Bounding Set Research Articles

Augmenting bi-objective branch and bound by scalarization-based information

While branch and bound based algorithms are a standard approach to solve single-objective (mixed-)integer optimization problems, multi-objective branch and bound methods are only rarely applied compared to the predominant objective space methods. In this paper we propose modifications to increase the performance of multi-objective branch and bound algorithms by utilizing scalarization-based information. We use the hypervolume indicator as a measure for the gap between lower and upper bound set to implement a multi-objective best-first strategy. By adaptively solving scalarizations in the root node to integer optimality we improve both, upper and lower bound set. The obtained lower bound can then be integrated into the lower bounds of all active nodes, while the determined solution is added to the upper bound set. Numerical experiments show that the number of investigated nodes can be significantly reduced by up to 83% and the total computation time can be reduced by up to 80%.

On the Benefit of Nonlinear Control for Robust Logarithmic Growth: Coin Flipping Games as a Demonstration Case

The takeoff point for this paper is the voluminous body of literature addressing recursive betting games with expected logarithmic growth of wealth being the performance criterion. Whereas almost all existing papers involve use of linear feedback, the use of nonlinear control is conspicuously absent. This is epitomized by the large subset of this literature dealing with Kelly Betting. With this as the high-level motivation, we study the potential for use of nonlinear control in this framework. To this end, we consider a “demonstration case” which is one of the simplest scenarios encountered in this line of research: repeated flips of a biased coin with probability of heads p, and even-money payoff on each flip. First, we formulate a new robust nonlinear control problem which we believe is both simple to understand and apropos for dealing with concerns about distributional robustness; i.e., instead of assuming that p is perfectly known as in the case of the classical Kelly formulation, we begin with a bounding set P0,1 for this probability. Then, we provide a theorem, our main result, which gives a closed-form description of the optimal robust nonlinear controller and a corollary which establishes that it robustly outperforms linear controllers such as those found in the literature. A second, less significant, contribution of this paper bears upon the computability of our solution. For an n-flip game, whereas an admissible controller has 2n-1 parameters, at the optimum only O(n2) of them turn out to be distinct. Finally, it is noted that the initial assumptions on payoffs and the use of the uniform distribution on p are made mainly for simplicity of the exposition and compliance with length requirements for a Letter. Accordingly, the paper also includes a new section with a discussion indicating how these assumptions can be relaxed.

Robust Adaptive Path Tracking Control Scheme for Safe Autonomous Driving via Predicted Interval Algorithm

This paper introduces a robust adaptive path-tracking control scheme via a predicted interval approach for safe autonomous driving tasks under uncertainties. Specifically, a recursive least squares-based set-membership mechanism is firstly designed to estimate a bounding set of acceptable values to depict the uncertain parameters. Based on the estimated system parameters, an interval predictor is deployed to improve the prediction accuracy in the primary control action design. Successively, the unconstrained output feedback-based robust controller is proposed to yield the closed-loop system stabilization by utilizing predicted interval output only. Meanwhile, a model predictive control technique is conceived from solving an optimization problem that is given in the interval predictor to ensure robust constraint satisfaction. The recursive feasibility of the controlled system is theoretically analyzed by applying the nonconservative Lyapunov function with a novel structure and the closed-loop system possesses the input-to-state stability criteria. Finally, simulation results are provided to verify the efficacy of the presented strategy under various intricate scenarios. The results show that the suggested controller always maintains its cross-tracking error and longitudinal velocity error at the lowest level even in the most challenging weather scenario.

An agent based force vector model of social influence that predicts strong polarization in a connected world.

