Robust Optimization Criterion Research Articles

Error-distribution-free kernel extreme learning machine for traffic flow forecasting

Traffic flow modeling plays a crucial role in intelligent transportation systems, which is of vital significance for mitigating traffic congestion and reducing carbon emissions. Owing to the uncertainties and nonlinear characteristics of traffic flow, it confronts a considerable challenge to establish a model to predict traffic flow efficiently and robustly. Kernel-based extreme learning machine (KELM), a natural extension of extreme learning machine (ELM) that incorporates kernel learning, has demonstrated excellent performance in traffic flow prediction. However, the performance of KELM may significantly decrease when the noise is non-Gaussian, as it was developed under the minimum mean square error (MMSE) criterion assuming Gaussian noise. To address this issue, we propose an error-distribution-free kernel extreme learning machine, termed ɛDFKELM, by embedding a more robust optimization criterion to guide the training. In addition, we further develop an online version of the ɛDFKELM model for continual forecasting, called ɛDFKELMv2. We perform extensive experiments on two widely-used public benchmark traffic flow datasets, which illustrate that the ɛDFKELM model outperforms the state-of-the-art approaches in terms of forecasting performance. ɛDFKELM model achieves RMESE values of 251.49 vehs/h, 196.27 vehs/h, 216.97 vehs/h, and 160.92 vehs/h on the A1, A2, A4, and A8 highways of Amsterdam dataset, respectively.

Robust generalized hyperbolic secant algorithm for nonlinear active noise control

In nonlinear active noise control (ANC) system where the primary noise obeys non-Gaussian distribution, the traditional filter-s least mean square (FsLMS) algorithm is hardly satisfactory. In order to surmount this deficiency, a novel generalized hyperbolic secant function with lp norm (FsGHSFP) algorithm is proposed for nonlinear ANC system, which designs a robust optimization criterion utilizing the generalized hyperbolic secant distribution function. The FsGHSFP algorithm has robust noise attenuation capability for non-Gaussian noise, but suffers from the problem of tuning the parameter. To address this problem, an improved FsGHSFP (IFsGHSFP) algorithm is developed by taking advantage of the fact that continuous mixed norm does not need to adjust norm parameter. Moreover, in order to strike a balance between convergence and misalignment, thereby further enhancing the robustness of the ANC system, a convex combination algorithm of IFsGHSFP is suggested. The robustness mechanism of the proposed algorithms is analyzed, as well as stability conditions and computational complexity. It is verified by simulations that the proposed algorithms accelerate the convergence speed and improve the noise attenuation performance. In addition, the experimental results further validate the effectiveness of the proposed algorithms.

Accounting for non-normal distribution of input variables and their correlations in robust optimization

In this work, metamodel-based robust optimization is performed using measured scatter of noise variables. Principal component analysis is used to describe the input noise using linearly uncorrelated principal components. Some of these principal components follow a normal probability distribution, others however deviate from a normal probability distribution. In that case, for more accurate description of material scatter, a multimodal distribution is used. An analytical method is implemented to propagate the noise distribution via metamodel and to calculate the statistics of the response accurately and efficiently. The robust optimization criterion as well as the constraints evaluation are adjusted to properly deal with multimodal response. Two problems are presented to show the effectiveness of the proposed approach and to validate the method. A basketball free throw in windy weather condition and forming of B-pillar component are presented. The significance of accounting for non-normal distribution of input variables using multimodal distributions is investigated. Moreover, analytical calculation of response statistics, and adjustment of the robust optimization problem are presented and discussed.

Recursive Geman–McClure Estimator for Implementing Second-Order Volterra Filter

The second-order Volterra (SOV) filter is a powerful tool for modeling the nonlinear system. The Geman-McClure estimator, whose loss function is non-convex and has been proven to be a robust and efficient optimization criterion for learning system. In this paper, we present a SOV filter, named SOV recursive Geman-McClure, which is an adaptive recursive Volterra algorithm based on the Geman-McClure estimator. The mean stability and mean-square stability (steady-state excess mean square error (EMSE)) of the proposed algorithm is analyzed in detail. Simulation results support the analytical findings and show the improved performance of the proposed new SOV filter as compared with existing algorithms in both Gaussian and impulsive noise environments.

