An algorithm of sequential systems of linear equations for nonlinear optimization problems with arbitrary initial point

For current sequential quadratic programming (SQP) type algorithms, there exist two problems: (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and the computation amount of this algorithm is very large. So they are not suitable for the large-scale problems; (ii) the SQP algorithms require that the related quadratic programming subproblems be solvable per iteration, but it is difficult to be satisfied. By using ε-active set procedure with a special penalty function as the merit function, a new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented. This new algorithm only needs to solve three systems of linear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence. To some extent, the new algorithm can overcome the shortcomings of the SQP algorithms mentioned above. Project partly supported by the National Natural Science Foundation of China and Tianyuan Foundation of China.     

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.     

非线性约束条件下一个超线性收敛的可行方法

在本文中，我们对非线性不等式约束条件下的非线性优化问题给出了一个新的ＳＱＰ类可行方法．此算法不但结构简单、易于计算，并且在适当的假设条件下，我们证明了算法具有全局收敛性及超线性收敛性     

A globally and superlinearly convergent modified SQP-filter method

In this paper, we presented a modified SQP-filter method based on the modified quadratic subproblem proposed by Zhou (J. Global Optim. 11, 193–2005, 1997). In contrast with the SQP methods, each iteration this algorithm only needs to solve one quadratic programming subproblems and it is always feasible. Moreover, it has no demand on the initial point. With the filter technique, the algorithm shows good numerical results. Under some conditions, the globally and superlinearly convergent properties are given.     

A ROBUST SQP METHOD FOR OPTIMIZATION WITH INEQUALITY CONSTRAINTS

A new algorithm for inequality constrained optimization is presented, which solves a linear programming subproblem and a quadratic subproblem at each iteration. The algorithm can circumvent the difficulties associated with the possible inconsistency of QP subproblem of the original SQP method. Moreover, the algorithm can converge to a point which satisfies a certain first-order necessary condition even if the original problem is itself infeasible. Under certain condition, some global convergence results are proved and local superlinear convergence results are also obtained. Preliminary numerical results are reported.     

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.     

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.     

A feasible filter SQP algorithm with global and local convergence

A feasible sequential quadratic programming (SQP) filter algorithm is proposed for general nonlinear programming. It is based on the modified quadratic programming (QP) subproblem in which each iteration proceeds in two phases. The first phase solves a general convex QP problem which does not require any feasibility restoration phase whose computation may be expensive. And, under some mild conditions, the global convergence is proved. The second phase can make the presented SQP method derive quadratic convergence by employing exact Hessian information.     

线性约束两分块非凸优化的ADMM-SQP算法

基于乘子交替方向法(ADMM)和序列二次规划(SQP)方法思想, 致力于研究线 性约束两分块非凸优化的新型高效算法. 首先, 以SQP思想为主线, 在其二次规划(QP)子问题的求解中引入ADMM思想, 将QP分解为两个相互独立的小规模QP求解. 其次, 借助增广拉格朗日函数和Armijo线搜索产生原始变量新迭代点. 最后, 以显式解析式更新对偶变量. 因此, 构建了一个新型ADMM-SQP算法. 在较弱条件下, 分析了算法通常意义下的全局收敛性, 并对算法进行了初步的数值试验.     

New sequential quadratic programming algorithm with consistent subproblems

One of the most interesting topics related to sequential quadratic programming algorithms is how to guarantee the consistence of all quadratic programming subproblems. In this decade, much work trying to change the form of constraints to obtain the consistence of the subproblems has been done. The method proposed by De O. Pantoja J.F. A. and coworkers solves the consistent problem of SQP method, and is the best to the authors’ knowledge. However, the scale and complexity of the subproblems in De O. Pantoja’s work will be increased greatly since all equality constraints have to be changed into absolute form. A new sequential quadratic programming type algorithm is presented by means of a special ε-active set scheme and a special penalty function. Subproblems of the new algorithm are all consistent, and the form of constraints of the subproblems is as simple as one of the general SQP type algorithms. It can be proved that the new method keeps global convergence and Local superlinear convergence. Project partly supported by the National Natural Science Foundation of China.     

