Nonlinear Inequality Constraints Research Articles

A target-free vision-based method for out-of-plane vibration measurement using projection speckle and camera self-calibration technology

In recent decades, computer vision-based methods have been introduced into engineering structures to provide full-field, non-contact vibration measurement. However, the vision-based vibration measurement method is challenging to measure the targetless structures. This paper proposed a target-free out-of-plane vibration measurement method using projection speckle and camera self-calibration technology. In this method, the camera extrinsic parameter estimation result based on the direct method is used as the initial value in the proposed method. Then, a nonlinear inequality constraint on the extrinsic parameters is established regarding the epipolar condition to minimize the reprojection error. Furthermore, the projection relationship between the projection speckle displacement and the actual displacement of the structure is studied so that the deviation of the out-of-plane vibration displacement is corrected. In the experimental verification, 3D reconstruction and rigid translation experiments are designed, respectively. The results show that the self-calibration accuracy of the binocular camera proposed in this paper is notably improved compared with the existing methods. At the same time, the deviation correction effect of the vibration displacement is significant. Finally, the typical flat plate structure and composite wing structure are taken as vibration measurement cases to demonstrate that the proposed approach is reliable and accurate in measuring the dynamic responses of engineering structures.

Distributed Nash Equilibrium Seeking for Multicluster Aggregative Game of Euler-Lagrange Systems With Coupled Constraints.

This article considers the distributed Nash equilibrium seeking problem of a multicluster aggregative game subject to local set constraints, consensus constraints in the same cluster, and coupled linear equality and nonlinear inequality constraints among all clusters. In the considered game, each cluster is composed of a group of players formulated by uncertain Euler-Lagrange (EL) dynamics, and its objective is to minimize its own cost function, which is the sum of the local functions of all players in the cluster. The local cost function of each player depends on its own decision and an aggregate of the decisions of all the players. An adaptive continuous-time distributed strategy is developed for uncertain EL systems to reach the generalized Nash equilibrium (GNE) of multicluster aggregative game. In particular, a new auxiliary system is synthesized using a projection operator, gradient descent, and dynamic average consensus to estimate the GNE. Based on the outputs of the auxiliary system, an adaptive tracking algorithm is developed for an EL system with uncertain parameters. Using the Lyapunov stability theory, it is shown that the developed distributed strategy achieves accurate convergence to the GNE. Finally, a numerical example is presented to demonstrate the theoretical results.

Real-time MPC with Control Barrier Functions for Autonomous Driving using Safety Enhanced Collocation

The autonomous driving industry is continuously dealing with safety-critical scenarios, and nonlinear model predictive control (NMPC) is a powerful control strategy for handling such situations. However, standard safety constraints are not scalable and require a long NMPC horizon. Moreover, the adoption of NMPC in the automotive industry is limited by the heavy computation of numerical optimization routines. To address those issues, this paper presents a real-time capable NMPC for automated driving in urban environments, using control barrier functions (CBFs). Furthermore, the designed NMPC is based on a novel collocation transcription approach, named RESAFE/COL, that allows to reduce the number of optimization variables while still guaranteeing the continuous time (nonlinear) inequality constraints satisfaction, through regional convex hull approximation. RESAFE/COL is proven to be 5 times faster than multiple shooting and more tractable for embedded hardware without a decrease in the performance, nor accuracy and safety of the numerical solution. We validate our NMPC-CBF with RESAFE/COL on digital twins of the vehicle and the urban environment and show the safe controller's ability to improve crash avoidance by 91%. Supplementary visual material can be found at https://youtu.be/_EnbfYwljp4.

Distributed Online Learning Algorithm for Noncooperative Games Over Unbalanced Digraphs.

This article investigates constrained online noncooperative games (NGs) of multiagent systems over unbalanced digraphs, where the cost functions of players are time-varying and are gradually revealed to corresponding players only after decisions are made. Moreover, in the problem, the players are subject to local convex set constraints and time-varying coupling nonlinear inequality constraints. To the best of our knowledge, no result about online games with unbalanced digraphs has been reported, let alone constrained online games. To seek the variational generalized Nash equilibrium (GNE) of the game online, a distributed learning algorithm is proposed based on gradient descent, projection, and primal-dual methods. Under the algorithm, sublinear dynamic regrets and constraint violations are established. Finally, online electricity market games illustrate the algorithm.

