Smoothing SQP algorithm for semismooth equations with box constraints

In this paper, in order to solve semismooth equations with box constraints, we present a class of smoothing SQP algorithms using the regularized-smooth techniques. The main difference of our algorithm from some related literature is that the correspondent objective function arising from the equation system is not required to be continuously differentiable. Under the appropriate conditions, we prove the global convergence theorem, in other words, any accumulation point of the iteration point sequence generated by the proposed algorithm is a KKT point of the corresponding optimization problem with box constraints. Particularly, if an accumulation point of the iteration sequence is a vertex of box constraints and additionally, its corresponding KKT multipliers satisfy strictly complementary conditions, the gradient projection of the iteration sequence finitely terminates at this vertex. Furthermore, under local error bound conditions which are weaker than BD-regular conditions, we show that the proposed algorithm converges superlinearly. Finally, the promising numerical results demonstrate that the proposed smoothing SQP algorithm is an effective method.     

A Strongly and Superlinearly Convergent SQP Algorithm for Optimization Problems with Linear Complementarity Constraints

This paper discusses a kind of optimization problem with linear complementarity constraints, and presents a sequential quadratic programming (SQP) algorithm for solving a stationary point of the problem. The algorithm is a modification of the SQP algorithm proposed by Fukushima et al. [Computational Optimization and Applications, 10 (1998), 5-34], and is based on a reformulation of complementarity condition as a system of linear equations. At each iteration, one quadratic programming and one system of equations needs to be solved, and a curve search is used to yield the step size. Under some appropriate assumptions, including the lower-level strict complementarity, but without the upper-level strict complementarity for the inequality constraints, the algorithm is proved to possess strong convergence and superlinear convergence. Some preliminary numerical results are reported.     

Sequential Systems of Linear Equations Algorithm for Nonlinear Optimization Problems with General Constraints

In Ref. 1, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints was proposed. At each iteration, this new algorithm only needs to solve four systems of linear equations having the same coefficient matrix, which is much less than the amount of computation required for existing SQP algorithms. Moreover, unlike the quadratic programming subproblems of the SQP algorithms (which may not have a solution), the subproblems of the SSLE algorithm are always solvable. In Ref. 2, it is shown that the new algorithm can also be used to deal with nonlinear optimization problems having both equality and inequality constraints, by solving an auxiliary problem. But the algorithm of Ref. 2 has to perform a pivoting operation to adjust the penalty parameter per iteration. In this paper, we improve the work of Ref. 2 and present a new algorithm of sequential systems of linear equations for general nonlinear optimization problems. This new algorithm preserves the advantages of the SSLE algorithms, while at the same time overcoming the aforementioned shortcomings. Some numerical results are also reported.     

Simultaneous Pseudo-Timestepping for PDE-Model Based Optimization Problems

In this paper we present a new method for the solution of optimization problems with PDE constraints. It is based on simultaneous pseudo-time stepping for evolution equations. The new method can be viewed as a continuous reduced SQP method in the sense that it uses a preconditioner derived from that method. The reduced Hessian in the preconditioner is approximated by a pseudo-differential operator, whose symbol can be investigated analytically. We apply our method to a boundary control model problem. The new optimization method needs 3.2-times the overall computational effort of the solution of simulation problem alone.     

A Comparison of Global Optimization Methods for the Design of a High-speed Civil Transport

The conceptual design of aircraft often entails a large number of nonlinear constraints that result in a nonconvex feasible design space and multiple local optima. The design of the high-speed civil transport (HSCT) is used as an example of a highly complex conceptual design with 26 design variables and 68 constraints. This paper compares three global optimization techniques on the HSCT problem and two test problems containing thousands of local optima and noise: multistart local optimizations using either sequential quadratic programming (SQP) as implemented in the design optimization tools (DOT) program or Snyman's dynamic search method, and a modified form of Jones' DIRECT global optimization algorithm. SQP is a local optimizer, while Snyman's algorithm is capable of moving through shallow local minima. The modified DIRECT algorithm is a global search method based on Lipschitzian optimization that locates small promising regions of design space and then uses a local optimizer to converge to the optimum. DOT and the dynamic search algorithms proved to be superior for finding a single optimum masked by noise of trigonometric form. The modified DIRECT algorithm was found to be better for locating the global optimum of functions with many widely separated true local optima.     

An SQP approach with line search for a system of nonlinear equations

In this paper, a line search sequential quadratic programming (SQP) approach to a system of nonlinear equations (SNE) is taken. In this method, the system of nonlinear equations is transformed into a constrained nonlinear programming problem at each step, which is then solved using SQP algorithms with a line search strategy. Furthermore, at each step, some equations, which are satisfied at the current point, are treated as constraints and the others act as objective functions. In essence, constrained optimization strategies are utilized to cope with the system of nonlinear equations.     

A method combining norm-relaxed QP subproblems with systems of linear equations for constrained optimization

Based on the ideas of norm-relaxed sequential quadratic programming (SQP) method and the strongly sub-feasible direction method, we propose a new SQP algorithm for the solution of nonlinear inequality constrained optimization. Unlike the previous work, at each iteration, the norm-relaxed quadratic programming subproblem (NRQPS) in our algorithm only consists of the constraints corresponding to an estimate of the active set, and the high-order correction direction (used to avoid the Maratos effect) is obtained by solving a system of linear equations (SLE) which also only consists of such a subset of constraints and gradients. Moreover, the line search technique can effectively combine the initialization process with the optimization process, and therefore (if the starting point is not feasible) the iteration points always get into the feasible set after a finite number of iterations. The global convergence is proved under the Mangasarian–Fromovitz constraint qualification (MFCQ), and the superlinear convergence is obtained without assuming the strict complementarity. Finally, the numerical experiments show that the proposed algorithm is effective and promising for the test problems.     

SEQUENTIAL SYSTEMS OF LINEAR EQUATIONS ALGORITHM FOR NONLINEAR OPTIMIZATION PROBLEMS-INEQUALITY CONSTRAINED PROBLEMS

AbstractIn this paper, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints is proposed. Since the new algorithm only needs to solve several systems of linear equations having a same coefficient matrix per iteration, the computation amount of the algorithm is much less than that of the existing SQP algorithms per iteration. Moreover, for the SQP type algorithms, there exist so-called inconsistent problems, i.e., quadratic programming subproblems of the SQP algorithms may not have a solution at some iterations, but this phenomenon will not occur with the SSLE algorithms because the related systems of linear equations always have solutions. Some numerical results are reported.     

Exploiting additional structure in equality constrained optimization by structured SQP secant algorithms

In 1988, Tapia (Ref. 1) developed and analyzed SQP secant methods in equality constrained optimization taking explicitly the additive structure of the problem setting into account. In this paper, we extend Tapia's augmented scale Lagrangian secant method to the case where additional structure coming from the objective function is available. Using the example of nonlinear least squares with equality constraints, we demonstrate these ideas and develop a convergence theory proving local and q-superlinear convergence for this kind of structured SQP-algorithms.This research was supported by the Studienstiftung des Deutschen Volkes.     

On-line optimization design of sliding mode guidance law with multiple constraints

The design of terminal guidance law with impact angle constraint is required for air-to-ground guided weapons to increase their warhead effect. The variable structure guidance law that consists of diving plane guidance and turning plane guidance equations with impact angle constraint is derived, and the saturation function is introduced into the design of reaching law control to weaken the chattering of the guidance system. The influence of four guidance parameters (i.e., reaching law factor, switching item gain, angle error item factor, and boundary layer thickness) on guidance performance is studied and three typical constraints (i.e., heating rate, normal load factor, and dynamic pressure) are analyzed. An optimization model is established for this problem and the feasibility of on-line optimization on guidance law parameters by the Sequential Quadratic Programming (SQP) algorithm is discussed as well. Simulation results show that the on-line optimization of the derived guidance law not only satisfies specified constraints, but also minimizes the fuel cost during the flying course. Moreover, the optimization process can be completed in a few seconds so that it is suitable for on-board applications.     

