SQP Algorithm Research Articles

A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer

It is well acknowledged that transport-theory-based reconstruction algorithm can provide the most accurate reconstruction results especially when small tissue volumes or high absorbing media are considered. However, these codes have a high computational burden and are often only slowly converging. Therefore, methods that accelerate the computation are highly desirable. To this end, we introduce in this work a partial-differential-equation (PDE) constrained approach to optical tomography that makes use of an all-at-once reduced Hessian sequential quadratic programming (rSQP) scheme. The proposed scheme treats the forward and inverse variables independently, which makes it possible to update the radiation intensities and the optical coefficients simultaneously by solving the forward and inverse problems, all at once. We evaluate the performance of the proposed scheme with numerical and experimental data, and find that the rSQP scheme can reduce the computation time by a factor of 10–25, as compared to the commonly employed limited memory BFGS method. At the same time accuracy and robustness even in the presence of noise are not compromised.

실험계획법을 이용한 인공위성 주반사경 플렉셔 마운트의 최적 설계

The primary mirror system in a satellite camera is an opto-mechanically coupled system for a reason that optical and mechanical behaviors are intricately interactive. In order to enhance the opto-mechanical performance of the primary mirror system, opto-mechanical behaviors should be thoroughly investigated by using various analysis procedures such as elastic, thermo-elastic, optical and eigenvalue analysis. In this paper, optimal design of the bipod flexure mounts for high opto-mechanical performance is performed. Optomechanical performances considered in this paper are RMS wavefront error under the gravity and thermal loading conditions and 1st natural frequency of the mirror system. The procedures of the flexure mounts design based on design of experiments and statistics is as follows. The experiments for opto-mechanical analysis are constructed based on the tables of orthogonal arrays and analysis of each experiment is carried out. In order to deal with the multiple opto-mechanical properties, MADM (Multiple-attribute decision making) is employed. From the analysis results, the critical design variables of the flexure mounts which have dominant influences on opto-mechanical performance are determined through analysis of variance and F-test. The regression model in terms of the critical design variables is constructed based on the response surface analysis. Then the critical design variables are optimized from the regression model by using SQP algorithm. Opto-mechanical performance of the optimal bipod flexure mounts is verified through analysis.

An improved SQP algorithm for solving minimax problems

In this work, an improved SQP method is proposed for solving minimax problems, and a new method with small computational cost is proposed to avoid the Maratos effect. In addition, its global and superlinear convergence are obtained under some suitable conditions.

Global convergence of a tri-dimensional filter SQP algorithm based on the line search method

In this paper, we propose a new filter line search SQP method in which the violations of equality and inequality constraints are considered separately. Thus the filter in our algorithm is composed by three components: objective function value, equality and inequality constraints violations. The filter with three components accepts reasonable steps flexibly, comparing that with two components. The new filter shares some features with the Chin and Fletcher's approach, namely the “slanting envelope” and the “inclusion property”. Under mild conditions, the filter line search SQP method is proven to be globally convergent. Numerical experiments also show the efficiency of our method.

A smoothing trust-region Newton-CG method for minimax problem

This paper presents a smooth approximate method with a new smoothing technique and a standard unconstrained minimization algorithm in the solution to the finite minimax problems. The new smooth approximations only replace the original problem in some neighborhoods of the kink points with a twice continuously differentiable function, its gradient and Hessian matrix are the combination of the first and the second order derivative of the original functions respectively. Compared to the other smooth functions such as the exponential penalty function, the remarkable advantage of the new smooth function is that the combination coefficients of its gradient and the Hessian matrix have sparse properties. Furthermore, the maximal possible difference value between the optimal values of the smooth approximate problem and the original one is determined by a fixed parameter selected previous. An algorithm to solve the equivalent unconstrained problem by using the trust-region Newton conjugate gradient method is proposed in the solution process. Finally, some numerical examples are reported to compare the proposed algorithm with SQP algorithm that implements in MATLAB toolbox and the algorithm in [E. Polak, J.O. Royset, R.S. Womersley, Algorithms with adaptive smoothing for finite minimax problems, Journal of Optimization Theory and Applications 119 (3) (2003) 459–484] based on the exponential penalty function, the numerical results prove that the proved algorithm is efficient.

An efficient feasible SQP algorithm for inequality constrained optimization

In this paper, an efficient feasible SQP method is proposed to solve nonlinear inequality constrained optimization problems. Here, a new modified method is presented to obtain the revised feasible descent direction. Per single iteration, it is only necessary to solve one QP subproblem and a system of linear equations with only a subset of the constraints estimated as active. In addition, its global and superlinear convergence are obtained under some suitable conditions.

