Robust Minimax Research Articles

Robust min-max (regret) optimization using ordered weighted averaging

In decision-making under uncertainty, several criteria have been studied to aggregate the performance of a solution over multiple possible scenarios. This paper introduces a novel variant of ordered weighted averaging (OWA) for optimization problems. It generalizes the classic OWA approach, which includes the robust min–max optimization as a special case, as well as the min–max regret optimization. We derive new complexity results for this setting, including insights into the inapproximability and approximability of this problem. In particular, we provide stronger positive approximation results that asymptotically improve the previously best-known bounds for the classic OWA approach.

PaVa: A novel path-based valley-seeking clustering algorithm

Clustering methods are being applied to a wider range of scenarios involving more complex datasets, where the shapes of clusters tend to be arbitrary. In this paper, we propose a novel Path-based Valley-seeking clustering algorithm for arbitrarily shaped clusters. This work aims to seek the valleys among clusters and then individually extract clusters. Three vital techniques are used in this algorithm. First, path distance (minmax distance) is employed to transform the irregular boundaries among clusters, that is density valleys, into perfect spherical shells. Second, a suitable density measurement, k-distance, is employed to make adjustment on Minimum Spanning Tree, by which a robust minmax distance is calculated. Third, we seek the transformed density valleys by determining their centers and radii. Based on the vital techniques, the main contributions of the proposed algorithm can be summarized as follows. First, the clusters are wrapped in spherical shells after the distance transformation, making the extraction process efficient even with clusters of arbitrary shape. Second, adjusted Minimum Spanning Tree enhances the robustness of minmax distance under different kinds of noise. Last, the number of clusters does not need to be inputted or decided manually due to the individual extraction process. After applying the proposed algorithm to several commonly used synthetic datasets, the results indicate that the Path-based Valley-seeking algorithm is accurate and efficient. The algorithm is based on the dissimilarity of objects, so it can be applied to a wide range of fields. Its performance on real-world datasets illustrates its versatility.

Robust Channel Estimation in Multiuser Downlink 5G Systems Under Channel Uncertainties

In wireless communication, the performance of the network highly relies on the accuracy of channel state information (CSI). On the other hand, the channel statistics are usually unknown, and the measurement information is lost due to the fading phenomenon. Therefore, we propose a channel estimation approach for downlink communication under channel uncertainty. We apply the Tobit Kalman filter (TKF) method to estimate the hidden state vectors of wireless channels. To minimize the maximum estimation error, a robust minimax minimum estimation error (MSE) estimation approach is developed while the QoS requirements of wireless users is taken into account. We then formulate the minimax problem as a non-cooperative game to find an optimal filter and adjust the best behavior for the worst-case channel uncertainty. We also investigate a scenario in which the actual operating point is not exactly known under model uncertainty. Finally, we investigate the existence and characterization of a saddle point as the solution of the game. Theoretical analysis verifies that our work is robust against the uncertainty of the channel statistics and able to track the true values of the channel states. Additionally, simulation results demonstrate the superiority of the model in terms of MSE value over related techniques.

A predictive fault tolerant control method for qLPV systems subject to input faults and constraints

In this paper, we investigate the use of Model Predictive Control (MPC) applications for quasi-Linear Parameter Varying (qLPV) systems subject to faults along the input channels. We propose a Fault Tolerant Control (FTC) mechanism based on a robust state-feedback MPC synthesis, considering polytopic inclusions. In order to alleviate the numerical burden of the robust min-max procedure, we use small prediction horizons, in such a way that the solution becomes viable for real-time systems. The FTC system is able to tolerate time-varying saturation of the actuator, which may happen due to malfunctions. Recursive feasibility and poly-quadratic stability guarantees are ensured through the synthesis of adequate terminal ingredients. Accordingly, we present a catalogue of three different LMI remedies, considering: (a) parameter-independent ingredients, (b) a parameter-dependent terms and (c) a parameter-dependent maps that take into account bounded rates of parameter variation. An autonomous driving car example is used to illustrate the performances of the proposed technique, which is compared to other MPCs from the literature. The proposed FTC method is able to ensure good performances, obtained with reduced computational demand.

