Minimax Optimization Research Articles

Robust parallel virtual sensing method for feedback active noise control in a headrest

Additional filter method (AFM) and remote microphone method (RMM) are the two common virtual sensing methods used to compensate for active noise control (ANC) in headrest systems where mounting the error microphone at the desired zone is inconvenient. AFM and RMM are divided into two stages: the training and control stages. In the training stage, a temporary microphone is installed at the target to generate the additional and observation filters for AFM and RMM, respectively. However, they are fully effective only when the disturbances are consistent during both the training and control stages. To solve this dilemma, we propose a robust parallel virtual sensing method (PVSM) for multichannel adaptive feedback ANC system to attenuate the varying narrowband noises. During the training stage, the potential primary noises are used to model a group of parallel additional filters. Subsequently, the synthetic observation filter required for estimating the virtual error signal is derived using minimax optimisation. In the control stage, the most eligible additional filter that minimises the energy of the estimated virtual error signal is matched per frame based on the minimum-mean-square-estimated-error-matching mechanism and applied to the adaptive controller. Theoretical analysis validates that when the primary source varies, PVSM is more robust than both AFM and RMM, and performs noise reduction similar to the direct control with the temporary error microphone placed at the target. The proposed PVSM is further analysed in terms of its convergence condition and computational complexity. Numerous simulations and real-time experiments conducted using a dual-channel ANC headrest validate the feasibility of the proposed algorithm and present a guideline for the selection of frame length in PVSM. Therefore, the proposed PVSM can be considered a robust virtual sensing method against varying primary disturbances.

Design of maximum-stability PID controllers for LTI systems based on a stabilizing-set construction method

BackgroundThe PID algorithm has been widely used in control of chemical processes. A maximum-stability PID controller can provide superior stability robustness towards plant's variations and maximum exponential decay rate for disturbance rejection. However, design of maximum-stability PID controller is a min-max optimization problem and an effective method is still lack in the literature. MethodsBased on the characterization of stabilizing PID controller set, an efficient algorithm is developed to test if a plant is σ-stabilizable, where σ is abscissa or stability degree of the Hurwitz stable closed-loop characteristic polynomial. This algorithm is then used along with a bisection strategy to find a σ-interval [σε*,σε*+ε] which contains the maximum stability degree σ* for a specified ε, and the PID controller parameter set for achieving the stability degree σε*. Significant findingsThis paper has presented a systematic and efficient approach to design PID controllers with maximum degree of stability. The principal results include: (i) an improved theorem is presented for identifying stabilizing kp-intervals such that unnecessary computations are avoided; (ii) a simple yet effective method has been adopted to provide a non-conservative interval of σ which facilitates the bisectional branch-and-bound operation; (iii) the design procedure does not involve the actual construction of stabilizing PID controller sets thus renders its efficiency.

Asymptotically optimal robust information-based quick detection for general stochastic models with nonparametric postchange uncertainty

By making use of Kullback-Leibler information, we develop a new approach for the quickest detection problem for general statistical models with dependent observations and unknown postchange distributions; the postchange distribution depends on either unknown informative parameters or unknown nonparametric infinite-dimensional nuisance functions. For such models, we introduce a robust risk as the supremum of the mean detection delay over the class of postchange distributions. On the basis of the window-limited cumulative sum rules developed by Lai in 1988, we propose new detection procedures, making use of the noise density that minimizes the Kullback-Leibler divergence. Then for the constructed detection procedures, we provide sufficient conditions on the considered statistical models that ensure minimax optimality properties with respect to the robust risk. We apply the developed methods to the quick detection problems for both scalar and multivariate autoregressive processes with unknown postchange parameters and unknown noise distributions.

Feature Optimization Method of Material Identification for Loose Particles Inside Sealed Relays.

