SQP Method Research Articles

Optimizing the Satellite Control Gains with Nonlinear Motion Equations using SQP Method

functions of control system are attitude maneuver and attitude stability. The capability to attitude-maneuver a satellite is based on using control torques. In this paper, the torques generated by Euler angle errors and quaternion error vector for small attitude commands are compared and then the same is done for large attitude commands. It's founded that Euler angle errors characterized by a fixed control do not produce the desirable response. The SQP optimization method also used to optimize calculated control gains. This SQP optimization method with the following provisions such as maximum settling time, maximum rise time, maximum overshoot and maximum steady state error causes that the Euler angles error method provides an appropriate responses.

Line search SQP method with a flexible step acceptance procedure

This paper describes a new algorithm for nonlinear programming with inequality constraints. The proposed approach solves a sequence of quadratic programming subproblems via the line search technique and uses a new globalization strategy. An increased flexibility in the step acceptance procedure is designed to promote long productive steps for fast convergence. Global convergence is proved under some reasonable assumptions and preliminary numerical results are presented.

Finite Element Discretization of State-Constrained Elliptic Optimal Control Problems with Semilinear State Equation

We study a class of semilinear elliptic optimal control problems with pointwise state constraints. The purpose of this paper is twofold. First, we present convergence results for the finite element discretization of this problem class similarly to known results with finite-dimensional control space, thus extending results that are---for control functions---only available for linear-quadratic convex problems. We rely on a quadratic growth condition for the continuous problem that follows from second order sufficient conditions. Second, we show that the second order sufficient conditions for the continuous problem transfer to its discretized version. This is of interest, for example, when considering questions of local uniqueness of solutions or the convergence of solution algorithms such as the SQP method.

A trust-region SQP method without a penalty or a filter for nonlinear programming

In this paper we present a new trust-region SQP algorithm for nonlinear programming. This method avoids using a penalty function, nor a filter, and instead establishes a new step acceptance mechanism. Under some reasonable assumptions, the method can be proved to be globally convergent to a KT point. Preliminary numerical experiments are presented that show the potential efficiency of the new approach.

HLRF–BFGS optimization algorithm for structural reliability

In this work we briefly discuss some concepts of structural reliability as well as the optimization algorithm that is commonly used in this context, called HLRF. We show that the HLRF algorithm is a particular case of the SQP method, in which the Hessian of the Lagrangian is approximated by an identity matrix. Motivated by this fact, we propose the HLRF–BFGS algorithm that considers the BFGS update formula to approximate the Hessian. The algorithm proposed herein is as simple as the HLRF algorithm, since it requires just one function and gradient evaluation at each iteration and the new iterate is given by a recursive formula. Comparative numerical experiments on a set of problems selected from the literature are presented to illustrate the performance of the algorithm and the results indicate that the HLRF–BFGS algorithm has the advantage of being more robust and efficient with respect to the function and gradient evaluation, than HLRF.

The Optimization Design on Transmission Mechanism of Tightening Machine

This article, to optimize the transmission mechanism of tightening machine, select the involute planetary gear with two teeth differences a research object, determine the design variables about modification coefficient, modulus and tooth thickness, based on the target of having the minimum volume, a mathematical model of optimization was established, the optimal solution of the objective function is obtained by using the SQP method on fmincon function of optimization toolbox of Matlab. The results showed, the volume of pinion decreased 15% after optimizated, the volume of the corresponding gear will also greatly reduced, achieved the optimization goal.

Shape Optimization for Arch Dam with Sequential Quadratic Programming Method

The sequential quadratic programming method is introduced in arch dam shape optimization design. In the optimization computation, the arch dam volume is taken as objective function, and the calculation is realized by automatically considering loading process, equivalent stress computation and simulation of shape optimization for arch dam based on FEM. A parabola arch dam is used as a numerical example. The result shows that the optimized figure is much safer and more practical, which indicates that the SQP method is an effective way for optimization design of arch dam shape.

A new SQP algorithm and numerical experiments for nonlinear inequality constrained optimization problem

In this paper, a new algorithm based on SQP method is presented to solve the nonlinear inequality constrained optimization problem. As compared with the other existing SQP methods, per single iteration, the basic feasible descent direction is computed by solving at most two equality constrained quadratic programming. Furthermore, there is no need for any auxiliary problem to obtain the coefficients and update the parameters. Under some suitable conditions, the global and superlinear convergence are shown. Keywords: Global convergence, Inequality constrained optimization, Nonlinear programming problem, SQP method, Superlinear convergence rate.