Robust Optimum Life-Testing Plans under Progressive Type-I Interval Censoring Schemes with Cost Constraint

This paper considers optimal design problems for the Weibull distribution, which can be used to model symmetrical or asymmetrical data, in the presence of progressive interval censoring in life-testing experiments. Two robust approaches, Bayesian and minimax, are proposed to deal with the dependence of the D-optimality and c-optimality on the unknown model parameters. Meanwhile, the compound design method is applied to ensure a compromise between the precision of estimation of the model parameters and the precision of estimation of the quantiles. Furthermore, to make the design become more practical, the cost constraints are taken into account in constructing the optimal designs. Two algorithms are provided for finding the robust optimal solutions. A simulated example and a real life example are given to illustrate the proposed methods. The sensitivity analysis is also studied. These new design methods can help the engineers to obtain robust optimal designs for the censored life-testing experiments.

Solving Robust Production Planning Problem with Interval Budgeted Uncertainty in Cumulative Demands

In this paper, a production planning problem with inventory and backordering levels is discussed. It is assumed that cumulative demands in periods are uncertain and an interval uncertainty representation with continuous budget is used to model this uncertainty. The robust minmax criterion is applied to compute an optimal production plan. A row and column generation algorithm is constructed for solving the problem. Results of some computational tests are shown which demonstrate that the algorithm is efficient for the instances with up to 100 periods and returns solutions that are robust against the uncertainty in demands.

Improved RSSD-Based Source Localization With Unknown Sensor Position Errors

In this letter, a received signal strength difference (RSSD) approach is presented to localize a source with unknown transmit power in the presence of sensor position errors. The performance of conventional least squares (LS) algorithms is degraded because they consider measurement errors rather than estimation errors. An algorithm is presented here to overcome this problem. First, a robust minimax mean squared error (MSE) estimator is developed based on the estimation error for bounded location estimation and sensor position errors to minimize the worst case sum of the variance and squared norm of the bias. This nonlinear problem is solved by transforming the nonconvex objective function into a convex optimization problem using the S-procedure, relaxation, and semidefinite programming. This problem is extended to the unknown path loss exponent case. Necessary and sufficient conditions are given for convergence of the proposed RSSD convex relaxation of MSE semidefinite programming (RCRM-SDP) approach. Simulation results are presented which confirm the robustness of this method for sufficiently large signal to noise ratios (SNRs).

Highly Efficient Robust and Stable M-Estimates of Location

This article is partially a review and partially a contribution. The classical two approaches to robustness, Huber’s minimax and Hampel’s based on influence functions, are reviewed with the accent on distribution classes of a non-neighborhood nature. Mainly, attention is paid to the minimax Huber’s M-estimates of location designed for the classes with bounded quantiles and Meshalkin-Shurygin’s stable M-estimates. The contribution is focused on the comparative performance evaluation study of these estimates, together with the classical robust M-estimates under the normal, double-exponential (Laplace), Cauchy, and contaminated normal (Tukey gross error) distributions. The obtained results are as follows: (i) under the normal, double-exponential, Cauchy, and heavily-contaminated normal distributions, the proposed robust minimax M-estimates outperform the classical Huber’s and Hampel’s M-estimates in asymptotic efficiency; (ii) in the case of heavy-tailed double-exponential and Cauchy distributions, the Meshalkin-Shurygin’s radical stable M-estimate also outperforms the classical robust M-estimates; (iii) for moderately contaminated normal, the classical robust estimates slightly outperform the proposed minimax M-estimates. Several directions of future works are enlisted.

A minimax regret approach for robust multi-objective portfolio selection problems with ellipsoidal uncertainty sets

Since security return cannot be accurately estimated using past data, in this paper it is assumed to take values in a given ellipsoidal uncertainty set. This paper aims to discuss a robust multi-objective portfolio selection problem based on the minimax regret criterion under an ellipsoidal uncertainty sets, in which the two objective functions are the portfolio return to be maximized and the mean absolute deviation as a risk measure to be minimized. The robust counterpart formulation for the proposed model is firstly presented, then an algorithm based on the relaxation procedure is designed to solve the robust counterpart formulation with second-order cone constraints and infinite constraints. Finally, a practical example based on real market data is presented to illustrate the effectiveness of the proposed model and the algorithm. Compared with the traditional robust portfolio model based on minimax robustness, the robust minimax regret optimal solutions proposed in this paper have better performance on several evaluation criteria.

