave fundamental diagram with a “double hump” (i.e. with a second peak) which shows up when body forces come into play. We present several numerical tests (some of them being inspired by the recent ISO 20414 standard), which show how the model can reproduce classical self-organizing patterns.

Generalized Smooth Functions for Modeling Steady-State Response of Controls in Transmission and Distribution

• Present generalized continuous functions to model steady-state response of grid controls. • Employ homotopy and limiting techniques for scalability and robustness. • Traditional controls as well as newer controls are mapped to smooth continuous functions. • Applied to both transmission and distribution power flow to scale to systems as large as 70k buses. The promise of renewables and the consequent fluctuations in the power grid necessitate a robust simulation framework to capture the steady-state behavior of new controls. However, standard non-differentiable models of control mechanisms produce divergence and/or numerical oscillations in large power flow simulations. In this paper, we describe a methodology that introduces two generalized class C 1 smooth basis functions to model the steady-state of various controls in power flow for transmission and three-phase distribution as well as optimization settings. These models are accompanied by homotopy methods and limiting techniques in the simulation engine that ensure scalable and robust convergence. We map standard power flow controls, typically modeled as non-differentiable functions in commercial tools, to the proposed class C 1 basis functions and demonstrate the benefits for robustness in comparison to commercial tools on test cases as large as the US Eastern Interconnection system. We further extend the approach to model non-standard devices such as STATCOMs and the steady-state effects of inverter-based generators in three-phase distribution and optimization problems.

FOCUS: Flexible Optimizable Counterfactual Explanations for Tree Ensembles

Model interpretability has become an important problem in machine learning (ML) due to the increased effect algorithmic decisions have on humans. Counterfactual explanations can help users understand not only why ML models make certain decisions, but also how these decisions can be changed. We frame the problem of finding counterfactual explanations as an optimization task and extend previous work that could only be applied to differentiable models. In order to accommodate non-differentiable models such as tree ensembles, we use probabilistic model approximations in the optimization framework. We introduce an approximation technique that is effective for finding counterfactual explanations for predictions of the original model and show that our counterfactual examples are significantly closer to the original instances than those produced by other methods specifically designed for tree ensembles.

Update on the Use of Nanocarriers and Drug Delivery Systems and Future Directions in Cervical Cancer.

Cervical cancer represents a major health problem among females due to its increased mortality rate. The conventional therapies are very aggressive and unsatisfactory when it comes to survival rate, especially in terminal stages, which requires the development of new treatment alternatives. With the use of nanotechnology, various chemotherapeutic drugs can be transported via nanocarriers directly to cervical cancerous cells, thus skipping the hepatic first-pass effect and decreasing the rate of chemotherapy side effects. This review comprises various drug delivery systems that were applied in cervical cancer, such as lipid-based nanocarriers, polymeric and dendrimeric nanoparticles, carbon-based nanoparticles, metallic nanoparticles, inorganic nanoparticles, micellar nanocarriers, and protein and polysaccharide nanoparticles. Nanoparticles have a great therapeutic potential by increasing the pharmacological activity, drug solubility, and bioavailability. Through their mechanisms, they highly increase the toxicity in the targeted cervical tumor cells or tissues by linking to specific ligands. In addition, a nondifferentiable model is proposed through holographic implementation in the dynamics of drug delivery dynamics. As any hologram functions as a deep learning process, the artificial intelligence can be proposed as a new analyzing method in cervical cancer.

Some Properties for Adjustable Entropy Function Method and Its Application to Power Systems Dynamic Economic Dispatch with Wind Integrated

With the continuous development of the world economy, the development and utilization of environmentally friendly and renewable energy have become the trend in many countries. In this paper, we study the dynamic economic dispatch with wind integrated. Firstly, we take advantage of the positive and negative spinning reserve to deal with wind power output prediction errors in order to establish a dynamic economic dispatch model of wind integrated. The existence of a min function makes the dynamic economic dispatch model nondifferentiable, which results in the inability to directly use the traditional mathematical methods based on gradient information to solve the model. Inspired by the aggregate function, we can easily transform the nondifferentiable model into a smooth model when parameter p tends to infinity. However, the aggregate function will cause data overflow when p tends to infinity. Then, for solving this problem, we take advantage of the adjustable entropy function method to replace of aggregate function method. In addition, we further discuss the adjustable entropy function method and point out that the solution generated by the adjustable entropy function method can effectively approximate the solution of the original problem without parameter p tending to infinity. Finally, simulation experiments are given, and the simulation results prove the effectiveness and correctness of the adjustable entropy function method.

Second-order multi-objective non–differentiable Schaible type model and its duality relation under (K xQ)–C–type–I functions

ABSTRACT In this article, we constructed second-order --type-I functions and also illustrated a non-trivial concrete numerical example for such existing types of functions. Furthermore, we have presented a second-order non-differentiable multi-objective Schaible type dual model with cone objectives as well as cone constraints. Finally, we established duality theorems under --type-I assumptions.