The model is based on a vector representation of each agent. The components of the vector are the key continuous “attributes” that determine the social behavior of the agent. A simple mathematical force vector model is used to predict the effect of each agent on all other agents. The force law used is motivated by gravitational force laws and electrical force laws for dipoles. It assumes that the force between two agents is proportional to the “similarity of attributes”, which is implemented mathematically as the dot product of the vectors representing the attributes of the agents, and the force goes as the inverse square of the difference in attributes, which is expressed as the Euclidean distance in attribute space between the two vectors. The force between the agents may be positive (attractive), zero, or negative (repulsive) depending on whether the angle between the corresponding vectors is less than, equal to, or greater than 90°. A positive force causes the attributes of the agents to become more similar and the corresponding vectors to become more nearly parallel. Interaction between all agents is allowed unless the distance between the attributes representing the agents exceeds a confidence limit (the Attribute Influence Bound) set in the simulation. Agents with similar attributes tend to form groups. For small values of the Attribute Influence Bound, numerous groups remain scattered throughout attribute space at the end of a simulation. As the Attribute Influence Bound is increased, and agents with increasingly different attributes can communicate, fewer groups remain at the end, and the remaining groups have increasingly different characteristic attributes and approximately equal sizes. With a large Attribute Influence Bound all agents are connected and extreme bi- or tri-polarization results. During the simulations, depending on the initial conditions, collective behaviors of grouping, consensus, fragmentation and polarization are observed as well as certain symmetries specific to the model, for example, the average of the attributes for all agents does not change significantly during a simulation.

Value-Set-Based Approach to Robust Stability Analysis for Ellipsoidal Families of Fractional-Order Polynomials with Complicated Uncertainty Structure

This paper presents the application of a value-set-based graphical approach to robust stability analysis for the ellipsoidal families of fractional-order polynomials with a complex structure of parametric uncertainty. More specifically, the article focuses on the families of fractional-order linear time-invariant polynomials with affine linear, multilinear, polynomic, and general uncertainty structure, combined with the uncertainty bounding set in the shape of an ellipsoid. The robust stability of these families is investigated using the zero exclusion condition, supported by the numerical computation and visualization of the value sets. Four illustrative examples are elaborated, including the comparison with the families of fractional-order polynomials having the standard box-shaped uncertainty bounding set, in order to demonstrate the applicability of this method.

State bounding estimation for a linear continuous-time singular system with time-varying delay

This paper investigates the problem of a state bounding estimation for a linear continuous-time singular system with time-varying delay. By employing the maximal Lyapunov–Krasovskii functional and applying the new free-matrix-based integral inequality, some proper conditions are derived in terms of LMIs and a bounding estimation lemma and set are obtained for the studied singular system.

Assessment of large-scale catastrophes in complex engineering systems

The paper considers a traditional approach to ensuring safety of complex technical system that is based on development of protection barriers against the bounding set of credible (design-basis) impacts and presents a modern approach. The approach is based on combined efforts to improve the system protection from design basis impacts and to increases the system resilience towards beyond design basis events allowing one to include low probability –high consequences (beyond design-basis) events into the scope of risk assessment and management framework

System reliability analysis of tunnels reinforced by rockbolts

System reliability analysis of tunnelling problems is illustrated by using an iterative closed-form formulation for a circular tunnel reinforced by end-anchored rockbolts. Three performance functions, namely, the mobilized tensile force in the rockbolt, the maximum displacement and the plastic zone size of the tunnel, are considered. The bimodal bounds method and the multivariate normal cumulative distribution function (mvncdf) method are applied to the solution. Results show that the system probability of failure (Pf) estimated from the mvncdf method lies within the lower bound and upper bound set by the bimodal bounds method for the cases studied. The second-order reliability method (SORM) is used to refine the reliability indices and to improve the accuracy of the estimated system Pf. The influence of the correlation coefficient between the cohesion and the friction angle and the influence of the rockbolt installation position on the system Pf are discussed. A modified hybrid approach using the linear response surface method (RSM) to locate the design point and the artificial neural network (ANN) to approximate the actual limit state surface (LSS) is suggested to be used in the system reliability analysis for problems where closed-form solutions are not available. Traditional second-order RSM without cross terms may be inaccurate and may suffer from the problem of low convergence rate and the second-order RSM with cross terms may encounter the “false branch” problem. It is also shown in this paper that the “false branch” problem can be overcome by the importance sampling technique. Comparison with the second-order RSM shows that the proposed approach is efficient, accurate and robust for the system reliability analysis.