Necessary optimality conditions for average cost minimization problems

Control systems involving unknown parameters appear a natural framework for applications in which the model design has to take into account various uncertainties. In these circumstances the performance criterion can be given in terms of an average cost, providing a paradigm which differs from the more traditional minimax or robust optimization criteria. In this paper, we provide necessary optimality conditions for a nonrestrictive class of optimal control problems in which unknown parameters intervene in the dynamics, the cost function and the right end-point constraint. An important feature of our results is that we allow the unknown parameters belonging to a mere complete separable metric space (not necessarily compact).

Quantile-Optimal Treatment Regimes

ABSTRACTFinding the optimal treatment regime (or a series of sequential treatment regimes) based on individual characteristics has important applications in areas such as precision medicine, government policies, and active labor market interventions. In the current literature, the optimal treatment regime is usually defined as the one that maximizes the average benefit in the potential population. This article studies a general framework for estimating the quantile-optimal treatment regime, which is of importance in many real-world applications. Given a collection of treatment regimes, we consider robust estimation of the quantile-optimal treatment regime, which does not require the analyst to specify an outcome regression model. We propose an alternative formulation of the estimator as a solution of an optimization problem with an estimated nuisance parameter. This novel representation allows us to investigate the asymptotic theory of the estimated optimal treatment regime using empirical process techniques. We derive theory involving a nonstandard convergence rate and a nonnormal limiting distribution. The same nonstandard convergence rate would also occur if the mean optimality criterion is applied, but this has not been studied. Thus, our results fill an important theoretical gap for a general class of policy search methods in the literature. The article investigates both static and dynamic treatment regimes. In addition, doubly robust estimation and alternative optimality criterion such as that based on Gini’s mean difference or weighted quantiles are investigated. Numerical simulations demonstrate the performance of the proposed estimator. A data example from a trial in HIV+ patients is used to illustrate the application. Supplementary materials for this article are available online.

The effect of using a robust optimality criterion in model based adaptive optimization

Optimizing designs using robust (global) optimality criteria has been shown to be a more flexible approach compared to using local optimality criteria. Additionally, model based adaptive optimal design (MBAOD) may be less sensitive to misspecification in the prior information available at the design stage. In this work, we investigate the influence of using a local (lnD) or a robust (ELD) optimality criterion for a MBAOD of a simulated dose optimization study, for rich and sparse sampling schedules. A stopping criterion for accurate effect prediction is constructed to determine the endpoint of the MBAOD by minimizing the expected uncertainty in the effect response of the typical individual. 50 iterations of the MBAODs were run using the MBAOD R-package, with the concentration from a one-compartment first-order absorption pharmacokinetic model driving the population effect response in a sigmoidal EMAX pharmacodynamics model. The initial cohort consisted of eight individuals in two groups and each additional cohort added two individuals receiving a dose optimized as a discrete covariate. The MBAOD designs using lnD and ELD optimality with misspecified initial model parameters were compared by evaluating the efficiency relative to an lnD-optimal design based on the true parameter values. For the explored example model, the MBAOD using ELD-optimal designs converged quicker to the theoretically optimal lnD-optimal design based on the true parameters for both sampling schedules. Thus, using a robust optimality criterion in MBAODs could reduce the number of adaptations required and improve the practicality of adaptive trials using optimal design.

The Data‐Reusing MCC‐Based Algorithm and Its Performance Analysis

Maximum correntropy criterion (MCC) provides a robust optimality criterion for non-Gaussian signal processing. In this paper, the weight update equation of the conventional MCC-based adaptive filtering algorithm is modified by reusing the past K input vectors, forming a class of data-reusing MCC-based algorithm, called DRMCC algorithm. Comparing with the conventional MCCbased algorithm, the DR-MCC algorithm provides a much better convergence performance when the input data is correlated. The mean-square stability bound of the DRMCC algorithm has been studied theoretically. For both Gaussian noise case and non-Gaussian noise case, the expressions for the steady-state Excess mean square error (EMSE) of DR-MCC algorithm have been derived. The relationship between the data-reusing order and the steadystate EMSEs is also analyzed. Simulation results are in agreement with the theoretical analysis.

Robust Optimization of Credit Portfolios

We introduce a dynamic credit portfolio framework where optimal investment strategies are robust against misspecifications of the reference credit model. The risk-averse investor models his fear of credit risk misspecification by considering a set of plausible alternatives whose expected log likelihood ratios are penalized. We provide an explicit characterization of the optimal robust bond investment strategy, in terms of default state dependent value functions associated with the max-min robust optimization criterion. The value functions can be obtained as the solutions of a recursive system of Hamilton-Jacobi-Bellman (HJB) equations. We show that each HJB equation is equivalent to a suitably truncated equation admitting a unique bounded regular solution. The truncation technique relies on estimates for the solution of the master HJB equation that we establish.