Ensemble slice sampling

Slice sampling has emerged as a powerful Markov Chain Monte Carlo algorithm that adapts to the characteristics of the target distribution with minimal hand-tuning. However, Slice Sampling’s performance is highly sensitive to the user-specified initial length scale hyperparameter and the method generally struggles with poorly scaled or strongly correlated distributions. This paper introduces Ensemble Slice Sampling (ESS), a new class of algorithms that bypasses such difficulties by adaptively tuning the initial length scale and utilising an ensemble of parallel walkers in order to efficiently handle strong correlations between parameters. These affine-invariant algorithms are trivial to construct, require no hand-tuning, and can easily be implemented in parallel computing environments. Empirical tests show that Ensemble Slice Sampling can improve efficiency by more than an order of magnitude compared to conventional MCMC methods on a broad range of highly correlated target distributions. In cases of strongly multimodal target distributions, Ensemble Slice Sampling can sample efficiently even in high dimensions. We argue that the parallel, black-box and gradient-free nature of the method renders it ideal for use in scientific fields such as physics, astrophysics and cosmology which are dominated by a wide variety of computationally expensive and non-differentiable models.

Parameter identification of bolted joint models by trust-region constrained sensitivity approach

Bolted joints are commonly used in mechanical connections and they pose the complex hysteretic behavior due to the existence of contact and friction. Various bolted joint models including the bilinear model, Iwan models etc have been established, and calibration of such models is a prerequisite to design, analysis and health monitoring with bolted joints. To this end, a novel trust-region constrained sensitivity approach is developed in this paper to identify bolted joint parameters from noisy time-domain response data. At first, parameter identification is formulated as a nonlinear least-squares problem, that finds the parameters to minimize the residual between the measured and predicted data. Then, the sensitivity analysis is conducted to solve the least-squares problem. As is noteworthy, bolted joints are often modelled as parallel union of Jenkins elements which are non-differentiable and the smoothing strategy is essentially called for sensitivity analysis of non-differentiable models. Moreover, to enhance the convergence of the sensitivity approach, the trust-region constraint is additionally introduced, notwithstanding, it is simply tackled by the Tikhonov regularization. Numerical examples and finite element simulations are studied to demonstrate the effectiveness and accuracy of the proposed approach.

SDTR: Soft Decision Tree Regressor for Tabular Data

Deep neural networks have been proved a success in multiple fields. However, researchers still favor traditional approaches to obtain more interpretable models, such as Bayesian methods and decision trees when processing heterogeneous tabular data. Such models are hard to differentiate, thus inconvenient to be integrated into end-to-end settings. On the other hand, traditional neural networks are differentiable but perform poorly on tabular data. We propose a hierarchical differentiable neural regression model, Soft Decision Tree Regressor (SDTR). SDTR imitates a binary decision tree by a differentiable neural network and is plausible for ensemble schemes like bagging and boosting. The SDTR method was evaluated on multiple tabular-based regression tasks (YearPredictionMSD, MSLR-Web10K, Yahoo LETOR, SARCOS and Wine quality). Its performance is comparable with non-differentiable models (gradient boosting decision trees) and better than uninterpretable models (regular FCNN). On top of that, it can produce fair results with a restricted number of parameters, only using a small forest or even a single tree. We also propose an “average entropy” metric to evaluate the level of interpretability of a trained, soft decision tree neural network. This metric also helps to select proper structure and hyperparameters for such networks.

ON A HIGH-PASS FILTER DESCRIBED BY LOCAL FRACTIONAL DERIVATIVE

The local fractional derivative (LFD) has gained much interest recently in the field of electrical circuits. This paper proposes a non-differentiable (ND) model of high-pass filter described by the LFD, where the ND transfer function is obtained with the help of the local fractional Laplace transform, and its parameters and properties are studied. The obtained results reveal the sufficiency of the LFD for analyzing circuit systems in fractal space.

Oscillatory regimes of Langmuir probe current in femtosecond laser-produced plasmas: Experimental and theoretical investigations

We investigate the oscillatory regimes of the Langmuir probe current recorded for the electrical characterization of femtosecond laser-produced plasma. The influence of metallic target biasing, which can perturb the ambipolar electric field generated through the charge separation at early stages of the expansion, is also studied. Two distinct behaviors are evidenced, and they point to the existence of two plasma structures: a fast one consisting in promptly ejected highly charged particles, and a slow “tail” of thermalized particles. Theoretically, a non-differentiable model which involves two scale resolutions is developed. Assuming the principle of scale resolution superposition, the current density at global scale is expressed as the sum of the current density at Coulomb and the thermal scale resolutions, in agreement with the experimental data. Interestingly, the theoretical estimation of oscillation frequencies shows a correspondence with the filling factor hierarchy in the fractional quantum Hall effect.