A Smooth Penalty Function Algorithm for Computing Nonlinear Inequality Constraint Optimization Problem

A Smooth Penalty Function Algorithm for Computing Nonlinear Inequality Constraint Optimization Problem

Online Distributed Learning for Aggregative Games With Feedback Delays

This paper proposes a new aggregative game (AG) model with feedback delays. The strategies of players are selected from given strategy sets and subject to global nonlinear inequality constraints. Both cost functions and constrained functions of players are time-varying, which reflects the changing nature of environments. At each time, each player only has access to its strategy set information, and the information of its current cost function and current constrained function is unknown. Due to feedback delays, the feedback information of corresponding cost functions and constrained functions is not revealed to players immediately after the selection of strategies. It would take a period of time for players to observe their feedback information. To address such an AG problem, a distributed learning algorithm is proposed with the local information from their neighbors and the delayed feedback information from environments, and it is applicable to time-varying weighted digraphs. We find that the two metrics of the algorithm grow sub-linearly with respect to the learning time. A simulation example is given to illustrate the performance of the proposed algorithm.

Two-timescale recurrent neural networks for distributed minimax optimization

In this paper, we present two-timescale neurodynamic optimization approaches to distributed minimax optimization. We propose four multilayer recurrent neural networks for solving four different types of generally nonlinear convex–concave minimax problems subject to linear equality and nonlinear inequality constraints. We derive sufficient conditions to guarantee the stability and optimality of the neural networks. We demonstrate the viability and efficiency of the proposed neural networks in two specific paradigms for Nash-equilibrium seeking in a zero-sum game and distributed constrained nonlinear optimization.

Event-Triggered Sliding-Mode Control for Polynomial Fuzzy Singular Systems

This article focuses on the problem of the observer-based sliding-mode control (SMC) for polynomial fuzzy singular systems via a novel dynamic event-triggered (DET) mechanism. First, the DET mechanism is introduced to reduce the transmission burden in the communication network. Second, the DET-based observer is constructed to estimate the unavailable state variables. On this basis, the integral sliding variable is provided, and the observer dynamics can be obtained. A sufficient condition is then proposed to make sure that the whole closed-loop system composed of the observer dynamics and the estimation error system is admissible and passive by utilizing the sum-of-squares approach. Meanwhile, the sliding surface parameter matrix, the observer gain, and the trigger weighting matrix are co-designed. It should be emphasized that the common equality constraints, nonstrict and nonlinear inequality constraints are removed through equivalent sets and several inequality techniques in this design process. Moreover, the DET observer-based SMC law is derived to assure the reachability of the specified sliding surface. Finally, the theoretical results are verified by simulation results.

A Nonsmooth Dynamical Systems Perspective on Accelerated Extensions of ADMM

Recently, there has been great interest in connections between continuous-time dynamical systems and optimization methods, notably in the context of accelerated methods for smooth and unconstrained problems. In this paper we extend this perspective to nonsmooth and constrained problems by obtaining differential inclusions associated to novel accelerated variants of the alternating direction method of multipliers (ADMM). Through a Lyapunov analysis, we derive rates of convergence for these dynamical systems in different settings that illustrate an interesting tradeoff between decaying versus constant damping strategies. We also obtain modified equations capturing fine-grained details of these methods, which have improved stability and preserve the leading order convergence rates. An extension to general nonlinear equality and inequality constraints in connection with singular perturbation theory is provided.

Hierarchical Real-Time Optimal Planning of Collision-Free Trajectories of Collaborative Robots

In collaborative robotics the manipulator trajectory has to be planned to avoid collisions, yet in real-time. In this paper we pose the problem as minimization of a quadratic functional among piecewise linear trajectories in the angular (joint) space. The minimization is subjected to novel nonlinear inequality constraints that simplify the original non-penetration constraints to become cheap to evaluate in real time while still preserving collision-avoidance. The very first and most critical step of the computation is to find an initial trajectory that is free of collisions. To that goal we minimize a weighted sum of the violated constraints until they become feasible or a maximal number of steps is reached. Sometimes an incremental growing of the obstacle helps. By incremental growing we mean that we sequentially solve auxiliary subproblems with obstacles growing from ground or falling from top and use as the initial trajectory the one optimized in the previous step. The initial trajectory is then optimized while preserving feasibility at each step. We solve a sequence of simple-bound constrained quadratic programming problems formulated in the dual space of Lagrange multipliers, which are related to the original linearized inequality constraints that are active or close-to-active. Finally, we refine the trajectory parameterization and repeat the optimization, which we refer to as an hierarchical approach, until an overall prescribed time limit, being well below a second, is reached.

