Minimax Approach Research Articles

The generalized equivalence of regularization and min\u2013max robustification in linear mixed models

The connection between regularization and min–max robustification in the presence of unobservable covariate measurement errors in linear mixed models is addressed. We prove that regularized model parameter estimation is equivalent to robust loss minimization under a min–max approach. On the example of the LASSO, Ridge regression, and the Elastic Net, we derive uncertainty sets that characterize the feasible noise that can be added to a given estimation problem. These sets allow us to determine measurement error bounds without distribution assumptions. A conservative Jackknife estimator of the mean squared error in this setting is proposed. We further derive conditions under which min-max robust estimation of model parameters is consistent. The theoretical findings are supported by a Monte Carlo simulation study under multiple measurement error scenarios.

Optimal Recovery of Missing Values for Non-Negative Matrix Factorization

Missing values imputation is often evaluated on some similarity measure between actual and imputed data. However, it may be more meaningful to evaluate downstream algorithm performance after imputation than the imputation itself. We describe a straightforward unsupervised imputation algorithm, a minimax approach based on optimal recovery, and derive probabilistic error bounds on downstream non-negative matrix factorization (NMF). Under certain geometric conditions, we prove upper bounds on NMF relative error, which is the first bound of this type for missing values. We also give probabilistic bounds for the same geometric assumptions. Experiments on image data and biological data show that this theoretically-grounded technique performs as well as or better than other imputation techniques that account for local structure. We also comment on imputation fairness.

ФОРМИРОВАНИЕ ПОРОГОВЫХ ЗНАЧЕНИЙ ИНДИКАТОРОВ БЕЗОПАСНОСТИ ЭКОНОМИЧЕСКОГО РАЗВИТИЯ РЕГИОНА (НА ПРИМЕРЕ РЕСПУБЛИКИ КОМИ)

The article discusses and analyzes the approaches available in science to the formation of threshold values of indicators of economic security. The author conducts a comparative study of the existing methods for determining threshold values, examines the advantages and disadvantages of each of the approaches. On the basis of the study, the threshold values for seven indicators of the security of the economic develop-ment of the region are substantiated. Further, the indicative system presented in the work is tested on the example of the Komi Republic using the indicators highlighted by the author and their threshold values using a specially developed technique based on the synthesis of the minimax approach and the indicative method, diagnostics of the state of security of economic development is carried out, the main threats arising in this area are highlighted. The application of the approach proposed in the work to the formation of threshold values of indicators, as well as the methodology used to assess threats, made it possible to diagnose threats that pose a significant danger to the development of the regional economy. The results obtained can be used by public authorities when they carry out activities to monitor the state of the region's economy.

Min-Max Approach for High-Speed Train Scheduling and Rescheduling Models

The definition of train scheduling is going to shift its current static approach, as generally resulting in a timetable, towards a dynamic approach where the timetable can be modified in order to enhance the satisfaction of passengers’ demand. In this paper, demand-oriented scheduling and rescheduling models are formalized in order to propose a dynamic timetable that can be modified to satisfy groups of passengers. More specifically, the assessment is based on a min max method. The proposed approach allows the definition of the train timetables for scheduling rail services along a multiline rail network according to operational constraints related to train capacity, train speed limits, passenger’s train transfers, possible conflict in the track section use, with the main objective to minimize the travel time of a set of passengers’ groups. The timetable resulted by the scheduling problem have been used as the input to a rescheduling model where predefined disturbances have been assumed to some nodes of the network. The proposed models are applied to the real case study of the rail network for high-speed trains in the Northern Italy.

A novel embedded min-max approach for feature selection in nonlinear Support Vector Machine classification

In recent years, feature selection has become a challenging problem in several machine learning fields, such as classification problems. Support Vector Machine (SVM) is a well-known technique applied in classification tasks. Various methodologies have been proposed in the literature to select the most relevant features in SVM. Unfortunately, all of them either deal with the feature selection problem in the linear classification setting or propose ad-hoc approaches that are difficult to implement in practice. In contrast, we propose an embedded feature selection method based on a min-max optimization problem, where a trade-off between model complexity and classification accuracy is sought. By leveraging duality theory, we equivalently reformulate the min-max problem and solve it without further ado using off-the-shelf software for nonlinear optimization. The efficiency and usefulness of our approach are tested on several benchmark data sets in terms of accuracy, number of selected features and interpretability.

