Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization

This chapter provides a literature review of optimization problems in the context of grey system theory, as proposed by various authors. The chapter explains the binary interactive algorithm approach as a problem-solving method for linear programming and quadratic programming problems with uncertainty and a genetic-algorithm-based approach as a second problem-solving scheme for linear programming, quadratic programming, and general nonlinear programming problems with uncertainty. In the chapter, details on the computation procedures involved for solving the aforementioned optimization problems with uncertainty are presented and results from these two approaches are compared and contrasted. Finally, possible future work area in the subject is suggested.

In this paper we deal with multiobjective linear and quadratic programming problem with uncertain information. So far in the field of statistical analysis and data mining, e.g., mean-variance portfolio problem, support vector machine and their varieties, we have encountered various kinds of quadratic and linear programming problems with multiple criteria. Moreover coefficients in such problems have uncertainty that is expressed by interval, probabilistic distribution or possibilistic (fuzzy) distribution. In this paper, we define a robust basis for all possible perturbation of coefficients within intervals in objective functions and constraints that is regarded as secure and conservative solution under uncertainty. According to the conventional multi-objective programming literature, it is required to solve test subproblem for each basis. Therefore, in case of our interval problem excessive computational demand is estimated. In this paper investigating the properties of robust basis by means of combination of interval extreme points we obtained the result that the robust basis can be examined by working with only a finite subset of possible perturbations of the coefficients

Aircraft assembly modeling requires mass solving of quadratic programming (QP) problems. Thus, reducing the computation time becomes a very urgent task. Earlier [12], it was shown that a significant reduction in time can be achieved by reformulating the original QP problem in the so-called form of the relative QP problem. The current paper discusses the aspects of using relative QP problem related to the accuracy of the computations. Data preparation for the relative QP problem involves the matrix inversions. Since the Hessian in the considered QP problems is ill-conditioned, this operation leads to a rapid accumulation of roundoff errors. The paper proposes to use the error-free operations of addition and multiplication to increase the accuracy of the QP problem solution and to maintain high computation speed for QP problem solving. Finally, a comparison of the primal, dual, and relative QP problem formulations in terms of computation time and result accuracy for assembly problems is presented.KeywordsQuadratic programmingRelative problem formulationAssembly problemsComputational accuracy

<p>There exist several methods for solving fuzzy linear or nonlinear programming problems under positivity fuzzy variables and restricted fuzzy coefficients. Due to the limitation of these methods, they can’t be applied for solving fully fuzzy linear or non- linear programming problems with unrestricted fuzzy coefficients and fuzzy variables. In this paper an efficient method to find the fuzzy optimal solution for fully fuzzy quadratic programming (FFQP) problem with unrestricted variables and parameters has been proposed. All the coefficients and decision variables of both objective functions and the constraints are triangular fuzzy numbers. The proposed method is based on converted FFQP problem into crisp quadratic programming (CQP) problem. Finally an illustrative numerical example has been given to clarify the proposed solution method.</p>

Many physical process systems have performance limitation regardless the input. This limitation usually occurs in the form of input saturation of actuator as the constraint of the system. To overcome this problem, Model Predictive Control (MPC) may be used due to its capability to compute optimal control signal in the presence of input saturation. The optimal control signal is obtained by solving a quadratic programming (QP) problem with variables' constraints at each time instants. QP problem with lower and upper constraint is known to be equivalent to an algebraic loop involving diagonal upper and lower saturation. Hence, QP can be iteratively computed until its solution converges. A slow computation of QP, however, limits its applications for fast systems with simple digital processor. Therefore, in order to be able to be implemented in real-time embedded applications, a fast algorithm of QP solver is in need. In this paper, a comparison study of two QP implementation, i.e. Projected Gauss Seidel (PGS) and so called Algorithm-2, using Arduino Mega 2560 is presented. In this case, MPC with QP solver using both algorithms is implemented to control the speed of lightly loaded BLDC motor. Before implementation, system modeling, controller designing, and simulation are undertaken. The performance of both iterative algorithms are then compared in terms of average computation time in the simulation and implementation. It can be observed from the results that Algorithm-2 gives a bit faster computation than PGS under same conditions.