Robust min–max optimal control design for systems with uncertain models: A neural dynamic programming approach

The design of an artificial neural network (ANN) based sub-optimal controller to solve the finite-horizon optimization problem for a class of systems with uncertainties is the main outcome of this study. The optimization problem considers a convex performance index in the Bolza form. The dynamic uncertain restriction is considered as a linear system affected by modeling uncertainties, as well as by external bounded perturbations. The proposed controller implements a min–max approach based on the dynamic neural programming approximate solution. An ANN approximates the Value function to get the estimate of the Hamilton–Jacobi–Bellman (HJB) equation solution. The explicit adaptive law for the weights in the ANN is obtained from the approximation of the HJB solution. The stability analysis based on the Lyapunov theory yields to confirm that the approximate Value function serves as a Lyapunov function candidate and to conclude the practical stability of the equilibrium point. A simulation example illustrates the characteristics of the sub-optimal controller. The comparison of the performance indexes obtained with the application of different controllers evaluates the effect of perturbations and the sub-optimal solution.

Scenario Approach for Minmax Optimization with Emphasis on the Nonconvex Case: Positive Results and Caveats

We treat the so-called scenario approach, a popular probabilistic approximation method for robust minmax optimization problems via independent and identically distributed (i.i.d.) sampling from the...

Robust Network-Wide Bus Scheduling With Transfer Synchronizations

Travel time and demand disturbances lead to unreliable bus operations and missed passenger transfers. This study formulates the multi-line synchronization problem as a robust min(i)max problem that considers the fluctuations of the travel and dwell times of bus trips. Given the infeasibility of the multi-line synchronization problem in extreme cases of travel/dwell time disturbances, we introduce a flexible problem formulation that incorporates the constraint violations into the objective function. To produce a robust schedule, the dispatching times of trips are our design variables and the travel and dwell time fluctuations are the environmental variables which have an adversarial role in our minimax problem. We validate our approach in the bus network of The Hague using 1 month of actual vehicle location and passenger counting data. There, we demonstrate the potential improvement in terms of service regularity and increased synchronizations in common case and extreme case conditions.

A Novel Network Traffic Prediction Method Based on RMPM

The network traffic prediction plays the key role in congestion control and bandwidth allocation. A variety of traditional learning models such as artificial neural networks (ANN) have been applied in prediction. To avoid the drawbacks of traditional models for prediction, a novel robust minimax probability machine (RMPM)-based traffic prediction method is proposed in this paper. The prediction performance is tested on two different types of traffic data, Ethernet data flow and MPEG4 video flow, at the timescale 1. The experiments demonstrate that the proposed method attains satisfactory performance in prediction accuracy. Therefore, the proposed method can be used for congestion control or bandwidth allocation, to meet the user QOS requirements.

Optimal unidirectional switch designs.

Stepped wedge designs and delayed start designs can all be considered as special cases of the so-called unidirectional switch design. This paper provides optimal proportions of clusters that are allocated to switch patterns in a unidirectional switch design to minimize the asymptotic variance of the treatment effect estimator. This unique optimal design applies to certain cross-sectional and longitudinal variance component models. When the intraclass correlation coefficient is zero, the optimal unidirectional switch design coincides with the classic (cluster) parallel group design. The optimal unidirectional switch design is more efficient than the optimal stepped wedge design and delayed start designs. Compared with the uniform unidirectional switch design, the efficiency gain of the optimal unidirectional switch design can be substantial, but it depends on the intraclass correlation and the cluster size. We also showed that augmenting the optimal stepped wedge design with pure control pattern is more efficient than the optimal stepped wedge design. In addition, robust minimax design for unidirectional switch design, delayed start design, and stepped wedge design are provided.

Robust adaptive efficient estimation for semi-Markov nonparametric regression models

We consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises. An adaptive model selection procedure is proposed. Under general moment conditions on the noise distribution a sharp non-asymptotic oracle inequality for the robust risks is obtained and the robust efficiency is shown. It turns out that for semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.

Robust combinatorial optimization under budgeted–ellipsoidal uncertainty

In the field of robust optimization, uncertain data are modeled by uncertainty sets which contain all relevant outcomes of the uncertain problem parameters. The complexity of the related robust problem depends strongly on the shape of the chosen set. Two popular classes of uncertainty are budgeted uncertainty and ellipsoidal uncertainty. In this paper, we introduce a new uncertainty class which is a combination of both. More precisely, we consider ellipsoidal uncertainty sets with the additional restriction that at most a certain number of ellipsoid axes can be used at the same time to describe a scenario. We define a discrete and a convex variant of the latter set and prove that in both cases the corresponding robust min–max problem is NP-hard for several combinatorial problems. Furthermore, we prove that for uncorrelated budgeted–ellipsoidal uncertainty in both cases the min–max problem can be solved in polynomial time for several combinatorial problems if the number of axes which can be used at the same time is fixed. We derive exact solution methods and formulations for the problem which we test on random instances of the knapsack problem and of the shortest path problem.

