A working set SQCQP algorithm with simple nonmonotone penalty parameters

In this paper, we present a new sequential quadratically constrained quadratic programming (SQCQP) algorithm, in which a simple updating strategy of the penalty parameter is adopted. This strategy generates nonmonotone penalty parameters at early iterations and only uses the multiplier corresponding to the bound constraint of the quadratically constrained quadratic programming (QCQP) subproblem instead of the multipliers of the quadratic constraints, which will bring some numerical advantages. Furthermore, by using the working set technique, we remove the constraints of the QCQP subproblem that are locally irrelevant, and thus the computational cost could be reduced. Without assuming the convexity of the objective function or the constraints, the algorithm is proved to be globally, superlinearly and quadratically convergent. Preliminary numerical results show that the proposed algorithm is very promising when compared with the tested SQP algorithms.     

Sequential quadratically constrained quadratic programming norm-relaxed algorithm of strongly sub-feasible directions

In this paper, we present a sequential quadratically constrained quadratic programming (SQCQP) norm-relaxed algorithm of strongly sub-feasible directions for the solution of inequality constrained optimization problems. By introducing a new unified line search and making use of the idea of strongly sub-feasible direction method, the proposed algorithm can well combine the phase of finding a feasible point (by finite iterations) and the phase of a feasible descent norm-relaxed SQCQP algorithm. Moreover, the former phase can preserve the “sub-feasibility” of the current iteration, and control the increase of the objective function. At each iteration, only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction. Without any other correctional directions, the global, superlinear and a certain quadratic convergence (which is between 1-step and 2-step quadratic convergence) properties are proved under reasonable assumptions. Finally, some preliminary numerical results show that the proposed algorithm is also encouraging.     

Exact penalty function algorithm with simple updating of the penalty parameter

A new globally convergent algorithm for minimizing an objective function subject to equality and inequality constraints is presented. The algorithm determines a search direction by solving a quadratic programming subproblem, which always has an optimal solution, and uses an exact penalty function to compute the steplength along this direction through an Armijo-type scheme. The special structure of the quadratic subproblem is exploited to construct a new and simple method for updating the penalty parameter. This method may increase or reduce the value of the penalty parameter depending on some easily performed tests. A new method for updating the Hessian of the Lagrangian is presented, and a Q-superlinear rate of convergence is established.This work was supported in part by the British Council and the Conselho Nacional de Desenvolvimento Cientifico & Tecnologico/CNPq, Rio de Janeiro, Brazil.The authors are very grateful to Mr. Lam Yeung for his invaluable assistance in computing the results and to a reviewer for constructive advice.     

A sequential quadratically constrained quadratic programming method with an augmented Lagrangian line search function

Based on an augmented Lagrangian line search function, a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems. Compared to quadratic programming solved in the traditional SQP methods, a convex quadratically constrained quadratic programming is solved here to obtain a search direction, and the Maratos effect does not occur without any other corrections. The “active set” strategy used in this subproblem can avoid recalculating the unnecessary gradients and (approximate) Hessian matrices of the constraints. Under certain assumptions, the proposed method is proved to be globally, superlinearly, and quadratically convergent. As an extension, general problems with inequality and equality constraints as well as nonmonotone line search are also considered.     

The global and superlinear convergence of a new nonmonotone MBFGS algorithm on convex objective functions

In this paper, a new nonmonotone MBFGS algorithm for unconstrained optimization will be proposed. Under some suitable assumptions, the global and superlinear convergence of the new nonmonotone MBFGS algorithm on convex objective functions will be established. Some numerical experiments show that this new nonmonotone MBFGS algorithm is competitive to the MBFGS algorithm and the nonmonotone BFGS algorithm.     

An RQP algorithm using a differentiable exact penalty function for inequality constrained problems

In this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. The algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. We prove that the algorithm possesses global and superlinear convergence properties. Numerical results are reported.     

On local convergence of sequential quadratically-constrained quadratic-programming type methods,with an extension to variational problems

We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadratic convergence of the primal-dual sequence was established by Fukushima, Luo and Tseng (SIAM J. Optim. 13, 1098–1119, 2003) under the assumption of convexity, the Slater constraint qualification, and a strong second-order sufficient condition. We obtain a new local convergence result, which complements the above (it is neither stronger nor weaker): we prove primal-dual quadratic convergence under the linear independence constraint qualification, strict complementarity, and a second-order sufficiency condition. Additionally, our results apply to variational problems beyond the optimization case. Finally, we provide a necessary and sufficient condition for superlinear convergence of the primal sequence under a Dennis-Moré type condition. Research of the second author is partially supported by CNPq Grants 300734/95-6 and 471780/2003-0, by PRONEX–Optimization, and by FAPERJ.     