ALESQP: An Augmented Lagrangian Equality-Constrained SQP Method for Optimization with General Constraints

We present a new algorithm for infinite-dimensional optimization with general constraints, called ALESQP. In short, ALESQP is an augmented Lagrangian method that penalizes inequality constraints and solves equality-constrained nonlinear optimization subproblems at every iteration. The subproblems are solved using a matrix-free trust-region sequential quadratic programming (SQP) method that takes advantage of iterative, i.e., inexact linear solvers, and is suitable for large-scale applications. A key feature of ALESQP is a constraint decomposition strategy that allows it to exploit problem-specific variable scalings and inner products. We analyze convergence of ALESQP under different assumptions. We show that strong accumulation points are stationary. Consequently, in finite dimensions ALESQP converges to a stationary point. In infinite dimensions we establish that weak accumulation points are feasible in many practical situations. Under additional assumptions we show that weak accumulation points are stationary. We present several infinite-dimensional examples where ALESQP shows remarkable discretization-independent performance in all of its iterative components, requiring a modest number of iterations to meet constraint tolerances at the level of machine precision. Also, we demonstrate a fully matrix-free solution of an infinite-dimensional problem with nonlinear inequality constraints.

Inequality constrained stochastic nonlinear optimization via active-set sequential quadratic programming

We study nonlinear optimization problems with a stochastic objective and deterministic equality and inequality constraints, which emerge in numerous applications including finance, manufacturing, power systems and, recently, deep neural networks. We propose an active-set stochastic sequential quadratic programming (StoSQP) algorithm that utilizes a differentiable exact augmented Lagrangian as the merit function. The algorithm adaptively selects the penalty parameters of the augmented Lagrangian, and performs a stochastic line search to decide the stepsize. The global convergence is established: for any initialization, the KKT residuals converge to zero almost surely. Our algorithm and analysis further develop the prior work of Na et al. (Math Program, 2022. https://doi.org/10.1007/s10107-022-01846-z). Specifically, we allow nonlinear inequality constraints without requiring the strict complementary condition; refine some of designs in Na et al. (2022) such as the feasibility error condition and the monotonically increasing sample size; strengthen the global convergence guarantee; and improve the sample complexity on the objective Hessian. We demonstrate the performance of the designed algorithm on a subset of nonlinear problems collected in CUTEst test set and on constrained logistic regression problems.

Rapid Suboptimal Low-Thrust Space Trajectory Design

The finite Fourier series shape-based approach for fast initial trajectory design satisfies the equations of motion and other problem constraints at discrete points by using a nonlinear programming solver to design the Fourier coefficients. In this paper, the finite Fourier series approach is extended to account for the necessary conditions for optimality during the Fourier coefficient design. The proposed method uses the approximate trajectories to analytically estimate the costates at these discrete points by employing a finite difference technique on the adjoint differential equations. Lagrange multipliers associated with any terminal state constraints are determined using an auxiliary function. Residual errors in the stationarity condition are reduced through the objective function, with the Legendre–Clebsch condition enforced as a nonlinear inequality constraint; applying these two conditions finds suboptimal solutions without an excessive loss of computational efficiency. Test cases use steering angle and thrust acceleration magnitude controls and span Earth-escape spirals, geostationary spacecraft rendezvous, and phasing maneuvers. The presented results demonstrate the proposed method’s ability to achieve trajectories closer to the optimal solution.

Trajectory optimization of space vehicle in rendezvous proximity operation with evolutionary feasibility conserving techniques

In this paper, a direct approach is developed for discovering optimal transfer trajectories of close-range rendezvous of satellites considering disturbances in elliptical orbits. The control vector representing the inputs is parameterized via different interpolation methods, and an Estimation of Distribution Algorithm (EDA) that implements mixtures of probability models is presented. To satisfy the terminal conditions, which are represented as non-linear inequality constraints, several feasibility conserving mechanisms associated with learning and sampling methods of the EDAs are proposed, which guarantee the feasibility of the explored solutions. They include a particular implementation of a clustering algorithm, outlier detection, and several heuristic mapping methods. The combination of the proposed operators guides the optimization process in achieving the optimal solution by surfing the regions of the search domain associated with feasible solutions. Numerical simulations confirm that space transfer trajectories with minimum-fuel consumption for the chaser spacecraft can be obtained with terminal condition satisfaction in rendezvous proximity operation.

H∞ Codesign for Uncertain Nonlinear Control Systems Based on Policy Iteration Method.