Research of Large-Scale Reduced SQP Algorithm for Chemical Process System Optimization

In chemical process system optimization, it is common that problem has large number of equality constraints and bound constraints, and relatively few degrees of freedom. Reduced space algorithms are well suited for this category of problem. In this paper, a new version of RSQP algorithm coded and implemented in Matlab language was developed. In the RSQP algorithm, the rule of basis selection was revised, and basis selection was realized by Matlab subroutine. Also, an integrated line search of filter method was performed to obtained steplengh. The method combines the advantages of filter line search and that of traditional line search, and was validated by some benchmark examples. The RSQP was applied to some chemical process optimization problems; computational results demonstrate its effectiveness and efficiency.

Merit-Function Piecewise SQP Algorithm for Mathematical Programs with Equilibrium Constraints

We propose a merit-function piecewise SQP algorithm for mathematical programs with equilibrium constraints (MPEC) formulated as mathematical programs with complementarity constraints. Under mild conditions, the new algorithm is globally convergent to a piecewise stationary point. Moreover, if the partial MPEC linear independence constraint qualification (LICQ) is satisfied at the accumulation point, then the accumulation point is an S-stationary point.

Heterogeneous batch distillation processes: Real system optimisation

In this paper, optimisation of batch distillation processes is considered. It deals with real systems with rigorous simulation of the processes through the resolution full MESH differential algebraic equations. Specific software architecture is developed, based on the BatchColumn ® simulator and on both SQP and GA numerical algorithms, and is able to optimise sequential batch columns as long as the column transitions are set. The efficiency of the proposed optimisation tool is illustrated by two case studies. The first one concerns heterogeneous batch solvent recovery in a single distillation column and shows that significant economical gains are obtained along with improved process conditions. Case two concerns the optimisation of two sequential homogeneous batch distillation columns and demonstrates the capacity to optimise several sequential dynamic different processes. For such multiobjective complex problems, GA is preferred to SQP that is able to improve specific GA solutions.

A sequential equality constrained quadratic programming algorithm for inequality constrained optimization

In this paper, the feasible type SQP method is improved. A new SQP algorithm is presented to solve the nonlinear inequality constrained optimization. As compared with the existing SQP methods, per single iteration, in order to obtain the search direction, it is only necessary to solve equality constrained quadratic programming subproblems and systems of linear equations. Under some suitable conditions, the global and superlinear convergence can be induced.

A trust region SQP algorithm for mixed-integer nonlinear programming

We propose a modified sequential quadratic programming method for solving mixed-integer nonlinear programming problems. Under the assumption that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value, successive quadratic approximations are applied. The algorithm is stabilized by a trust region method with Yuan’s second order corrections. It is not assumed that the mixed-integer program is relaxable or, in other words, function values are evaluated only at integer points. The Hessian of the Lagrangian function is approximated by a quasi-Newton update formula subject to the continuous and integer variables. Numerical results are presented for a set of 80 mixed-integer test problems taken from the literature. The surprising result is that the number of function evaluations, the most important performance criterion in practice, is less than the number of function calls needed for solving the corresponding relaxed problem without integer variables.

A simple feasible SQP algorithm for inequality constrained optimization

In this paper, an SQP method generating feasible iterates is presented to solve the nonlinear inequality constrained optimization problems. A new modified method which ensures the global and superlinear convergence is proposed by solving systems of linear equation, instead of QP subproblems and linear squares problems. Here the search direction is a suitable combination of a descent direction, a feasible direction and a second-order revised direction. The theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions.

MODELING OF NLP PROBLEMS OF CHEMICAL PROCESSES DESCRIBED BY ODEs

Both real-time and off-line optimizations are commonly performed in order to enhance productivity. The optimization problem is often posed as a nonlinear programming (NLP) problem solved by a SQP algorithm. When processes need to be described by differential equations, difficulties will arise in using SQP algorithms, since Jacobians of constraints described by differential equations will have to be evaluated. In this paper, we show how to derive analytical expressions for both Jacobian and Hessian matrices for the constraints described by ordinary differential equations, without increasing the dimension of the resultant NLP problem to be solved.

A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with nonlinear complementarity constraints, its goal is to present a globally and superlinearly convergent algorithm for the discussed problems. We first reformulate the complementarity constraints as a standard nonlinear equality and inequality constraints by making use of a class of generalized smoothing complementarity functions, then present a new SQP algorithm for the discussed problems. At each iteration, with the help of a pivoting operation, a master search direction is yielded by solving a quadratic program, and a correction search direction for avoiding the Maratos effect is generated by an explicit formula. Under suitable assumptions, without the strict complementarity on the upper-level inequality constraints, the proposed algorithm converges globally to a B-stationary point of the problems, and its convergence rate is superlinear.