A variety of realistic industrial problems can be constructed into quadratic programming (QP) problems, especially their time-varying versions. A zeroing neural network (ZNN) as a good approach for dynamic problems can solve QP problems subject to equality constraints in the past. In this article, we propose a unified predefined-time convergent and robust ZNN (PTCR-ZNN) model for solving time-varying QP problems subject to equality or inequality constraints. Compared with the normal ZNN model, the PTCR-ZNN model mainly has advantages in the following three aspects: 1) solving QP problems with or without inequality constraints in a unified model; 2) converging to the optimal solution of QP problems within a predefined time that can be determined in advance; and 3) resisting many external noises with tiny and predictable residual error. These improvements have been rigorously proved in theory. By conducting both qualitative and quantitative simulations with comparisons, the superior properties of the PTCR-ZNN model are further validated. Finally, the application of the PTCR-ZNN model to image fusion task illustrates the efficiency together with its applicability.

In the model predictive control (MPC) framework a controller is computed using a finite horizon optimal control cost. When linear constraints are imposed on the system with a quadratic cost function, then the MPC problem can be reformulated as a constrained quadratic programming (QP) problem. In this paper active set methods are used in order to solve the QP problem associated with MPC, and it is shown that it is possible to use information from the previous sample interval in order to provide improved initial conditions for the QP problem which is applied to the subsequent sample interval. Simulations show that when real-time considerations force the used of suboptimal intermediate control values, then improved initial conditions can allow for control values which are closer to the true optimal solution than obtained when using standard initial conditions. These ideas can allow for the implementation of MPC schemes on a greater number of real industrial applications, where standard MPC control cannot be applied due to the excessive CPU time requirements

Quadratic dynamic matrix control consists of the on-line solution of a quadratic programming (QP) problem. Blocking and condensing (B & C) are techniques that consider a reduced set of future manipulated variable and predicted output values in the QP problem by assuming them to be constant over intervals. B & C improve the controller robustness and reduce the QP computation time. There is a need for a systematic design methodology for the selection of B & C intervals. Wavelet transforms provide a suitable framework for developing a B & C design methodology. B & C in the time domain and in the wavelet domain are compared. It is shown that time domain B & C is edge driven, while that using wavelets is interval driven. A new design procedure for interval-driven B & C using information theory and sensitivity analysis is developed. The design procedure is applied to the Shell process control problem. Simulation results are discussed and some design guidelines are provided. B & C are also examined in the context of using move suppression factors

This paper is aimed at the remedy of a discontinuity problem arising in the infinity-norm acceleration minimization (INAM) of robot manipulators. Three important matters are involved. 1) A new acceleration-level bi-criteria scheme is proposed for preventing the INAM solution discontinuities and joint torques instability problem. It combines the minimum infinity-norm and minimum two-norm solutions by using a new problem formulation. 2) Such a bi-criteria scheme is then reformulated as a quadratic programming (QP) problem with its coefficient matrix being positive semi-definite. 3) The LVI-based primal-dual neural network is finally chosen to solve online such a QP problem as well as the bi-criteria weighting scheme. This is in view of the fact that the LVI-based primal-dual neural network has a simple piecewise-linear dynamics and higher computational efficiency. Simulation results based on PMUA560 robot manipulator also illustrate the advantages of using such a neural weighting scheme proposed in this paper.