Optimization of single mixed-refrigerant natural gas liquefaction processes described by nondifferentiable models

A new strategy for the optimization of natural gas liquefaction processes is presented, in which flowsheets formulated using nondifferentiable process models are efficiently and robustly optimized using an interior-point algorithm. The constraints in the optimization formulation lead to solutions that ensure optimal usage of the area of multistream heat exchangers in the processes in order to minimize irreversibilities. The process optimization problems are solved reliably without the need for a complex initialization procedure even when highly accurate descriptions of the process stream cooling curves are required. In addition to the well-studied PRICO liquefaction process, two significantly more complex single mixed-refrigerant processes are successfully optimized and results are reported for each process subject to constraints imposed by several different operating scenarios.

A dual quaternion algorithm of the Helmert transformation problem

Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Závoti and Kalmár (Acta Geod Geophys 51:245–256, 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77:66–76, 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.org/10.1186/s40623-015-0263-6).

Reliable Flash Calculations: Part 2. Process Flowsheeting with Nonsmooth Models and Generalized Derivatives

This article presents new methods for robustly simulating process flowsheets containing nondifferentiable models, using recent advances in exact sensitivity analysis for nonsmooth functions. Among other benefits, this allows flowsheeting problems to be equipped with newly developed nonsmooth inside-out algorithms for nonideal vapor–liquid equilibrium calculations that converge reliability, even when the phase regime at the results of these calculations is unknown a priori. Furthermore, process models for inherently nonsmooth unit operations may be seamlessly integrated into process flowsheets, so long as computationally relevant generalized derivative information is computed correctly and communicated to the flowsheet convergence algorithm. These techniques may be used in either sequential-modular simulations or simulations in which the most challenging modules are solved using tailored external procedures, while the remaining flowsheet equations are solved simultaneously. This new nonsmooth flowsheeting ...

A continuous updating weighted least squares estimator of tail dependence in high dimensions

Likelihood-based procedures are a common way to estimate tail dependence parameters. They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models. Moreover, they can be hard to compute in higher dimensions. An adaptive weighted least-squares procedure matching nonparametric estimates of the stable tail dependence function with the corresponding values of a parametrically specified proposal yields a novel minimum-distance estimator. The estimator is easy to calculate and applies to a wide range of sampling schemes and tail dependence models. In large samples, it is asymptotically normal with an explicit and estimable covariance matrix. The minimum distance obtained forms the basis of a goodness-of-fit statistic whose asymptotic distribution is chi-square. Extensive Monte Carlo simulations confirm the excellent finite-sample performance of the estimator and demonstrate that it is a strong competitor to currently available methods. The estimator is then applied to disentangle sources of tail dependence in European stock markets.

Pairs Generating as a Consequence of the Fractal Entropy: Theory and Applications

In classical concepts, theoretical models are built assuming that the dynamics of the complex system’s stuctural units occur on continuous and differentiable motion variables. In reality, the dynamics of the natural complex systems are much more complicated. These difficulties can be overcome in a complementary approach, using the fractal concept and the corresponding non-differentiable theoretical model, such as the scale relativity theory or the extended scale relativity theory. Thus, using the last theory, fractal entropy through non-differentiable Lie groups was established and, moreover, the pairs generating mechanisms through fractal entanglement states were explained. Our model has implications in the dynamics of biological structures, in the form of the “chameleon-like” behavior of cholesterol.

On a fractal LC-electric circuit modeled by local fractional calculus

A non-differentiable model of the LC-electric circuit described by a local fractional differential equation of fractal dimensional order is addressed in this article. From the fractal electrodynamics point of view, the relaxation oscillator, defined on Cantor sets in LC-electric circuit, and its exact solution using the local fractional Laplace transform are obtained. Comparative results among local fractional derivative, Riemann–Liouville fractional derivative and conventional derivative are discussed. Local fractional calculus is proposed as a new tool suitable for the study of a large class of electric circuits.

A Continuous Updating Weighted Least Squares Estimator of Tail Dependence in High Dimensions

Likelihood-based procedures are a common way to estimate tail dependence parameters. They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models. Moreover, they can be hard to compute in higher dimensions. An adaptive weighted least-squares procedure matching nonparametric estimates of the stable tail dependence function with the corresponding values of a parametrically specified proposal yields a novel minimum-distance estimator. The estimator is easy to calculate and applies to a wide range of sampling schemes and tail dependence models. In large samples, it is asymptotically normal with an explicit and estimable covariance matrix. The minimum distance obtained forms the basis of a goodness-of-fit statistic whose asymptotic distribution is chi-square. Extensive Monte Carlo simulations confirm the excellent finite-sample performance of the estimator and demonstrate that it is a strong competitor to currently available methods. The estimator is then applied to disentangle sources of tail dependence in European stock markets.

Global sensitivity analysis of terrain effects in geolocation systems

The design of geolocation systems has traditionally used linearized models or one-at-a-time sensitivity analyses, which ignore most of the design space. Global sensitivity analysis algorithms enable more complete system characterization for more complex, nondifferentiable models, such as a digital elevation model (DEM). This paper develops the necessary framework for applying global sensitivity analysis methods to geolocation systems and demonstrates the algorithm’s enhanced fidelity by applying it to an example angle of arrival and DEM geolocation system. Terrain roughness is found to be a significant parameter solely through interaction effects with other parameters, effects that would have been missed by simpler analyses.