Existing material identification for loose particles inside sealed relays focuses on the selection and optimization of classification algorithms, which ignores the features in the material dataset. In this paper, we propose a feature optimization method of material identification for loose particles inside sealed relays. First, for the missing value problem, multiple methods were used to process the material dataset. By comparing the identification accuracy achieved by a Random-Forest-based classifier (RF classifier) on the different processed datasets, the optimal direct-discarding method was obtained. Second, for the uneven data distribution problem, multiple methods were used to process the material dataset. By comparing the achieved identification accuracy, the optimal min–max standardization method was obtained. Then, for the feature selection problem, an innovative multi-index–fusion feature selection method was designed, and its superiority was verified through several tests. Test results show that the identification accuracy achieved by RF classifier on the dataset was improved from 59.63% to 63.60%. Test results of ten material verification datasets show that the identification accuracies achieved by RF classifier were greatly improved, with an average improvement of 3.01%. This strongly promotes research progress in loose particle material identification and is an important supplement to existing loose particle detection research. This is also the highest loose particle material identification accuracy achieved to in aerospace engineering, which has important practical value for improving the reliability of aerospace systems. Theoretically, it can be applied to feature optimization in machine learning.

Topological synthesis of fluidic pressure-actuated robust compliant mechanisms

This paper presents a robust density-based topology optimization approach for synthesizing pressure-actuated compliant mechanisms. To ensure functionality under manufacturing inaccuracies, the robust or three-field formulation is employed, involving dilated, intermediate and eroded realizations of the design. Darcy’s law in conjunction with a conceptualized drainage term is used to model the pressure load as a function of the design vector. The consistent nodal loads are evaluated from the obtained pressure field using the standard finite element method. The objective and load sensitivities are obtained using the adjoint-variable approach. A multi-criteria objective involving both the stiffness and flexibility of the mechanism is employed in the robust formulation, and min–max optimization problems are solved to obtain pressure-actuated inverter, gripper, and contractor compliant mechanisms with different minimum feature sizes. Limitations of the linear elasticity assumptions while designing mechanisms are identified with high pressure loads. Challenges involved in designing finite deformable pressure-actuated compliant mechanisms are presented.

Optimal Design of Honeycomb Beams with Unit Cell Structure Based on Multiobjective Optimization Algorithm

In order to promote the development of honeycomb composite reinforcements, the mechanical properties and optimization methods of honeycomb beams are studied. A structural optimization design method is proposed based on natural frequency and stiffness and the lightest weight. According to the vibration frequencies and mechanical analysis, the equivalent density and equivalent stiffness of the section in the unit length are deduced, which is considered a typical example. The geometric parameters of the structure system are trained by multiobjective optimization minimax algorithm (MOMA), and the Pareto optimal solution of variable combination is obtained. The results show that optimized objects are significantly better than that previously. Therefore, the optimized scheduling algorithm has some reference value and prospects for practical application.

Robust Power Allocation in Optical Satellite MIMO Links With Pointing Jitter

In this letter, an optical satellite system is considered comprised of a geostationary satellite with multiple transmitters on-board and an optical ground station with multiple receiving telescopes. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {\mathrm {A}}$ </tex-math></inline-formula> robust power allocation strategy for the downlink is investigated incorporating the effects of the atmospheric impairments as well as the optical beam pointing jitter. Firstly, a theoretical analysis for the optical satellite system under pointing jitter is given and a maximin optimization problem is formulated maximizing the worst-case network capacity. The power allocation is carried out under peak and total power constraints using standard convex techniques. The lower bound of the ergodic network capacity for independent log-normal optical channels over variable jitter variance is derived. The proposed methodologies are evaluated via simulations using experimental channel measurements from the ARTEMIS-optical satellite campaign and their performances are compared to baseline power allocation strategies. The presented results show the effects of jitter on the network capacity and confirm the superiority of the proposed allocation scheme.

Wide-area damping power system control based on robust self-triggered model predictive control

Unavoidable random disturbances during the operation of large-scale interconnected power systems may undermine their stability. To prevent this problem, we propose a damping control method based on robust self-triggered model predictive control. First, we linearize the power system model and determine the controller installation location and feedback signal through geometric measurements. Subsequently, the model order is reduced by applying Schur method for balanced truncation. According to the assumed distribution and boundary range of random disturbance, the multi-cell robust predictive control minimax optimization problem described by the vertex is established. By deriving the upper bound of the maximum problem, the minimax problem is transformed into minimizing the upper bound problem to deal with the uncertain problem, which is transformed into linear matrix inequality for controller design. Finally, we introduce a self-triggered control strategy that avoids solving a constrained optimization problem at every time step. This strategy reduces the number of controller responses and resource requirements for sampling and communication, thereby improving computational efficiency. Selecting the two zone four-machine system as the test system, the simulation results demonstrate that the proposed control strategy can effectively eliminate the effect of random disturbances on the stable operation of power systems, improve the damping characteristics of power systems, and suppress low-frequency oscillations.