A line search SQP method without a penalty or a filter

This paper describes a new algorithm for nonlinear programming problems with inequality constraints. The proposed approach first solves a sequence of quadratic programming subproblems via the line search technique and to induce the global convergence, it establishes a new step acceptance mechanism that is neither a penalty function nor a filter. A nonmonotone line search technique from the unconstraint optimization is applied to accelerate the algorithm. Under some reasonable assumptions, the method can be proved to be globally convergent to a KT point. Preliminary numerical results are presented that show the potential efficiency of the new approach.

On the Local Convergence of a Penalty-Function-Free SQP Method

In this article, we propose a nonmonotone linesearch sequential quadratic programming method for general constrained optimization problems without a penalty function or a filter. The algorithm proposed here is a development of the algorithm in Xue et al. [17]. Compared with the former, the novelty of the method we propose is that the new algorithm will achieve the local convergence under weaker assumptions. In order to avoid the Maratos effect, we use the second-order correction in this method, which need not be computed at each iteration. In other words, after a certain number of iterations, there is no need to compute the second-order correction step any more. The global convergence and the locally superlinear convergence of our method are proved under some suitable conditions.

Model hierarchy-based optimal control of radiative heat transfer

We present a model hierarchy multilevel optimisation approach to solve an optimal boundary control problem in glass manufacturing. The process is modelled by radiative heat transfer and formulated as an optimal control problem restricted by partial differential algebraic equations (PDAEs) and additional control constraints. We consider a sequence of model approximations given by space-time dependent non-linear PDAEs of ascending accuracy. The different models allow for a model hierarchy-based optimisation approach, where the models are shifted automatically as the optimisation proceeds. We present a realisation of a multilevel generalised SQP method within the fully space-time adaptive optimisation environment Kardos using linearly implicit methods of Rosenbrock type and multilevel finite elements. We apply the optimal control algorithm to a glass cooling problem and present numerical experiments for the model hierarchy-based approach in two spatial dimensions and for a fully space-time adaptive optimisation in three spatial dimensions.

Superlinearly Convergent Norm-Relaxed SQP Method Based on Active Set Identification and New Line Search for Constrained Minimax Problems

In this paper, the minimax problems with inequality constraints are discussed, and an alternative fast convergent method for the discussed problems is proposed. Compared with the previous work, the proposed method has the following main characteristics. First, the active set identification which can reduce the scale and the computational cost is adopted to construct the direction finding subproblems. Second, the master direction and high-order correction direction are computed by solving a new type of norm-relaxed quadratic programming subproblem and a system of linear equations, respectively. Third, the step size is yielded by a new line search which combines the method of strongly sub-feasible direction with the penalty method. Fourth, under mild assumptions without any strict complementarity, both the global convergence and rate of superlinear convergence can be obtained. Finally, some numerical results are reported.

A superlinearly convergent norm-relaxed method of quasi-strongly sub-feasible direction for inequality constrained minimax problems

In this paper, nonlinear minimax problems with inequality constraints are discussed. Combined the norm-relaxed SQP method with the idea of strongly sub-feasible directions method, a new method of quasi-strongly sub-feasible directions (MQSSFD) with arbitrary initial point for the discussed problems is presented. At each iteration of the proposed algorithm, an improved search direction is obtained by solving a quadratic program (QP) which always has a solution, and a high-order correction direction is yielded via a system of linear equations (SLE) to avoid the Maratos effect. After finite iterations, the iteration point always get into the feasible set by introducing a new non-monotone curve search. Under some mild conditions including the weak Mangasarian–Fromovitz constraint qualification (MFCQ), the proposed algorithm possesses global convergence, and the superlinear convergence is obtained without the strict complementarity. Finally, some elementary numerical experiments are implemented and reported.

Topological optimization design of structures under random excitations using SQP method

More and more stringent structural performance requirements are imposed in advanced engineering application, only a limited number of works have been devoted to the topology optimization of the structures with random vibration response requirements. In this study, the topology optimization problem with the objective function being the structural weight and the constraint functions being structural random vibration responses is investigated. An approximate topological optimization model for suppressing ‘localized modes’ of vibrating Cauchy solids is established in this paper. Based on moving asymptotes approximate functions, approximated–approximations expressions of the dynamic responses are constructed. In order to control the change quantity of topologic design variables, new dynamic response constraint limits are formed and introduced into the optimization model at the beginning of each sub-loop iteration. Then, an optimization sequential quadratic programming is introduced, and a set of iteration formulas for Lagrange multipliers is developed. Two examples are provided to demonstrate that the proposed method is feasible and effective for obtaining optimal topology.