Continuous-time probabilistic ultimate bounds and invariant sets: Computation and assignment

The concepts of ultimate bounds and invariant sets play a key role in several control theory problems, as they replace the notion of asymptotic stability in the presence of unknown disturbances. However, when the disturbances are unbounded, as in the case of Gaussian white noise, no ultimate bounds nor invariant sets can in general be found. To overcome this limitation we introduced, in previous work, the notions of probabilistic ultimate bound (PUB) and probabilistic invariant set (PIS) for discrete-time systems. This article extends the notions of PUB and PIS to continuous-time systems, studying their main properties and providing tools for their calculation. In addition, the use of these concepts in robust control design by covariance assignment is presented.

Shock capturing data assimilation algorithm for 1D shallow water equations

We propose a new data assimilation algorithm for shallow water equations in one dimension. The algorithm is based upon Discontinuous Galerkin spatial discretization of shallow water equations (DG-SW model) and the continuous formulation of the minimax filter. The latter allows for construction of a robust estimation of the state of the DG-SW model and computes worst-case bounds for the estimation error, provided the uncertain parameters belong to a given bounding set. Numerical studies show that, given sparse observations from numerical or physical experiments, the proposed algorithm quickly reconstructs the true solution even in the presence of shocks, rarefaction waves and unknown values of model parameters. The minimax filter is compared against the ensemble Kalman filter (EnKF) for a benchmark dam-break problem and the results show that the minimax filter converges faster to the true solution for sparse observations.

Surrogate upper bound sets for bi-objective bi-dimensional binary knapsack problems

The paper deals with the definition and the computation of surrogate upper bound sets for the bi-objective bi-dimensional binary knapsack problem. It introduces the Optimal Convex Surrogate Upper Bound set, which is the tightest possible definition based on the convex relaxation of the surrogate relaxation. Two exact algorithms are proposed: an enumerative algorithm and its improved version. This second algorithm results from an accurate analysis of the surrogate multipliers and the dominance relations between bound sets. Based on the improved exact algorithm, an approximated version is derived. The proposed algorithms are benchmarked using a dataset composed of three groups of numerical instances. The performances are assessed thanks to a comparative analysis where exact algorithms are compared between them, the approximated algorithm is confronted to an algorithm introduced in a recent research work.

Conformal behavior at four loops and scheme (in)dependence

We search for infrared zeros of the beta function and evaluate the anomalous dimension of the mass at the associated fixed point for asymptotically free vectorlike fermionic gauge theories with gauge group $SU(N)$. The fixed points of the beta function are studied at the two, three and four loop level in two different explicit schemes. These are the modified regularization invariant scheme and the minimal momentum subtraction scheme. The search is performed in Landau gauge where the beta function of the gauge parameter vanishes. We then compare our findings to earlier identical investigations performed in the modified minimal subtraction scheme. It is found that the value of the anomalous dimension of the mass is smaller at three and four loops than at two loops. This seems to be a generic pattern that is observed in all three different schemes. We then estimate the value of the anomalous dimension to be $\ensuremath{\gamma}\ensuremath{\sim}0.225--0.375$ for twelve fundamental flavors and three colors, $\ensuremath{\gamma}\ensuremath{\sim}0.500--0.593$ for two adjoint flavors and two colors and finally $\ensuremath{\gamma}\ensuremath{\sim}1.12--1.70$ for two two-indexed flavors and three colors with the lower and upper bound set by the minimum and maximum value respectively over all three schemes and at three and four loops. Our analysis suggests that the former two theories lie in the conformal window while the latter belongs to the chirally broken phase.

Computation of Continuous-Time Probabilistic Invariant Sets and Ultimate Bounds

The concepts of ultimate bounds and invariant sets play a key role in several control theory problems, as they replace the notion of asymptotic stability in the presence of unknown disturbances. However, when the disturbances are unbounded, as in the case of Gaussian white noise, no ultimate bounds nor invariant sets can in general be found. To overcome this limitation we introduced, in previous work, the notions of probabilistic ultimate bound (PUB) and probabilistic invariant set (PIS) for discrete-time systems. This article extends the notions of PUB and PIS to continuous-time systems, studying their main properties and providing tools for their calculation. An application of these concepts to robust control design is also presented.