A superlinearly convergent SQP algorithm for mathematical programs with linear complementarity constraints

Abstract In this paper, MPEC problems with linear complementarity constraints are considered. By means of Fischer–Burmeister function, the linear complementarity condition is transformed into a nonsmooth equation. Then, during the iteration, a corresponding smooth system approximates the nonsmooth equation. The smooth optimization is solved by SQP algorithm for standard constrained optimization. Global convergence and superlinear convergence rate are established under some suitable assumptions. Moreover, we conclude that the current iterate point is an exact stationary point of the MPEC problem when the proposed algorithm stops in finite iteration.

A feasible descent SQP algorithm for general constrained optimization without strict complementarity

In this paper, a class of optimization problems with equality and inequality constraints is discussed. Firstly, the original problem is transformed to an associated simpler problem with only inequality constraints and a parameter. The later problem is shown to be equivalent to the original problem if the parameter is large enough (but finite), then a feasible descent SQP algorithm for the simplified problem is presented. At each iteration of the proposed algorithm, a master direction is obtained by solving a quadratic program (which always has a feasible solution). With two corrections on the master direction by two simple explicit formulas, the algorithm generates a feasible descent direction for the simplified problem and a height-order correction direction which can avoid the Maratos effect without the strict complementarity, then performs a curve search to obtain the next iteration point. Thanks to the new height-order correction technique, under mild conditions without the strict complementarity, the globally and superlinearly convergent properties are obtained. Finally, an efficient implementation of the numerical experiments is reported.

Parameter Estimation for a Polymerization Reactor Model with a Composite-Step Trust-Region NLP Algorithm

We consider an application of the trust-region SQP algorithm described by Arora and Biegler (Comput. Optim. Appl., in press) to the parameter estimation for a polymerization reactor. In particular, we are interested in the robustness of this algorithm in solving ill-conditioned optimization problems with differential algebraic (DAE) models. For this system, it is possible to measure a variety of states, such as production and concentrations. Some of these states contain more information and lead to more reliable parameter estimates than would be obtained if the states were not measured. To minimize sensor costs, it is thus beneficial to select the smallest combination of state measurements that contains the most information. Because the polymer process model is nonlinear, we decide to test the information content of measurements by analyzing properties of the reduced Hessian and constructing joint-confidence regions. This approach is related to the observability analysis (Albuquerque, J. S.; Biegler, L. T. AIChE J. 1996, 42, 2841. Narasimhan, S.; Jordache, C. Data Reconciliation and Gross Error Detection; Gulf Publishing Company: Houston, TX, 2000. Crowe, C. M. Chem. Eng. Sci. 1989, 44, 2909). However, for nonlinear problems, a reliable and efficient parameter estimation algorithm is essential for the measurement selection procedure. Finally, whereas inputs to the process are often held constant in parameter estimation problems, these inputs often contain noise. It is therefore necessary to reconcile inputs along with states in a simultaneous optimization framework. This problem becomes an errors in variables model (EVM) and can be considerably larger and more difficult to solve. In this study, we also consider the related EVM problem and compare results to the standard parameter estimation problem.

An improved SQP algorithm for inequality constrained optimization

In this paper, the feasible type SQP method is improved. A new algorithm is proposed to solve nonlinear inequality constrained problem, in which a new modified method is presented to decrease the computational complexity. It is required to solve only one QP subproblem with only a subset of the constraints estimated as active per single iteration. Moreover, a direction is generated to avoid the Maratos effect by solving a system of linear equations. The theoretical analysis shows that the algorithm has global and superlinear convergence under some suitable conditions. In the end, numerical experiments are given to show that the method in this paper is effective.

Quadratic programming solver for structural optimisation using SQP algorithm

The dual quadratic programming algorithm of Goldfarb and Idnani is implemented as a solver for a sequential quadratic programming algorithm. Initially the algorithm is briefly described. As the algorithm requires the inverse of the Cholesky factor of the Hessian matrix at each iteration a procedure is presented to directly obtain a matrix that multiplied by its transpose gives the BFGS update of the Hessian. A procedure is then presented to triangularise the updated factor using two series of Givens rotations. In order to increase efficiency a ‘warm start’ strategy is proposed whereby the choice of constraints to enter the active set is based on information of previous SQP iterations. Finally two examples are given to demonstrate the efficiency and robustness of the implementation.

횡방향 기동을 하는 위성발사체의 3차원 궤적최적화와 직접식 유도기법

Ascent trajectory optimization and explicit guidance problems for a satellite launch vehicle with yaw maneuver in a 3-dimension are considered. The trajectory optimization problem with boundary conditions is formulated as a nonlinear programming problem by parameterizing the inertial pitch and yaw attitude control variables, and is solved by using the SQP algorithm. The flight constraints such as gravity-turn and range safety conditions are imposed. An explicit inertial guidance algorithm in the exoatmospheric phase is also presented. The guidance algorithm provides steering command and time-to-go value directly using the current states of the vehicle and the desired orbit insertion conditions. The liquid propelled Delta 2910 launch vehicle is used as a numerical model.