Robust optimization of base isolation devices under uncertain parameters

Base isolator devices are widely used for mitigation of vibrations induced in structures by seismic actions. In order to achieve high performances in the mitigation of seismic effects, base isolator mechanical properties should be designed by an optimum criterion. In common approaches, the nature of dynamic loads is assumed as the only source of uncertainty. In the present paper a robust optimization criterion for base isolator devices design is proposed, considering the unavoidable effects of uncertainty in structural properties and seismic action. Uncertain parameters are modeled as random variables and are represented by bounded independent probability density function, with uniform law. The structure is described by a single-degree-of-freedom model and is protected by a linear base isolator in order to reduce vibration levels induced by base acceleration, here modeled by the stationary Kanai-Tajimi stochastic process. The optimal design is formulated as a constrained minimization problem, assuming as an objective function a suitable measure of the isolator efficiency and imposing a constraint on the maximum isolator displacement. A sensitivity analysis is carried out on the robust solution in order to assess characteristics and differences with respect to the conventional deterministic solution.

An Adaptive Prediction Algorithm Based on Maximum Correntropy Criterion

The traditional cost function, minimization mean square prediction error is a second order statistic, and it is based on the error Gaussian distribution and linear assumption. But chaotic signals are non-Gaussian, so the optimization criterion is not suitable. Then we present using the robust optimization criterion, maximum correntropy to replace the popular minima mean square error criterion minimization error. In simulation, the algorithm shows an improved performance to a common three-order Volterra prediction.

A Culture-Based Particle Swarm Optimization Framework for Dynamic, Constrained Multi-Objective Optimization

Real-world optimization problems are often dynamic, multiple objective in nature with various constraints and uncertainties. This work proposes solving such problems by systematic segmentation via heuristic information accumulated through Cultural Algorithms. The problem is tackled by maintaining 1) feasible and infeasible best solutions and their fitness and constraint violations in the Situational Space, 2) objective space bounds for the search in the Normative Space, 3) objective space crowding information in the Topographic Space, and 4) function sensitivity and relocation offsets (to reuse available information on optima upon change of environments) in the Historical Space of a cultural framework. The information is used to vary the flight parameters of the Particle Swarm Optimization, to generate newer individuals and to better track dynamic and multiple optima with constraints. The proposed algorithm is validated on three numerical optimization problems. As a practical application case study that is computationally intensive and complex, parameter tuning of a PID (Proportional–Integral–Derivative) controller for plants with transfer functions that vary with time and imposed with robust optimization criteria has been used to demonstrate the effectiveness and efficiency of the proposed design.

Improving robustness of filtered- x least mean p-power algorithm for active attenuation of standard symmetric- α-stable impulsive noise

The paper concerns active control of impulsive noise having peaky distribution with heavy tail. Such impulsive noise can be modeled using non-Gaussian stable process for which second order moments do not exist. The most famous filtered- x least mean square (FxLMS) algorithm for active noise control (ANC) systems is based on the minimization of variance (second order moment) of error signal, and hence, becomes unstable for the impulsive noise. In order to improve the robustness of adaptive algorithms for processes having distributions with heavy tails (i.e. signals with outliers), either (1) a robust optimization criterion may be used to derive the adaptive algorithm or (2) the large amplitude samples may be ignored or replaced by an appropriate threshold value. Among the existing algorithms for ANC of impulsive noise, one is based on the minimizing least mean p-power (LMP) of the error signal, resulting in FxLMP algorithm (approach 1). The other is based on modifying; on the basis of statistical properties; the reference signal in the update equation of the FxLMS algorithm (approach 2). In this paper we propose two solutions to improve the robustness of the FxLMP algorithm. In first proposed algorithm, the reference and the error signals are thresholded before being used in the update equation of FxLMP algorithm. As another solution to improve the performance of FxLMP algorithm, a modified normalized step size is proposed. The computer simulations are carried out, which demonstrate the effectiveness of the proposed algorithms.