Insulin infusion rate control in type 1 diabetes patients using information-theoretic model predictive control

The present paper examines the glucose regulation system in patients with type 1 diabetes. To this end, the nonlinear Bergman's minimal model with parametric uncertainty, which represents this system, is converted to a fractional-order model using Caputo’s definition. As the recent literature suggests, the noise in the interstitial glucose concentration continuous control sensors is considered non-Gaussian. With this in mind, first an optimal non-fragile H∞ observer is designed for a Lipschitz nonlinear fractional-order system including parametric uncertainty and input disturbance in order to estimate the unknown states. Second, feedback linearization method for nonlinear fractional-order system with parametric uncertainty is used, after which the linearized system is used to design controller for diabetes mellitus patients. Third, robust fractional model predictive control (RFMPC) with a minimax optimization approach is presented for closed-loop insulin delivery. Finally, the performance of the proposed controller under non-Gaussian measurement noise is enhanced with a suitable choice of a cost function based on generalized correntropy (GC), whereas the performance of a mean square error (MSE)-based controller is simulated. The results indicate that not only the proposed controller performs better under non-Gaussian conditions but also it effectively recovers and maintains the glucose concentration within the desired range by appropriately infusing insulin under parametric uncertainty and meal disturbances.

Generative Adversarial Training for Supervised and Semi-supervised Learning.

Neural networks have played critical roles in many research fields. The recently proposed adversarial training (AT) can improve the generalization ability of neural networks by adding intentional perturbations in the training process, but sometimes still fail to generate worst-case perturbations, thus resulting in limited improvement. Instead of designing a specific smoothness function and seeking an approximate solution used in existing AT methods, we propose a new training methodology, named Generative AT (GAT) in this article, for supervised and semi-supervised learning. The key idea of GAT is to formulate the learning task as a minimax game, in which the perturbation generator aims to yield the worst-case perturbations that maximize the deviation of output distribution, while the target classifier is to minimize the impact of this perturbation and prediction error. To solve this minimax optimization problem, a new adversarial loss function is constructed based on the cross-entropy measure. As a result, the smoothness and confidence of the model are both greatly improved. Moreover, we develop a trajectory-preserving-based alternating update strategy to enable the stable training of GAT. Numerous experiments conducted on benchmark datasets clearly demonstrate that the proposed GAT significantly outperforms the state-of-the-art AT methods in terms of supervised and semi-supervised learning tasks, especially when the number of labeled examples is rather small in semi-supervised learning.

On approximate optimality conditions for robust multi-objective convex optimization problems

In this paper, we are interested in the study of approximate optimality conditions for weakly ϵ-efficient solutions to robust multi-objective optimization problems ((RMOP) for short) in view of its associated minimax optimization problem (MMOP). To this end, we first establish the relationship between a weakly ϵ-efficient solution to the problem (RMOP) and an α-solution to the problem (MMOP), where and . Then, we explore the representation of the so-called β-normal set (where is a given parameter) to a closed convex set at some reference point by two methods. At last, by employing the α-subdifferential of the max-function and the obtained representation of the β-normal set, we establish an approximate necessary optimality condition for the problem (RMOP). Moreover, we also give an example to illustrate our results.

Secrecy‐oriented user association in ultra dense heterogeneous networks against strategically colluding adversaries