This study focuses on real-time model predictive control (MPC) implementation for a point-absorber-type wave energy converter (WEC) to achieve maximum wave power extraction. To address the MPC dependence on current and future wave excitation force knowledge, the control framework includes a square-root unscented Kalman filter (SR-UKF) to estimate the wave excitation force, whereas an autoregressive (AR) model forecasts the excitation force over the prediction horizon. The MPC utilizes a linear model of the point absorber, and its cost function is formulated as a quadratic programming (QP) problem with the objective of maximizing the wave energy extraction subject to physical limitations and constraints. A set of Laguerre functions was used to reduce the MPC computational complexity by decreasing the number of decision variables in the QP problem. The MATLAB/Simulink simulation results showed that the embodiment of point absorber nonlinearities by the SR-UKF enhanced the estimation accuracy compared with that of linear Kalman filters in terms of maximum error, root mean square error, and goodness of fit. Moreover, an AR model of order 100 provided satisfactory prediction performance for a 4 s prediction. The MPC strategy was simulated to evaluate its efficiency with respect to the absorbed wave power and generated electrical power under different sea-state conditions and disturbance uncertainties. Depending on sea state, it was found that the proposed MPC strategy generated 27.8% to 80% more electrical power than resistive loading control. Finally, a hardware-in-the-loop experimental setup was used to validate the computational viability of the proposed control framework in real-time WEC applications.

This paper presents an innovative path tracking method of model predictive control based on multi-point preview (MPP-MPC) that the original equation of state is augmented by considering a dynamics model of the multiple preview points. The path tracking issue is formulated as an optimal control problem with dynamic disturbance, i.e., the future road curvature. Hence, an ameliorative Kalman filter is adopted to fuse the information about camera and vehicle dynamics to accurately estimate the road curvature. To ensure lateral control performance, dynamic constraints are defined by the estimated road curvature to guarantee comfort and safety. In the control domain of model predictive control (MPC), the model can be approximated as a linear time-invariant model (LTIM) to reduce the computational complexity. The MPC problem can be transformed into a standard quadratic programming (QP) problem with dynamic constraints of guaranteed comfort and safety. Finally, the complex path tracking problem can be solved by the QP problem. Through lane keeping experiments, it is shown that the tracking accuracy and steering smoothness can be significantly improved by the proposed method.

Trajectory following for a flexible probe with state/input constraints: An approach based on model predictive control

In consideration of the limitation of the communication and the possibility that redundant robots might deliver information at different power levels, cases under weight-unbalanced directed graphs from the network topology perspective are in larger accordance with those in multiple redundant robot systems. By moving forward along this direction, a distributed controller is proposed in this article to handle circumstances of collaborative control of multiple redundant robots under weight-unbalanced directed graphs. This kind of control problem is modeled into generalized quadratic programming (QP) problems with equality and inequality constraints. Then, the above QP problems are solved by a proposed neural-dynamics-based method, whose stability and convergence are theoretically proved subsequently. Besides, several experimental examples are conducted, and related comparisons are provided to demonstrate the feasibility of the proposed controller.

Redundant robot manipulators are usually applied to various complex scenarios where harmful noise especially periodic calculating noise always exists. In order to avoid the task failure affected by the periodic noise, a circadian rhythms neural network (CRNN) is applied to solve motion planning problems of redundant robot manipulators suffering from the periodic noise. First, in this article, we formulate a tracking problem of redundant manipulators as a convex quadratic programming (QP) problem. Second, the QP problem is converted into a matrix equation. Third, based on the neural dynamic design method, a CRNN model is exploited and developed to solve the matrix equation of the tracking problem. Comparative simulations between the proposed CRNN and the traditional zeroing neural network show that the CRNN has better robustness and performance on solving the end-effector tracking task. Two physical experiments are conducted to further certify the effectiveness, robustness, and practicability of the proposed CRNN.

This paper concerns the use of iterative solvers in interior-point methods for linear and quadratic programming problems. We state an adaptive termination rule for the inner iterative scheme and we prove the global convergence of the obtained algorithm, exploiting the theory developed for inexact Newton methods. This approach is promising for problems with special structure on parallel computers. We present an application on Cray T3E/256 and SGI Origin 2000/64 arising in stochastic linear programming and robust optimization, where the constraint matrix is block-angular and extremely large.