Dynamic programming algorithms for the bi-objective integer knapsack problem

This paper presents two new dynamic programming (DP) algorithms to find the exact Pareto frontier for the bi-objective integer knapsack problem. First, a property of the traditional DP algorithm for the multi-objective integer knapsack problem is identified. The first algorithm is developed by directly using the property. The second algorithm is a hybrid DP approach using the concept of the bound sets. The property is used in conjunction with the bound sets. Next, the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set. It means that the upper bound set is underestimated. Then, an extended upper bound set is proposed on the basis of the set of linear relaxation solutions. The efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set. The numerical results obtained from different types of bi-objective instances show the effectiveness of the proposed approach.

Global attitude and gyro bias estimation based on set-valued observers

The problem of attitude and rate gyro bias estimation is addressed by resorting to measurements acquired from rate gyros and vector observations. A Set-Valued Observer (SVO) is proposed that has no singularities and that, for any initial conditions, provides a bounding set with guarantees of containing the actual (unknown) rotation matrix. The sensor readings are assumed to be corrupted by bounded measurement noise and constant gyro bias. Conditions for the boundedness of the estimated sets are established and implementation details are discussed. The feasibility of the technique is demonstrated in simulation.

Practical perfect sampling using composite bounding chains: the Dirichlet-multinomial model

A discrete data augmentation scheme together with two different parameterizations yields two Gibbs samplers for sampling from the posterior distribution of the hyperparameters of the Dirichlet-multinomial hierarchical model under a default prior distribution. The finite-state space nature of this data augmentation permits us to construct two perfect samplers using bounding chains that take advantage of monotonicity and anti-monotonicity in the target posterior distribution, but both are impractically slow. We demonstrate that a composite algorithm that strategically alternates between the two samplers’ updates can be substantially faster than either individually. The speed gains come because the composite algorithm takes a divide-and-conquer approach in which one update quickly shrinks the bounding set for the augmented data, and the other update immediately coalesces on the parameter, once the augmented-data bounding set is a singleton. We theoretically bound the expected time until coalescence for the composite algorithm, and show via simulation that the theoretical bounds can be close to actual performance.

Simultaneous design of explicit/multi-parametric constrained moving horizon estimation and robust model predictive control

In this work we present a rigorous methodology for the simultaneous design of moving horizon estimation (MHE) and robust model predictive control based on multi-parametric programming. First, an explicit/multi-parametric solution of the MHE is derived. Then, a novel method is presented that allows for the derivation of the estimation error dynamics, the bounding set of the estimation error, and the state estimate dynamic equations of constrained MHE. A framework is then presented for the design of robust explicit/multi-parametric model predictive control (MPC) controllers, based on tube-based MPC methods, which ensures that no constraints are violated due to the estimation error and the process noise in the system. This framework is first shown for the Kalman filter and unconstrained MHE and is then extended to the constrained MHE.

Moving horizon estimation: Error dynamics and bounding error sets for robust control

In this work we present a method for the derivation of the estimation error dynamics, the bounding set of the estimation error, and the state estimate dynamic equations of the constrained Moving Horizon Estimator (MHE).

A new hyperchaotic Lorenz‐type system: Generation, analysis, and implementation

AbstractA new four‐dimensional continuous‐time autonomous hyperchaotic Lorenz‐type system is introduced and analyzed. This hyperchaotic system is not only visualized by computer simulation but also verified with bifurcation analysis and realized with an electronic circuit. Moreover, explicit formulae for estimating the ultimate bound and positive invariant set of the system are derived by constructing a family of generalized Lyapunov functions. The findings and results of this paper have good potential in control and synchronization of hyperchaos and their engineering applications. Copyright © 2010 John Wiley & Sons, Ltd.

Componentwise ultimate bound and invariant set computation for switched linear systems

We present a novel ultimate bound and invariant set computation method for continuous-time switched linear systems with disturbances and arbitrary switching. The proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties. The method provides ultimate bounds and invariant sets in the form of polyhedral and/or mixed ellipsoidal/polyhedral sets, is completely systematic once the aforementioned transformation is obtained, and provides a new sufficient condition for practical stability. We show that the transformation required by our method can easily be found in the well-known case where the subsystem matrices generate a solvable Lie algebra, and we provide an algorithm to seek such transformation in the general case. An example comparing the bounds obtained by the proposed method with those obtained from a common quadratic Lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases.