An interior point type QP-free algorithm with superlinear convergence for inequality constrained optimization

A feasible interior point type algorithm is proposed for the inequality constrained optimization. Iterate points are prevented from leaving to interior of the feasible set. It is observed that the algorithm is merely necessary to solve three systems of linear equations with the same coefficient matrix. Under some suitable conditions, superlinear convergence rate is obtained. Some numerical results are also reported.     

On combining feasibility,descent and superlinear convergence in inequality constrained optimization

Extension of quasi-Newton techniques from unconstrained to constrained optimization via Sequential Quadratic Programming (SQP) presents several difficulties. Among these are the possible inconsistency, away from the solution, of first order approximations to the constraints, resulting in infeasibility of the quadratic programs; and the task of selecting a suitable merit function, to induce global convergence. In ths case of inequality constrained optimization, both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints, and that yield monotonically decreasing objective function values. (Feasibility of the successive iterates is in fact required in many contexts such as in real-time applications or when the objective function is not well defined outside the feasible set.) It has been recently shown that this can be achieved while preserving local two-step superlinear convergence. In this note, the essential ingredients for an SQP-based method exhibiting the desired properties are highlighted. Correspondingly, a class of such algorithms is described and analyzed. Tests performed with an efficient implementation are discussed.This research was supported in part by NSF's Engineering Research Centers Program No. NSFD-CDR-88-03012, and by NSF grants No. DMC-84-51515 and DMC-88-15996.     

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 filled function method applied to nonsmooth constrained global optimization

The filled function method is an effective approach to find a global minimizer. In this paper, based on a new definition of the filled function for nonsmooth constrained programming problems, a one-parameter filled function is constructed to improve the efficiency of numerical computation. Then a corresponding algorithm is presented. It is a global optimization method which modify the objective function as a filled function, and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. Illustrative examples are provided to demonstrate the efficiency and reliability of the proposed filled function method.     

A variant of SQP method for inequality constrained optimization and its global convergence

In this paper, a variant of SQP method for solving inequality constrained optimization is presented. This method uses a modified QP subproblem to generate a descent direction as each iteration and can overcome the possible difficulties that the QP subproblem of the standard SQP method is inconsistency. Furthermore, the method can start with an infeasible initial point. Under mild conditions, we prove that the algorithm either terminates as KKT point within finite steps or generates an infinite sequence whose accumulation point is a KKT point or satisfies certain first-order necessary condition. Finally, preliminary numerical results are reported.     

A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS：Ⅰ. ALGORITHM AND GLOBAL CONVERGENCE

A class of trust region methods tor solving linear inequality constrained problems is propo6ed in this paper. It is shown that the algorithm is of global convergence. The algorithm uses a version of the two-slded projection and the strategy of the unconstrained trust region methods. It keeps the good convergence properties of the unconstrained case and has the merits of the projection method. In some sense, our algorithm can be regarded as an extension and improvement of the projected type algorithm.     

A modified SLP algorithm and its global convergence

This paper concerns a filter technique and its application to the trust region method for nonlinear programming (NLP) problems. We used our filter trust region algorithm to solve NLP problems with equality and inequality constraints, instead of solving NLP problems with just inequality constraints, as was introduced by Fletcher et al. [R. Fletcher, S. Leyffer, Ph.L. Toint, On the global converge of an SLP-filter algorithm, Report NA/183, Department of Mathematics, Dundee University, Dundee, Scotland, 1999]. We incorporate this filter technique into the traditional trust region method such that the new algorithm possesses nonmonotonicity. Unlike the tradition trust region method, our algorithm performs a nonmonotone filter technique to find a new iteration point if a trial step is not accepted. Under mild conditions, we prove that the algorithm is globally convergent.     

A cut-peak function method for global optimization

A new method is proposed for solving box constrained global optimization problems. The basic idea of the method is described as follows: Constructing a so-called cut-peak function and a choice function for each present minimizer, the original problem of finding a global solution is converted into an auxiliary minimization problem of finding local minimizers of the choice function, whose objective function values are smaller than the previous ones. For a local minimum solution of auxiliary problems this procedure is repeated until no new minimizer with a smaller objective function value could be found for the last minimizer. Construction of auxiliary problems and choice of parameters are relatively simple, so the algorithm is relatively easy to implement, and the results of the numerical tests are satisfactory compared to other methods.     

A filled function method for global optimization

A novel filled function with one parameter is suggested in this paper for finding a global minimizer for a general class of nonlinear programming problems with a closed bounded box. A new algorithm is presented according to the theoretical analysis. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.     

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 type of efficient feasible SQP algorithms for inequality constrained optimization

In this paper, motivated by Zhu et al. methods [Z.B. Zhu, K.C. Zhang, J.B. Jian, An improved SQP algorithm for inequality constrained optimization, Math. Meth. Oper. Res. 58 (2003) 271-282; Zhibin Zhu, Jinbao Jian, An efficient feasible SQP algorithm for inequality constrained optimization, Nonlinear Anal. Real World Appl. 10(2) (2009) 1220-1228], we propose a type of efficient feasible SQP algorithms to solve nonlinear inequality constrained optimization problems. By solving only one QP subproblem with a subset of the constraints estimated as active, a class of revised feasible descent directions are generated per single iteration. These methods are implementable and globally convergent. We also prove that the algorithms have superlinear convergence rate under some mild conditions.     