Discrete-Time Robust Hierarchical Linear-Quadratic Dynamic Games

In this paper, the theory of robust min–max control is extended to hierarchical and multiplayer dynamic games for linear quadratic discrete time systems. The proposed game model consists of one leader and many followers, while the performance of all players is affected by disturbance. The Stackelberg–Nash-saddle equilibrium point of the game is derived and a necessary and sufficient condition for the existence and uniqueness of such a solution is obtained. In the infinite time horizon, it is shown that the solution of the Riccati equation is upper bounded under a condition that is called individual controllability. By assuming such a condition and using a time-varying Lyapunov function the input-to-state stability of the hierarchical dynamic game is achieved, considering the optimal feedback strategies of the players and an arbitrary disturbance as the input.

A Novel Application of Minimax LQG Control Technique for High‐speed Spiral Imaging

AbstractOver the last two decades, increasing the scanning speed of an atomic force microscopy (AFM) has been attempted either by applying novel controllers, using alternative scanning methods, or by modifying the hardware setup. This paper demonstrates, the first two approaches to achieve high‐speed AFM image scanning. A robust minimax linear quadratic Gaussian (LQG) controller is designed and spiral scanning is considered as an alternative scanning method rather than conventional raster scanning. The minimax LQG controller is designed based on an uncertain system model which is constructed by measuring the plant variations due to variations in sample mass and also modeling error between the measured and model frequency responses. This controller is also robust against uncertainties introduced as a result of variations of sample mass, spillover dynamics of the scanner at frequencies higher than the first resonance frequency of the scanner, and variation in plant transfer functions due to temperature and humidity. The designed controller is experimentally implemented on an AFM using a dSPACE ds‐1103 real‐time prototyping system and open‐loop and closed‐loop spiral imaging performances are evaluated. The proposed controller provides sufficient damping at the resonant modes to accurately track the sinusoidal reference signal and generate vibration free images. Also, creep, hysteresis, and cross‐coupling effects are significantly reduced. The experimental results show that the proposed scheme outperforms the open‐loop case and some other existing approaches.

Robust decentralised mean field control in leader following multi‐agent systems

This study addresses a robust counterpart of the deterministic mean field control in a multi-agent system. A decentralised mean field algorithm is proposed to solve a min–max control problem for a large population of heterogeneous agents. In the proposed leader following scheme, the leader tracks a reference signal which is unknown to the followers and each follower tracks a convex combination of the population state average and the leader's state. The leader plays a robust min–max game against disturbance and the followers play a mean field e N -Nash game against each other and at the same time, each follower plays a robust min–max game against the disturbance. For all the players, finite horizon quadratic cost is considered. In the proposed decentralised algorithm, followers do not need the knowledge about each of leader's and other followers' states and they only use an estimate of the population state average. In this way, propose a policy iteration method which guarantees the convergence to the saddle point mean field e N -Nash solution. The proposed method is applied to a large population of agents and compared with centralised algorithm to demonstrate the results.

Fuel cell cars in a microgrid for synergies between hydrogen and electricity networks

Fuel cell electric vehicles convert chemical energy of hydrogen into electricity to power their motor. Since cars are used for transport only during a small part of the time, energy stored in the on-board hydrogen tanks of fuel cell vehicles can be used to provide power when cars are parked. In this paper, we present a community microgrid with photovoltaic systems, wind turbines, and fuel cell electric vehicles that are used to provide vehicle-to-grid power when renewable power generation is scarce. Excess renewable power generation is used to produce hydrogen, which is stored in a refilling station. A central control system is designed to operate the system in such a way that the operational costs are minimized. To this end, a hybrid model for the system is derived, in which both the characteristics of the fuel cell vehicles and their traveling schedules are considered. The operational costs of the system are formulated considering the presence of uncertainty in the prediction of the load and renewable energy generation. A robust min-max model predictive control scheme is developed and finally, a case study illustrates the performance of the designed system.