Network densification is recognized as the key technology to meet the ever growing demand of data traffic in next generation wireless networks. However the proliferation of small cell base stations (SCBSs) introduces vulnerabilities to information security, as they are prone to eavesdropping attacks. In this paper, it is studied how user association strategies can be specifically tailored to meet this security challenge in a ultra dense heterogeneous cellular network(UDHCN) with a group of colluding eavesdroppers. In particular, the situation is considered where eavesdroppers may have limited capabilities in intercepting and decoding data traffic. In this setting each eavesdropper has to make its own decision on eavesdropping targets, and they are able to take concerted actions to sabotage information security. To address this challenge, a zero sum game framework to reflect the conflicting interests of users and eavesdroppers is proposed and a user association scheme is devised that aims to maximize system sum secrecy rate against these adversaries whose actions intend to minimize sum secrecy rate. The case that all adversaries have limited eavesdropping capabilities is first considered, and it is shown that the corresponding zero sum game is indeed a bilinear game, and its Nash equilibrium solution can be readily approximated using a saddle point Frank Wolfe(SP-FW)algorithm. Then the framework is extended to the case where adversaries with a diverse configuration of eavesdropping capabilities exist. In this case, it is shown that the underlying minimax optimization problem is indeed a nonsmooth convex-nonconcave one. A two stage method is proposed by first smoothing the objective function with a Hermite cubic polynomial approximation, and then obtaining a nearly stationary solution via a recently proposed proximal point gradient descent method. Simulation results show that the proposed framework leads to significant increases in sum secrecy rate and secrecy probability against both the capability-limited adversaries and a hybrid type of adversaries.

A novel privacy-aware model for nonparametric decentralized detection

In recent years, the increasing development of the Internet of Things (IoT) demands the enhancement of both network security and user privacy protection. In the decentralized IoT network, multiple sensors send local observations to a fusion center for data aggregation and authorized hypothesis detection. But at the same time, private information might be inferred illegally, which would cause privacy leakage. In this paper, a novel privacy-aware model named AL-UP is proposed for the nonparametric decentralized detection in the IoT network. It aims to design a local differential privacy and data projection based sanitization mechanism for sensors, to hide the sensitive information in raw observations and protect the data and inference privacy. Based on the adversarial learning framework and linear discriminant analysis, we propose a max-min optimization problem to design parameters of the sanitization mechanism and hypothesis detection rules. The problem is solved via the block coordinate descend method. Numerical results on various public datasets indicate that the proposed model achieves better utility-privacy trade-off than the state of the arts.

Multiplicative deconvolution in survival analysis under dependency

We study the non-parametric estimation of an unknown survival function S with support on R + based on a sample with multiplicative measurement errors. The proposed fully data-driven procedure is based on the estimation of the Mellin transform of the survival function and a regularization of the inverse of the Mellin transform by a spectral cut-off. The upcoming bias-variance trade-off is handled by a data-driven choice of the cut-off parameter. In order to discuss the bias term, we consider the Mellin–Sobolev spaces which characterize the regularity of the unknown survival function S through the decay of its Mellin transform. For the analysis of the variance term, we consider the independent and identically distributed case and incorporate dependent observations in form of Bernoulli shift processes and β-mixing sequences. Additionally, we show minimax optimality over Mellin–Sobolev spaces of the spectral cut-off estimator.

Framework for Mapping and Optimizing the Solar Rooftop Potential of Buildings in Urban Systems

The accurate prediction of the solar energy that can be generated using the rooftops of buildings is an essential tool for many researchers, decision makers, and investors for creating sustainable cities and societies. This study is focused on the development of an automated method to extract the useable areas of rooftops and optimize the solar PV panel layout based on the given electricity loading of a building. In this context, the authors of this article developed two crucial methods. First, a special pixel-based rooftop recognition methodology was developed to analyze detailed and complex rooftop types while avoiding the challenges associated with the nature of the particular building rooftops. Second, a multi-objective enveloped min–max optimization algorithm was developed to maximize solar energy generation and minimize energy cost in terms of payback based on the marginal price signals. This optimization algorithm facilitates the optimal integration of three controlled variables—tilt angle, azimuth angle, and inter-row spacing—under a non-linear optimization space. The performance of proposed algorithms is demonstrated using three campus buildings at the University of Alberta, Edmonton, Alberta, Canada as case studies. It is shown that the proposed algorithms can be used to optimize PV panel distribution while effectively maintaining system constraints.