Approximate Robust Policy Iteration Using Multilayer Perceptron Neural Networks for Discounted Infinite-Horizon Markov Decision Processes With Uncertain Correlated Transition Matrices

We study finite-state, finite-action, discounted infinite-horizon Markov decision processes with uncertain correlated transition matrices in deterministic policy spaces. Existing robust dynamic programming methods cannot be extended to solving this class of general problems. In this paper, based on a robust optimality criterion, an approximate robust policy iteration using a multilayer perceptron neural network is proposed. It is proven that the proposed algorithm converges in finite iterations, and it converges to a stationary optimal or near-optimal policy in a probability sense. In addition, we point out that sometimes even a direct enumeration may not be applicable to addressing this class of problems. However, a direct enumeration based on our proposed maximum value approximation over the parameter space is a feasible approach. We provide further analysis to show that our proposed algorithm is more efficient than such an enumeration method for various scenarios.

Fuzzy-entropy based robust optimization criteria for tuned mass dampers

Tuned mass dampers (TMD) are well known as one of the most widely adopted devices in vibration control passive strategies. In the past few decades, many methods have been developed to find the optimal parameters of a TMD installed on a structure and subjected to a random base excitation process, but most of them are usually based on an implicit assumption that all of the structural parameters are deterministic. However, in many real cases this simplification is unacceptable, so robust optimal design criteria becomes a viable alternative to better support engineers in the design process. In Robust Design Optimization (RDO) approaches, indeed the solution must be able to not only minimize the performance but also to limit its variation induced by uncertainty. Most of the currently available RDO methods are based on a probabilistic description of the model uncertainty, even if in many cases they are not able to explicitly include the influence of all the possible sources of uncertainties. Therefore, in this study, a fuzzy version of the robust TMD design optimization problem is proposed. The consistency of the fuzzy approach is studied with respect to the available non-probabilistic formulations reported in the literature and an application to an example of a robust design of a linear TMD subjected to base random vibrations in the presence of fuzzy uncertainties. The results show that the proposed fuzzy-based approach is able to give a set of optimal solutions both in terms of structural efficiency and sensitivity to mechanical and environmental uncertainties.

Robust Optimality for Discounted Infinite-Horizon Markov Decision Processes With Uncertain Transition Matrices

We study finite-state, finite-action, discounted infinite-horizon Markov decision processes with uncertain transition matrices in the deterministic policy space. The transition matrices are classified as either independent or correlated. A generalized robust optimality criterion which can be degenerated to some popular optimality criteria is proposed, under which an optimal or near-optimal policy exists for any uncertain transition matrix. Theorems are developed to guarantee a stationary policy being optimal or near-optimal in the deterministic policy space.

Conformal Radiotherapy and Intensity-modulated Radiotherapy: Clinical Data

Conformal radiotherapy (CRT) is based on three hypotheses: (i) a higher rate of local control can improve the survival rate; (ii) dose escalation can increase tumor control; and (iii) CRT allows the delivery of higher doses by decreasing the incidence of late effects. These postulates are now supported by several data. Three-dimensional conformal radiotherapy (3D-CRT) has markedly progressed since its introduction two decades ago. However, there are situations for which 3D-CRT cannot produce a satisfactory treatment plan because of complex target volume shapes or the close proximity of sensitive normal tissues. This is why intensity-modulated radiation therapy (IMRT) was introduced. Its aim is to overcome the limitations of 3D-CRT by adding modulators of beam intensity to beam shaping. IMRT can achieve nearly any dose distribution; however, the role of the planner remains crucial. CRT has been investigated mainly for prostate cancers and head and neck cancers. By and large, the clinical data, although still limited, seem to confirm the advantages of this type of radiotherapy. Dose escalation in prostate cancers improves the local control rate without increasing late effects and for this cancer site IMRT appears to be a significant advance over conventional 3D-CRT. In head and neck cancers the clinical data are still scarce but encouraging. CRT should be investigated in breast cancers with the aim of reducing the incidence of late effects. The available data underline the great potential for major progress in 3D-CRT and IMRT. The techniques are still costly and time consuming, nevertheless they merit investigation since their cost should decrease. Efforts should be concentrated on the specification of robust optimization criteria, taking into account clinical and radiobiological data.

Robust estimation and optimal selection of polynomial parameters for the interpolation of GPS geoid heights

The polynomial interpolation of least squares and interpolation moving least squares based on control stations with known GPS (global positioning system) ellipsoidal heights and levelling orthometric heights are the most often used methods for the interpolation of the geoid heights. But in their applications there occur two problems: one lies in selecting the suitable polynomial parameters; the other in reducing the influences of some possibly abnormal data points. To solve both of the problems, without emphasizing a sound theoretical basis, a heuristic solution with the help of robust estimation technique and optimization criteria for the regression equation is presented. Through two actual numerical examples it is shown that the new solution concept is efficient and can be realized easily on computers.