Research on Helicopter Stabilator Control System

to deal with the problems caused by the fixed stabilator, the movable stabilator and its installation angle control system have been researched in this paper. Base on the analysis of the influence of movable stabilator on helicopter longitudinal flying qualities, the control structure of the installation angle control system has been designed and its parameters are figured out with SQP method. The control system raised in this paper could be useful to instruct the movable stabilator installation angle control system design.

Global convergence of a general filter algorithm based on an efficiency condition of the step

In this work we discuss global convergence of a general filter algorithm that depends neither on the definition of the forbidden region, which can be given by the original or slanting filter rule, nor on the way in which the step is computed. This algorithm basically consists of calculating a point not forbidden by the filter from the current point. Assuming that this step must be efficient, in the sense that near a feasible non-stationary point the decrease in the objective function is relatively large, we prove the global convergence of the algorithm. We also discuss that such a condition is satisfied if the step is computed by the SQP or Inexact Restoration methods. For SQP we present a general proof of this result that is valid for both the original and the slanting filter criterion. In order to compare the performance of the general filter algorithm according to the method used to calculate the step and the filter rule regarded, we present numerical experiments performed with problems from CUTEr collection.

Structural and sensitivity reanalyses based on singular value decomposition

The structural and sensitivity reanalysis of a structure subjected to modifications has a significant practical value. If the structural modifications result in low rank changes in the associated system matrices, the reanalysis of a structure could be completed with a computational load that is less than that of the complete analysis of the structure. To this end, the Sherman---Morrison---Woodbury formulas along with the singular value decomposition are employed to compute the extremum sensitivity values and optimum perturbations of design variables such that desired changes in the responses are achieved, which are difficult to be obtained by using the response derivatives. Numerical examples are presented to show the advantages of the proposed approach. Accuracy of the solutions is checked analytically and comparisons between the CPU times of the SVD-based reanalysis and SQP method are made, which show the advantage of the proposed SVD-based approach over conventional methods.

Improved Feasible SQP Algorithm for Nonlinear Programs with Equality Constrained Sub-Problems

This paper is concerned with an improved feasible sequential quadratic programming (FSQP) method for nonlinear programs. As compared with the existing SQP methods which required solving the QP sub-problem with inequality constraints in single iteration, in order to obtain the feasible direction, the method of this paper is only necessary to solve an equality constrained quadratic programming sub-problems. Combined the generalized projection technique, a height-order correction direction is yielded by explicit formulas, which can avoids Maratos effect. Furthermore, under some mild assumptions, the algorithm is globally convergent and its rate of convergence is one-step superlinearly. Numerical results reported show that the algorithm in this paper is effective.

Parallel Variable Distribution Algorithm for Constrained Optimization with Nonmonotone Technique

A modified parallel variable distribution (PVD) algorithm for solving large-scale constrained optimization problems is developed, which modifies quadratic subproblemQPlat each iteration instead of theQPl0of the SQP-type PVD algorithm proposed by C. A. Sagastizábal and M. V. Solodov in 2002. The algorithm can circumvent the difficulties associated with the possible inconsistency ofQPl0subproblem of the original SQP method. Moreover, we introduce a nonmonotone technique instead of the penalty function to carry out the line search procedure with more flexibly. Under appropriate conditions, the global convergence of the method is established. In the final part, parallel numerical experiments are implemented on CUDA based on GPU (Graphics Processing unit).

An Improved Nonmonotone Filter Trust Region Method for Equality Constrained Optimization

Motivated by the method of Su and Pu (2009), we present an improved nonmonotone filter trust region algorithm for solving nonlinear equality constrained optimization. In our algorithm a modified nonmonotone filter technique is proposed and the restoration phase is not needed. At every iteration, in common with the composite-step SQP methods, the step is viewed as the sum of two distinct components, a quasinormal step and a tangential step. A more relaxed accepted condition for trial step is given and a crucial criterion is weakened. Under some suitable conditions, the global convergence is established. In the end, numerical results show our method is effective.