A multivariate spectral projected gradient method for bound constrained optimization

In this paper, we consider a multivariate spectral projected gradient (MSPG) method for bound constrained optimization. Combined with a quasi-Newton property, the multivariate spectral projected gradient method allows an individual adaptive step size along each coordinate direction. On the basis of nonmonotone line search, global convergence is established. A numerical comparison with the traditional SPG method shows that the method is promising.     

A reduced Hessian SQP method for inequality constrained optimization

This paper develops a reduced Hessian method for solving inequality constrained optimization problems. At each iteration, the proposed method solves a quadratic subproblem which is always feasible by introducing a slack variable to generate a search direction and then computes the steplength by adopting a standard line search along the direction through employing the l _∞ penalty function. And a new update criterion is proposed to generate the quasi-Newton matrices, whose dimensions may be variable, approximating the reduced Hessian of the Lagrangian. The global convergence is established under mild conditions. Moreover, local R-linear and superlinear convergence are shown under certain conditions.     

Local and superlinear convergence of quasi-Newton methods based on modified secant conditions

For solving unconstrained minimization problems, quasi-Newton methods are popular iterative methods. The secant condition which employs only the gradient information is imposed on these methods. Several researchers paid attention to other secant conditions to get a better approximation of the Hessian matrix of the objective function. Recently, Zhang et al. [New quasi-Newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl. 102 (1999) 147–167] and Zhang and Xu [Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations, J. Comput. Appl. Math. 137 (2001) 269–278] proposed the modified secant condition which uses both gradient and function value information in order to get a higher order accuracy in approximating the second curvature of the objective function. They showed the local and q-superlinear convergence property of the BFGS-like and DFP-like updates based on their proposed secant condition. In this paper, we incorporate one parameter into this secant condition to smoothly switch the standard secant condition and the secant condition of Zhang et al. We consider a modified Broyden family which includes the BFGS-like and the DFP-like updates proposed by Zhang et al. We prove the local and q-superlinear convergence of our method.     

A feasible and superlinear algorithm for inequality constrained minimization problems

1. IntroductionConsider the optimization problemmin {f(x): gi(x) 5 0, j E I; x E R"}, (l)where f(x), gi(x): Rad - R, j E I ~ {1, 2,...,m}.It is well known that one of the most effective methods to solve problem (1) is thesequential quadratic programming (i.e., SoP) (see [1--6]), due to its property of superlinearconvergence. Especially in recent years, in order to perfect SoP both in theory and application, there have many papers, such as [7--10], been published. These papers focus mainly…     

An SQP-filter method for inequality constrained optimization and its global convergence

In this paper, we combine the filter technique with a modified sequential quadratic programming (SQP) method. The optimization solution is obtained by reducing step length, which is obtained by an exact linear search. Furthermore, this method can start with an infeasible initial point. The method uses a filter to promote global convergence.     

Accelerating method of global optimization for signomial geometric programming

Signomial geometric programming (SGP) has been an interesting problem for many authors recently. Many methods have been provided for finding locally optimal solutions of SGP, but little progress has been made for global optimization of SGP. In this paper we propose a new accelerating method for global optimization algorithm of SGP using a suitable deleting technique. This technique offers a possibility to cut away a large part of the currently investigated region in which the globally optimal solution of SGP does not exist, and can be seen as an accelerating device for global optimization algorithm of SGP problem. Compared with the method of Shen and Zhang [Global optimization of signomial geometric programming using linear relaxation, Appl. Math. Comput. 150 (2004) 99–114], numerical results show that the computational efficiency is improved obviously by using this new technique in the number of iterations, the required saving list length and the execution time of the algorithm.     

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.     

Feasible generalized monotone line search SQP algorithm for nonlinear minimax problems with inequality constraints

In this paper, the nonlinear minimax problems with inequality constraints are discussed, and a sequential quadratic programming (SQP) algorithm with a generalized monotone line search is presented. At each iteration, a feasible direction of descent is obtained by solving a quadratic programming (QP). To avoid the Maratos effect, a high order correction direction is achieved by solving another QP. As a result, the proposed algorithm has global and superlinear convergence. Especially, the global convergence is obtained under a weak Mangasarian–Fromovitz constraint qualification (MFCQ) instead of the linearly independent constraint qualification (LICQ). At last, its numerical effectiveness is demonstrated with test examples.     

Linearly constrained global optimization via piecewise-linear approximation

This paper considers the problem of optimizing a continuous nonlinear objective function subject to linear constraints via a piecewise-linear approximation. A systematic approach is proposed, which uses a lattice piecewise-linear model to approximate the nonlinear objective function on a simplicial partition and determines an approximately globally optimal solution by solving a set of standard linear programs. The new approach is applicable to any continuous objective function rather than to separable ones only and could be useful to treat more complex nonlinear problems. A numerical example is given to illustrate the practicability.