Cooperative Game Approach to Robust Control Design for Fuzzy Dynamical Systems.

There is uncertainty in the system, and we consider that uncertainty is (possibly fast) time varying, but with definite bound. Fuzzy set theory is used to describe the inexact boundary and then the problem of robust control of uncertain dynamical systems is studied. Based on two adjustable design parameters, a robust control method for general mechanical systems is proposed. The control is deterministic, not the conventional IF-THEN rule based. By using the Lyapunov minimax approach, it is proved that the proposed control can guarantee system performance to be uniformly bounded and uniformly ultimately bounded. In order to find the optimal solution in the prescribed range, a two-player cooperative game is used. To reduce costs while ensuring control performance, two performance indices are developed, each of which is controlled by an adjustable parameter (i.e., player). Both necessary and sufficient conditions for Pareto-optimality are established. Using these conditions, the Pareto-optimal solution can be obtained. The effectiveness of the control design is demonstrated by the simulation of the two-body pendulum.

A Hybrid Nash Min-Max Approach for Solving Continuous Static Games

A Hybrid Nash Min-Max Approach for Solving Continuous Static Games

Constrained optimal design of seismic base absorbers based on an extended KDamper concept

In this paper, an extended KDamper (EKD) concept is proposed, for implementation as a seismic base absorber. The complexity of the EKD renders the conventional minmax (H∞) approaches ineffective. For this reason, a dedicated constrained optimization problem is proposed for the selection of the optimal system parameters. The EKD is designed according to seismic design codes and the constraints and limitations of the EKD components are evaluated based on engineering criteria. In order to better observe the efficiency of the proposed configuration, an acceleration filter is placed, as an additional constraint. Therefore, each set of optimized parameters of the EKD refers to the maximum structure acceleration, expressed as a percentage of PGA. A database of artificial accelerograms, compatible with the EC8 response spectra is generated. Sensitivity analysis is performed, and the effect of detuning is observed. Subsequently, the EKD is implemented as a seismic base absorber in a multi-story building. The performance of the proposed control strategy is compared not only to the original structure but also to the cases of conventional and highly damped base isolation systems, as well as to the KDamper. The EKD presents an improved superstructure dynamic behavior, comparable with a base isolated system. At the same time, the base relative displacement is drastically reduced, in the order of a few centimeters, rendering the implementation of the EKD feasible using conventional structural elements, without the need for special types of bearings. In addition, the base shear is reduced, as compared to the original structure, with the implementation of the proposed configuration, and thus, EKD can be considered a possible retrofitting option for seismic protection.

Optimal Design of Maxwell-Viscous Coulomb Air Damper With a Modified Fixed Point Theory

Abstract Air damper dynamic vibration absorber (DVA) is modeled using Maxwell transformed element and coulomb element. This damper serves to minimize vibration at resonant and operation of constant speed machine. Its stiffness and damping factor are transformed from Maxwell to Voigt arrangement. Meanwhile, viscous equivalent Coulomb damping is expressed by absolute relative motion. System transmissibility contours are plotted by min–max approach. Its optimal parameters are determined using this approach. Contour operation minimization is obtained from minimum system transmissibility. Moreover, exact solution of fixed points and optimal natural frequency ratio are obtained by a modified fixed point theory. Optimal design curve is derived by Coulomb damping derivative and maximum condition. Operational vibration level is minimized by 7% at the operation minimization using minimum condition. On the experimental side, test platform of the air damper is constructed using linear slide block system. Computational model of the air damper is established by its physical details and experimental data. Linear relationship is obtained between viscous and Coulomb damping angles. Modified fixed points are validated by frequency response function resonant peaks. Experimental vibration level is minimized by 5%, which being close to the minimization result. The model is validated within 5% accuracy by its optimal experimental curve.