A superlinearly convergent method of quasi-strongly sub-feasible directions with active set identifying for constrained optimization

Combining the norm-relaxed sequential quadratic programming (SQP) method and the idea of method of quasi-strongly sub-feasible directions (MQSSFD) with active set identification technique, a new SQP algorithm for solving nonlinear inequality constrained optimization is proposed. Unlike the previous work, at each iteration of the proposed algorithm, the norm-relaxed quadratic programming (QP) subproblem only consists of the constraints corresponding to an active identification set. Moreover, the high-order correction direction (used to avoid the Maratos effect) is yielded by solving a system of linear equations (SLE) which also includes only the constraints and their gradients corresponding to the active identification set, therefore, the scale and the computation cost of the high-order correction directions are further decreased. The arc search in our algorithm can effectively combine the initialization processes with the optimization processes, and the iteration points can get into the feasible set after a finite number of iterations. Furthermore, the arc search conditions are weaker than the previous work, and the computation cost is further reduced. The global convergence is proved under the Mangasarian–Fromovitz constraint qualification (MFCQ). If the strong second-order sufficient conditions are satisfied, then the active constraints are exactly identified by the identification set. Without the strict complementarity, superlinear convergence can be obtained. Finally, some elementary numerical results are reported.     

A Constrained Optimization Approach for LCP

In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system H(x,y) = 0 by using the famous NCP function-Fischer-Burmeister function. Note that some equations in H(x, y) = 0 are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system H(x, y) = 0 to an optimization problem with linear equality constraints. After that we study the conditions under which the K-T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point, However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: M is P0 matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.     

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.     

A globally convergent trust region algorithm for optimization with general constraints and simple bounds

1.IntroductionInthispaper,weconsiderthefollowingnonlinearprogr~ngproblemwherec(x)=(c,(x),c2(2),',We(.))',i(x)andci(x)(i=1,2,',m)arerealfunctions*ThisworkissupPOrtedbytheNationalNaturalScienceFOundationofChinaandtheManagement,DecisionandinformationSystemLab,theChineseAcademyofSciences.definedinD={xEReIISx5u}.Weassumethath相似文献   

An optimization method to estimate models with store-level data: A case study

The quality of the estimation of a latent segment model when only store-level aggregate data is available seems to be dependent on the computational methods selected and in particular on the optimization methodology used to obtain it. Following the stream of work that emphasizes the estimation of a segmentation structure with aggregate data, this work proposes an optimization method, among the deterministic optimization methods, that can provide estimates for segment characteristics as well as size, brand/product preferences and sensitivity to price and price promotion variation estimates that can be accommodated in dynamic models. It is shown that, among the gradient based optimization methods that were tested, the Sequential Quadratic Programming method (SQP) is the only that, for all scenarios tested for this type of problem, guarantees of reliability, precision and efficiency being robust, i.e., always able to deliver a solution. Therefore, the latent segment models can be estimated using the SQP method when only aggregate market data is available.     

An improved sequential quadratic programming algorithm for solving general nonlinear programming problems

In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of the sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic programming (QP) subproblem and a SLE, respectively. Moreover, in order to overcome the Maratos effect, the higher-order correction direction is obtained by solving another SLE. The two SLEs have the same coefficient matrices, and we only need to solve the one of them after a finite number of iterations. By a new line search technique, the proposed algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some comparative numerical results are reported to show that the proposed algorithm is effective and promising.     

Numerical solution of a nonlinear parabolic control problem by a reduced SQP method

We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.     

Global convergence on an active set SQP for inequality constrained optimization

Sequential quadratic programming (SQP) has been one of the most important methods for solving nonlinearly constrained optimization problems. In this paper, we present and study an active set SQP algorithm for inequality constrained optimization. The active set technique is introduced which results in the size reduction of quadratic programming (QP) subproblems. The algorithm is proved to be globally convergent. Thus, the results show that the global convergence of SQP is still guaranteed by deleting some “redundant” constraints.     

Parameter estimation of ordinary differential equations

This paper addresses the development of a new algorithm forparameter estimation of ordinary differential equations. Here,we show that (1) the simultaneous approach combined with orthogonalcyclic reduction can be used to reduce the estimation problemto an optimization problem subject to a fixed number of equalityconstraints without the need for structural information to devisea stable embedding in the case of non-trivial dichotomy and(2) the Newton approximation of the Hessian information of theLagrangian function of the estimation problem should be usedin cases where hypothesized models are incorrect or only a limitedamount of sample data is available. A new algorithm is proposedwhich includes the use of the sequential quadratic programming(SQP) Gauss–Newton approximation but also encompassesthe SQP Newton approximation along with tests of when to usethis approximation. This composite approach relaxes the restrictionson the SQP Gauss–Newton approximation that the hypothesizedmodel should be correct and the sample data set large enough.This new algorithm has been tested on two standard problems.