In this article, the problem of H∞ codesign for nonlinear control systems with unmatched uncertainties and adjustable parameters is investigated. The main purpose is to solve the adjustable parameters and H∞ controller simultaneously so that better robust control performance can be achieved. By introducing a bounded function and defining a special cost function, the problem of solving the Hamilton-Jacobi-Isaacs equation is transformed into an optimization problem with nonlinear inequality constraints. Based on the sum of squares technique, a novel policy iteration algorithm is proposed to solve the problem of the H∞ codesign. Moreover, one modified algorithm for optimizing the robust performance index is given. The convergence and the performance improvement of new iteration policy algorithms are proved. Simulation results are presented to demonstrate the effectiveness of the proposed algorithms.

Implementation and Analyses of an Eco-Driving Algorithm for Different Battery Electric Powertrain Topologies Based on a Split Loss Integration Approach

Eco-driving algorithms optimize the speed profile to reduce the energy consumption of a vehicle. This paper presents an eco-driving algorithm for battery electric powertrains that applies a split loss integration approach to incorporate the component losses. The algorithm consistently uses loss models to overcome the drawbacks of efficiency maps, which cannot represent no-load losses at zero torque. The use of loss models is crucial since the optimal solution includes gliding, during which there are no-load losses. An analysis shows, that state-of-the-art nonlinear programming algorithms cannot represent these no-load losses at zero torque with a small modeling error. To effectively compute the powertrain losses with only a small error in comparison to the measurement data, we introduce a tailored combination of nonlinear inequality constraints that interleave two polynomial fits. This approach can properly represent reality. We parameterize the algorithm and validate the vehicle model used with real-world measurement data. Furthermore, we investigate the influence of the proposed interleaved fits by comparing them to a single continuous high-order polynomial fit and to the state of the art. The algorithm is published open source.

A υ-Constrained Matrix Adaptation Evolution Strategy With Broyden-Based Mutation for Constrained Optimization.

To solve the nonconvex constrained optimization problems (COPs) over continuous search spaces by using a population-based optimization algorithm, balancing between the feasible and infeasible solutions in the population plays an important role over different stages of the optimization process. To keep this balance, we propose a constraint handling technique, called the υ -level penalty function, which works by transforming a COP into an unconstrained one. Also, to improve the ability of the algorithm in handling several complex constraints, especially nonlinear inequality and equality constraints, we suggest a Broyden-based mutation that finds a feasible solution to replace an infeasible solution. By incorporating these techniques with the matrix adaptation evolution strategy (MA-ES), we develop a new constrained optimization algorithm. An extensive comparative analysis undertaken using a broad range of benchmark problems indicates that the proposed algorithm can outperform several state-of-the-art constrained evolutionary optimizers.

Gross error detection in steam turbine measurements based on data reconciliation of inequality constraints

Maintaining the online calculation accuracy of isentropic efficiency of a steam turbine stage is challenging due to widely existing gross errors in steam turbine measurements. They invalidate modelling results and hinder model-based monitoring and optimization. Data reconciliation is a mathematical method for gross error detection and has been applied in various industrial systems. In power plant systems, previous studies focus on gross error detection of flow rate measurements in regenerative systems based on equality constraints, which is insufficient for gross error detection of steam turbine systems. We propose a data reconciliation model adding inequality constraints to solve the problem. Statistical test is used to detect gross errors in steam turbine systems. Then an in-service 660 MW ultra-supercritical double reheat power plant is selected as a case study. Gross errors of flow rate measurements are detected and eliminated firstly. Then nonlinear inequality constraints, entropy increase of each stage, are added for further detection. Results show that the proposed model effectively detects gross errors in the steam turbine system and further improve the thermal calculation accuracy by 3.1–5.7%. It provides quantitative guidance for the calibration and maintenance of measurement instruments and facilitates performance monitoring and operation optimization in in-service power plants.

Necessary and sufficient conditions of optimality for second order discrete and differential inequalities

Abstract The present paper studies the optimization of the Bolza problem with a system of convex and nonconvex, discrete and differential state variable second-order inequality constraints by deriving necessary and sufficient conditions of optimality. The problem with a system of discrete-approximation inequalities is investigated using the proposed method of discretization and equivalence theorems for subdifferential inclusions, which greatly contributes to the derivation of adjoint discrete inclusions generated by a given system of nonlinear inequality constraints. Furthermore, we formulate sufficient conditions of optimality for the continuous problem by passing to the case of limit. A numerical example is provided to illustrate the theoretical approach’s effectiveness.

Semicontinuity of Solutions and Well-Posedness Under Perturbations for Equilibrium Problems with Nonlinear Inequality Constraints

Semicontinuity of Solutions and Well-Posedness Under Perturbations for Equilibrium Problems with Nonlinear Inequality Constraints