Stackelberg Game Theory-Based Optimization of High-Order Robust Control for Fuzzy Dynamical Systems

A novel class of high-order robust controls is presented for uncertain fuzzy systems in this article. The optimal tunable parameters are also obtained based on the Stackelberg game theory. First, a dynamical system structure is formulated with uncertainty. The uncertain portion in the system is bounded, nonlinear, and time-varying which is within prescribed fuzzy set. Thus, this is a fuzzy system. Then, the proof based on the Lyapunov minimax approach shows that the novel high-order robust controls are able to assure deterministic system performance, which is uniform boundedness and ultimate uniform boundedness. Furthermore, the optimal choice of the tunable parameters in the control is considered. We creatively apply the Stackelberg strategy to solving the optimization problem. Two parameters are regarded as leader and follower in a sequential game, respectively. Based on the Stackelberg game rules, we are able to design different cost functions for different parameters. The cost functions include both performance measures and control cost. Finally, the simulations of an electronic throttle system are presented for demonstration.

Approximate Minimax Estimation of Functionals of Solutions to the Wave Equation under Nonlinear Observations

The authors consider the problem of minimax estimation of a functional of the solution to the wave equation with rapidly oscillating coefficients. The observation (output signal) is nonlinear (has the operator of superposition type). For the small parameter e > 0, the existence of the solution of the original problem is proved using the traditional minimax approach. Transition to a homogenized parameter problem allows us to remove the nonlinearity in the observation. The main result of the paper is that the minimax estimate of the problem with homogenized parameters is an approximate minimax estimate of the original problem.

Risk-based optimal operation of hybrid power system using multiobjective optimization

ABSTRACT This paper solves an optimal generation scheduling problem of hybrid power system considering the risk factor due to uncertain/intermittent nature of renewable energy resources (RERs) and electric vehicles (EVs). The hybrid power system considered in this work includes thermal generating units, RERs such as wind and solar photovoltaic (PV) units, battery energy storage systems (BESSs) and electric vehicles (EVs). Here, the two objective functions are formulated, i.e., minimization of operating cost and system risk, to develop an optimum scheduling strategy of hybrid power system. The objective of proposed approach is to minimize operating cost and system risk levels simultaneously. The operating cost minimization objective consists of costs due to thermal generators, wind farms, solar PV units, EVs, BESSs, and adjustment cost due to uncertainties in RERs and EVs. In this work, Conditional Value at Risk (CVaR) is considered as the risk index, and it is used to quantify the risk due to intermittent nature of RERs and EVs. The main contribution of this paper lies in its ability to determine the optimal generation schedules by optimizing operating cost and risk. These two objectives are solved by using a multiobjective-based nondominated sorting genetic algorithm-II (NSGA-II) algorithm, and it is used to develop a Pareto optimal front. A best-compromised solution is obtained by using fuzzy min-max approach. The proposed approach has been implemented on modified IEEE 30 bus and practical Indian 75 bus test systems. The obtained results show the best-compromised solution between operating cost and system risk level, and the suitability of CVaR for the management of risk associated with the uncertainties due to RERs and EVs.

Multiobjective reactive power planning considering the uncertainties of wind farms and loads using Information Gap Decision Theory

This study deals with multiobjective reactive power planning, considering the uncertainties of load demand and wind power generation. The main feature of the current study is to examine the impact of multiple uncertainties on Reactive Power Planning (RPP), while several objectives exist. To fulfill this goal, the Information Gap Decision Theory (IGDT) is used to handle the uncertainties of load demand and wind power production. In order to cope with the probabilistic optimal RPP problem and to create Pareto optimal solutions, the ε-Constraint method is utilized. Fuzzy Decision Maker (FDM) and min-max approach are jointly applied to find the Best Compromise Solution (BCS). To evaluate the efficiency and the proficiency of the proposed multiobjective RPP model, it is implemented on the IEEE-30 bus test system via the GAMS software environment. To prove the superiority of the proposed model, the obtained results are compared with the scenario-based approach. The results imply that for specific amounts of uncertainty, the IGDT method performs reasonably towards the scenario-based approach.

Optimal strategy for resource deployment in performing the ship key operation under conditions of uncertainty

The study considers the problem of determining the rational strategy of ship resources deployment under the conditions when the goal of key operation depends on the parameters, the values of which are known only at the level of nebulous assessment. Thus in a number of operations to ensure safe ships running there arises a problem of available resources deployment aimed at their most efficient using. Formalization of this arrangements causes some difficulty, since the purpose of the operation often depends on the future values of some parameters specified at interval level. The minimax approach is aplied, which allows to choose a rational strategy of resources deployment and to ensure the safe running of the vessel when performing key operations. Drawing on the minimax risk criterion, the article shows that it is possible to estimate the reliability of most ship operations, in which only a set of possible values is known for the state vector. Besides, the same criterion could be applied to select the acceptable strategy of resource deployment which is to provide ship’s effectiveness and safety while performing key operations. It is assumed that the search of optimal strategy for resource deployment could be minimized to the search of a pair of values, which is the solution to a radical problem and allows to find the ultimate formula for choosing a rational and effective strategy of resource allocation needed by the ship’s authority, when performing the ship key operations. In conclusion, the paper describes a numerical example of choosing the strategy of the rational resource deployment, illustrating specific but meaningful actions of the fishing vessel’s authority undertaken in the process of implementing a regular commercial job.

Controlling tractor‐semitrailer vehicles in automated highway systems: Adaptive robust and Lyapunov minimax approach

AbstractIn this paper, we design a coordinated steering and braking control scheme to ensure the tractor‐semitrailer vehicle system is practically stable. The tractor's steering input is designed to realize lane tracking and the semitrailer's differential braking torque is designed to improve the stability of the semitrailer. This control methodology is developed in two steps. Firstly, the expected steering input and the expected differential braking force are derived by introducing a set of given tracking constraints and considering the possible initial condition derivation from the constraints based on the Udwadia‐Kalaba approach. Secondly, we develop an adaptive robust control law to tackle the parameter uncertainty, which may be (possibly) time‐varying, and design the required braking torques, which are the actual inputs, to generate the desired braking forces. Furthermore, this control methodology can deal with nonlinear mechanical systems with both holonomic constraints and nonholonomic constraints. Numerical simulations demonstrate that the control algorithm could guarantee the vehicle dynamics are practically stable and achieve lane following maneuvering.

Min max min robust (relative) regret combinatorial optimization

We consider combinatorial optimization problems with uncertainty in the cost vector. Recently, a novel approach was developed to deal with such uncertainties: instead of a single one robust solution, obtained by solving a min max problem, the authors consider a set of solutions obtained by solving a min max min problem. In this new approach, the set of solutions is computed once and we can choose the best one in real time each time a cost vector occurs yielding better solutions compared to the min max approach. In this paper, we apply the new approach to the absolute and relative regret cases. Algorithms to solve the min max min robust (relative) regret problems are presented with computational experiments.

Sustainability of European agricultural holdings

European agricultural activity has an important contribution to European and global food security and also in the preservation of countryside and livelihood. The main goal of this paper is to analyze the agricultural sustainability of the twenty-eight Member States of the European Union in order to promote sustainable agriculture, food security and rural development in the European region. Information and data used are obtained from the European Commission database. The methodology includes the min–max approach and multivariate techniques, namely, Cluster and Factorial Analysis. The findings show the existence of three different clusters of farms in European countries, namely, Central European countries, New Eastern Member States, and the Mediterranean countries. The results also conclude that European agriculture and their respective farms have moderate sustainability where the subsidies from Common Agricultural Policy have a positive impact on. Moreover, agricultural activity in the European Central countries presents the highest value in economic contribution, whereas the Mediterranean countries present more contribution in environmental terms. The highest contribution in terms of creating rural jobs is found in Eastern countries. The main conclusion highlights the need to better adjust agricultural policies among the European Member States in order to better promote the sustainability of agriculture in Europe.

A minimax approach for inverse variational inequalities

In this work, we characterize the existence of a solution for a certain variational inequality by means of a classical minimax theorem. In addition, we propose a numerical algorithm for the solution of an inverse problem associated with a variational inequality. To this end we state a collage-type result in this variational framework.

Compiling Investment Attractiveness Rating of the Largest Russian Companies Using Minimax Approach

Purpose of research.The actuality of the problem of the integrated (complex) assessment of investment attractiveness of large Russian companies is due to the need of increasing production volumes and improves processing technologies of the largest sectors of the Russian economy, as well as the requirements of the accelerated growth of high-tech exports. The purpose of this study is to develop rating of investment attractiveness of companies using minimax approach and hierarchical data analysis. Decision about share structure of investment distribution contributes to the implementation of important knowledge-intensive projects for the development of major industry companies. Integral rating of companies plays a leading role in making investment decisions. Research hypothesis of the minimax approach application and hierarchical data analysis will allow developing an intelligent computerized system that will perform real-time calculations and make recommendations on share distribution of investments of the largest Russian companies using a hierarchical rating procedure and minimax approach.Materials and methods.Currently, methodological tools for rating of investment attractiveness of the largest Russian companies are at initial stage of development, although the relevance of such studies in world practice has reached the highest level, thanks to leading rating agencies that have their own methodology and justified methodology for rating companies. Until middle of 2017 when developing investment strategies for the development of Russian industry giants, the opinions of the international rating agencies of “big three”: Standard & Poor's, Fitch Ratings, Moody's and experts of Forbes magazine were taken into account. However, due to reduction in the activity of foreign expertise in Russia, it became necessary to develop and implement its own rating methodology for assessing the investment attractiveness of companies. Current rating approach (accredited Agency Expert RA), as well as young and promising rating agencies need new and mathematically sound approach to compile rating, which optimizes investment process for the most important high-tech projects by priority indicators and industry sign companies. Author's methodology is based at analysis of statistical indicators of companies in the most important sectors of economy. The important feature of the developed method is using author's mathematical apparatus, including a hierarchical analysis of the ranked indicators of financial-economic activity of companies according to their priority and application of minimax approach to obtain rating of companies based at industry characteristics.Results. Method developed in the paper is based at logical rules of grouping companies, allows obtaining rating of companies and rating of leading Russian industries, which companies were considered in data analysis. By using obtained ratings and minimax approach, the rating of investment attractiveness of the companies is made (taking into account industry affiliation). Proposed method has scientific novelty, due to hierarchical procedure of ranking indicators and minimax problem. More that author’s method allows obtaining rating assessment of investment attractiveness of companies, taking into account industry characteristic. In this paper, computational experiments were performed, which used indicators on volume of output and profit of the largest (by volume of output) companies in Russia. Computational experiments have shown the leading role of oil and gas and banking companies in Russian economy.Conclusion. The practical use of the developed method of integral ranking of companies allows optimizing process of allocation the investment resources and contributes to expansion of production of high-tech products produced by leading companies of the most important sectors of Russian economy. Proposed method is advisable for using in investment strategies for the development of high-tech projects. Efficiency of the developed rating of Russian companies has been demonstrated at practice of assessing the largest Russian companies; results are complemented, expanded and enriched by existing research of specialists and leading rating agencies. The author recommends using presented tools for optimization the investment resources for development of the most important industry companies in Russia.

Detecting dental problem related brain disease using intelligent bacterial optimized associative deep neural network

Nowadays, a lot of people have the oral health problems due to continuous changes in the lifestyle such as the person’s speech which can be affected by the malocclusion in teeth and the crooked teeth. The dental problems can cause cavity and bacterial infection. The dental and speech problems mostly can be related to the Alzheimer disease, and cognitive changes. Therefore, the dental information is collected from patients and analyzed by applying intelligent machine learning techniques. The gathered dental information is normalized by standardized min max approach. Further, different statistical parameters are derived which are huge in dimension. The optimal features are selected using grey wolf optimized approach. The method effectively selects the optimum dental features and the selected features are processed using bacterial optimized associative deep neural network. The network collects the Alzheimer disease features and compare them with the collected dental features to establish the brain related issues with dental features. The efficiency of the system is evaluated using experimental results and discussion. Thus, the introduced intelligent bacterial optimized associative deep neural network recognizes the relationship up to 98.98% of accuracy which is the maximum accuracy compared to other methods. Further, IBADNN-based Alzheimer detection system approach attains maximum predicting and selecting disease features (precision 98.65% and recall 99.03%) whereas other approaches such as OLVQ (precision 95.03% and recall 96.23%), HACANN (precision 96.36% and recall 96.91%) and GCNN (precision 97.47% and recall 97.512%) and attains low predicting and selecting accuracy.