Optimal minimax random designs for weighted least squares estimators

Summary This work studies an experimental design problem where the values of a predictor variable, denoted by $x$, are to be determined with the goal of estimating a function $m(x)$, which is observed with noise. A linear model is fitted to $m(x)$, but it is not assumed that the model is correctly specified. It follows that the quantity of interest is the best linear approximation of $m(x)$, which is denoted by $\ell(x)$. It is shown that in this framework the ordinary least squares estimator typically leads to an inconsistent estimation of $\ell(x)$, and rather weighted least squares should be considered. An asymptotic minimax criterion is formulated for this estimator, and a design that minimizes the criterion is constructed. An important feature of this problem is that the $x$ values should be random, rather than fixed. Otherwise, the minimax risk is infinite. It is shown that the optimal random minimax design is different from its deterministic counterpart, which was studied previously, and a simulation study indicates that it generally performs better when $m(x)$ is a quadratic or a cubic function. Another finding is that, when the variance of the noise goes to infinity, the random and deterministic minimax designs coincide. The results are illustrated for polynomial regression models and a generalization is given in the Supplementary Material.

A budget allocation model for domestic airport network protection

In order to protect a domestic airport network from terrorist attacks, we investigate how to allocate a limited amount of budget into airports with different characteristics. We propose a minimax optimization model, where the Transportation Security Administration decides the budget allocation of each airport in the upper level, while the terrorist group chooses which flights to attack in the lower level. The objective of the terrorist group is to maximize the expected damage and the objective of the Transportation Security Administration is to minimize the corresponding maximum damage. We adopt a generalized cutting-plane algorithm to solve the proposed bilevel optimization problem. Moreover, we provide an illustrative example, and present some detailed sensitivity analysis and comparison results.

TELET: A monotonic algorithm to design large dimensional equiangular tight frames for applications in compressed sensing

An Equiangular tight frame (ETF) - also known as the Welch-bound-equality sequences - consists of a sequence of unit norm vectors whose absolute inner product is identical and minimal. Due to this unique property, these frames are preferred in different applications such as in constructing sensing matrices for compressed sensing systems, robust transmission and quantum computing. Construction of ETFs involves solving a challenging non-convex minimax optimization problem, and only a few methods were successful in constructing them, albeit only for smaller dimensions. In this paper, we propose an iterative algorithm named TEchnique to devise Large dimensional Equiangular Tight-frames (TELET-frames) based on the majorization minimization (MM) procedure - in which we design and minimize a tight upper bound for the ETF cost function at every iteration. Since TELET is designed using the MM approach, it inherits useful properties of MM such as monotonicity and guaranteed convergence to a stationary point. Subsequently, we use the derived frames to construct optimized sensing matrix for compressed sensing systems. In the numerical simulations, we show that the proposed algorithm can generate complex and real frames (in the order of hundreds) with very low mutual coherence value when compared to the state-of-the-art algorithm, with a slight increase in computational cost. Experiments using synthetic data and real images reveal that the optimized sensing matrix obtained through the frames constructed by TELET performs better, in terms of image reconstruction accuracy, than the sensing matrix constructed using state-of-the-art methods.

The Maximax Minimax Quotient Theorem

We present an optimization problem emerging from optimal control theory and situated at the intersection of fractional programming and linear max-min programming on polytopes. A naïve solution would require solving four nested, possibly nonlinear, optimization problems. Instead, relying on numerous geometric arguments we determine an analytical solution to this problem. In the course of proving our main theorem, we also establish another optimization result stating that the minimum of a specific minimax optimization is located at a vertex of the constraint set.

High-dimensional asymptotics of likelihood ratio tests in the Gaussian sequence model under convex constraints

In the Gaussian sequence model Y=μ+ξ, we study the likelihood ratio test (LRT) for testing H0:μ=μ0 versus H1:μ∈K, where μ0∈K, and K is a closed convex set in Rn. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair (μ0,K), in the high-dimensional regime where the estimation error of the associated least squares estimator diverges in an appropriate sense. The normal approximation further leads to a precise characterization of the power behavior of the LRT in the high-dimensional regime. These characterizations show that the power behavior of the LRT is in general nonuniform with respect to the Euclidean metric, and illustrate the conservative nature of existing minimax optimality and suboptimality results for the LRT. A variety of examples, including testing in the orthant/circular cone, isotonic regression, Lasso and testing parametric assumptions versus shape-constrained alternatives, are worked out to demonstrate the versatility of the developed theory.