A modified SQP-filter method for nonlinear complementarity problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming. For solving nonlinear programming, sequential quadratic programming (SQP) type method is very effective. Moreover, filter method, for its good numerical results, are extensively studied to handle nonlinear programming problems recently. In this paper, a modified quadratic subproblem is proposed. Based on it, we employ filter technique to tackle nonlinear complementarity problem. This method has no demand on initial point. The restoration phase, which is always used in traditional filter method, is not needed. Global convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.     

Global convergence of a robust filter SQP algorithm

We present a robust filter SQP algorithm for solving constrained optimization problems. This algorithm is based on the modified quadratic programming proposed by Burke to avoid the infeasibility of the quadratic programming subproblem at each iteration. Compared with other filter SQP algorithms, our algorithm does not require any restoration phase procedure which may spend a large amount of computation. The main advantage of our algorithm is that it is globally convergent without requiring strong constraint qualifications, such as Mangasarian–Fromovitz constraint qualification (MFCQ) and the constant rank constraint qualification (CRCQ). Furthermore, the feasible limit points of the sequence generated by our algorithm are proven to be the KKT points if some weaker conditions are satisfied. Numerical results are also presented to show the efficiency of the algorithm.     

A feasible SQP method for nonlinear programming

In this paper, a new SQP algorithm is presented to solve the general nonlinear programs with mixed equality and inequality constraints. Quoted from P. Spellucci (see [9]), this method maybe be named sequential equality constrained quadratic programming (SECQP) algorithm. Per single iteration, based on an active set strategy ( see [9]), this SECQP algorithm requires only to solve equality constrained quadratic programming subproblems or system of linear equations. The theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions.     

A NEW INEXACT SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM

This paper represents an inexact sequential quadratic programming (SQP) algorithm which can solve nonlinear programming (NLP) problems. An inexact solution of the quadratic programming subproblem is determined by a projection and contraction method such that only matrix-vector product is required. Some truncated criteria are chosen such that the algorithm is suitable to large scale NLP problem. The global convergence of the algorithm is proved.     

An Affine Scaling Interior Trust-Region Algorithm Combining Backtracking Line Search with Filter Technique for Nonlinear Constrained Optimization

In this article, an affine scaling interior trust-region algorithm which employs backtracking line search with filter technique is presented for solving nonlinear equality constrained programming with nonnegative constraints on variables. At current iteration, the general full affine scaling trust-region subproblem is decomposed into a pair of trust-region subproblems in vertical and horizontal subspaces, respectively. The trial step is given by the solutions of the pair of trust-region subproblems. Then, the step size is decided by backtracking line search together with filter technique. This is different from traditional trust-region methods and has the advantage of decreasing the number of times that a trust-region subproblem must be resolved in order to determine a new iteration point. Meanwhile, using filter technique instead of merit function to determine a new iteration point can avoid the difficult decisions regarding the choice of penalty parameters. Under some reasonable assumptions, the new method possesses the property of global convergence to the first-order critical point. Preliminary numerical results show the effectiveness of the proposed algorithm.     

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.     

An SQP-type algorithm for nonlinear second-order cone programs

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm. This work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

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 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.     

A Modified SQP Method and Its Global Convergence

The sequential quadratic programming method developed by Wilson, Han andPowell may fail if the quadratic programming subproblems become infeasibleor if the associated sequence of search directions is unbounded. In [1], Hanand Burke give a modification to this method wherein the QP subproblem isaltered in a way which guarantees that the associated constraint region isnonempty and for which a robust convergence theory is established. In thispaper, we give a modification to the QP subproblem and provide a modifiedSQP method. Under some conditions, we prove that the algorithm eitherterminates at a Kuhn–Tucker point within finite steps or generates aninfinite sequence whose every cluster is a Kuhn–Tucker point.Finally, we give some numerical examples.