Explicit Surrogate Dual Formulation and Algorithm for Quadratic Programming

ABSTRACTThe rapid growth of interconnected devices and data traffic necessitates robust mobile networks, especially in rural areas with unreliable grid power. This paper introduces an optimisation framework for the off‐grid green open radio access network architecture utilising renewable energy to meet this demand. The framework aims to optimise power and bandwidth allocation, ensuring high data rates and reliability from sustainable sources. In addition, the CO2 emission factor and the quality of service are also optimised in which several assumptions and network metrics have been incorporated such as power consumption (PC), load demand, number of virtual machines (VMs) and maintenance cost. The problem is formulated as the weighted nonlinear problem. First, the model has been formulated using simplified data rate assumption and has been solved mathematically using Lagrange multipliers. Subsequently, the mathematical problem has been solved using the gradient descent numerical solution and then compared with sequential quadratic programming (SQP) and interior point algorithms. The second assumption is formulated as the data rate considering the noise and interference factors; the comparative analysis of SQP, active set and interior point algorithms has been presented. The third case assumed the dynamic behaviour of the network of users and VMs, where the problem is also solved mathematically and numerically. Empirical evidence from rural deployments confirms the enhancement of mobile coverage and service provision through the integration of solar and wind‐powered base stations.

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

In this paper, a new algorithm for convex quadratic programming (QP) is presented. Firstly, the surrogate problem for QP is developed, and the Karush-Kuhn-Tucker conditions of the surrogate problem hold if the unconstrained minimum of the objective function does not satisfy any constraints. Then, Karmarkar's algorithm for linear programming (LP) is introduced to solve the surrogate dual problem. In addition, the case of general constraints is also discussed, and some examples of optimum truss sizing problems show that the proposed algorithm is robust and efficient.

: Research is presented on an algorithm for quadratic programming when the matrix of the quadratic part of the maxim and is semi-definite. When m is the number of linear inequality constraints the algorithm leads to a simplex tableau of order m x 2m. In one section an algorithm is given which is valid when the matrix of the quadratic part is strictly definitive. The second section deals with linear programming, which is considered as a quadratic programming problem with a zero matrix for the quadratic part of the maximand. It is an introduction to the third section, which deals with singular quadratic programming. The algorithm solves explicitly for the dual problem as well.

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.

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

Recently, with the wide adoption of electric vehicles, transportation network has shown different characteristics and been further developed. In this paper, we present a new time-flow impedance function, which may be more realistic than the existing time-flow impedance functions. Based on this new impedance function, we present an optimization model for a mixed-mode traffic network in which battery electric vehicles (BEVs) and gasoline vehicles (GVs) are chosen. We suggest two approaches to handle the model: One is to use the interior point (IP) algorithm and the other is to employ the sequential quadratic programming (SQP) algorithm. Three numerical examples are presented to illustrate the efficiency of these approaches. In particular, our numerical results show that more travelers prefer to choosing BEVs when the distance limit of BEVs is long enough and the unit operating cost of GVs is higher than that of BEVs, and the SQP algorithm is faster than the IP algorithm.

By using conjugate directions a method for solving convex quadratic programming problems is developed. The algorithm generates a sequence of feasible solutions and terminates after a finite number of iterations. Extensions of the algorithm for nonconvex and large structured quadratic programming problems are discussed.

This paper presents a reactive power optimization model that is based on successive quadratic programming (SQP) methods. Mathematical formulation and unified algorithm suppose different objective functions (OF) of reactive power optimization, depending on type and purposes of current reactive power control or planning problem. A bicriterion reactive power optimization model, that represents compromise between economical and security objective functions, is proposed. An efficient algorithm for approximation of initial problem by quadratic programming problem is described. The quadratic programming problem (QP) is solved on the basis of the Newton type quadratic programming method. A modified successive quadratic programming method was developed, that provides reliable convergence of the SQP method.

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

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.

We describe the design of version 1.0 of GALAHAD, a library of Fortran 90 packages for large-scale nonlinear optimization. The library particularly addresses quadratic programming problems, containing both interior point and active set algorithms, as well as tools for preprocessing problems prior to solution. It also contains an updated version of the venerable nonlinear programming package, LANCELOT.

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 paper is concerned with relative localization-based optimal area coverage placement using multiple unmanned aerial vehicles (UAVs). It is assumed that only one of the UAVs has its global position information before performing the area coverage task and that ranging measurements can be obtained among the UAVs by using ultra-wide band (UWB) sensors. In this case, multi-UAV relative localization and cooperative coverage control have to be run simultaneously, which is a quite challenging task. In this paper, we propose a single-landmark-based relative localization algorithm, combined with a distributed coverage control law. At the same time, the optimal multi-UAV placement problem was formulated as a quadratic programming problem by compromising between optimal relative localization and optimal coverage control and was solved by using Sequential Quadratic Programming (SQP) algorithms. Simulation results show that our proposed method can guarantee that a team of UAVs can efficiently localize themselves in a cooperative manner and, at the same time, complete the area coverage task.