A recursive quadratic programming algorithm that uses differentiable exact penalty functions

In this paper, a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied. The line search functions used are approximations to Fletcher's differentiable exact penalty function. Global convergence and local superlinear convergence results are proved, and some numerical results are given.     

约束最优化的一个使用非单调线搜索的超线性收敛算法

Abstract. In the paper, a new mixed algorithm combined with schemes of nonmonotone line search, the systems of linear equations for higher order modification and sequential quadratic programming for constrained optimizations is presented. Under some weaker assumptions,without strict complementary condition, the algorithm is globally and superlinearly convergent.     

不等式约束优化一个新的SQP算法

本文提出了一个处理不等式约束优化问题的新的SQP算法．和传统的SQP算法相比,该算法每步只需求解一个仅含等式约束的子二次规划,从而减少了算法的计算工作量．在适当的条件下,证明算法是全局收敛的且具有超线性收敛速度．数值实验表明算法是有效的．     

Global and local convergence of a nonmonotone trust region algorithm for equality constrained optimization

This paper presents a nonmonotone trust region algorithm for equality constrained optimization problems. Under certain conditions, we obtain not only the global convergence in the sense that every limit point is a stationary point but also the one step superlinear convergence rate. Numerical tests are also given to show the efficiency of the proposed algorithm.     

等式约束优化问题的一类新的简单光滑精确罚函数

精确罚函数方法是求解优化问题的一类经典方法,传统的精确罚函数不可能既是简单的又是光滑的,这里简单的是指罚函数中不包含目标函数和约束函数的梯度信息。针对等式约束问题提出了不同与传统罚函数的一类新的简单光滑罚函数并证明了它是精确的。给出了以新的罚函数为基础的罚函数方法并用数值例子说明算法是可行的。     

A modified SQP algorithm for minimax problems

In this paper, a modified nonmonotone line search SQP algorithm for nonlinear minimax problems is presented. During each iteration of the proposed algorithm, a main search direction is obtained by solving a reduced quadratic program (QP). In order to avoid the Maratos effect, a correction direction is generated by solving the reduced system of linear equations. Under mild conditions, the global and superlinear convergence can be achieved. Finally, some preliminary numerical results are reported.     

Convergence to a second-order point of a trust-region algorithm with a nonmonotonic penalty parameter for constrained optimization

In a recent paper (Ref. 1), the author proposed a trust-region algorithm for solving the problem of minimizing a nonlinear function subject to a set of equality constraints. The main feature of the algorithm is that the penalty parameter in the merit function can be decreased whenever it is warranted. He studied the behavior of the penalty parameter and proved several global and local convergence results. One of these results is that there exists a subsequence of the iterates generated by the algorithm that converges to a point that satisfies the first-order necessary conditions.In the current paper, we show that, for this algorithm, there exists a subsequence of iterates that converges to a point that satisfies both the first-order and the second-order necessary conditions.This research was supported by the Rice University Center for Research on Parallel Computation, Grant R31853, and the REDI Foundation.     

Nonlinear programming algorithms using trust regions and augmented Lagrangians with nonmonotone penalty parameters

The strategy for obtaining global convergence is based on the trust region approach. The merit function is a type of augmented Lagrangian. A new updating scheme is introduced for the penalty parameter, by means of which monotone increase is not necessary. Global convergence results are proved and numerical experiments are presented. Received May 31, 1995 / Revised version received December 12, 1997 Published online October 21, 1998     

A Class of penalty functions for optimization problema with bound constraints

In this paper we propose a new class of continuously differentiable globally exact penalty functions for the solution of minimization problems with simple bounds on some (all) of the variables. The penalty functions in this class fully exploit the structure of the problem and are easily computable. Furthermore we introduce a simple updating rule for the penalty parameter that can be used in conjunction with unconstrained minimization techniques to solve the original problem.     

等式约束优化一个新的SQP算法

提出了一个处理等式约束优化问题新的SQP算法,该算法通过求解一个增广Lagrange函数的拟Newton方法推导出一个等式约束二次规划子问题,从而获得下降方向．罚因子具有自动调节性,并能避免趋于无穷．为克服Maratos效应采用增广Lagrange函数作为效益函数并结合二阶步校正方法．在适当的条件下,证明算法是全局收敛的,并且具有超线性收敛速度．     

A successive quadratic programming algorithm with global and superlinear convergence properties

This paper presents a successive quadratic programming algorithm for solving general nonlinear programming problems. In order to avoid the Maratos effect, direction-finding subproblems are derived by modifying the second-order approximations to both objective and constraint functions of the problem. We prove that the algorithm possesses global and superlinear convergence properties.This work was supported in part by a Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

A globally convergent,implementable multiplier method with automatic penalty limitation

This paper deals with penalty function and multiplier methods for the solution of constrained nonconvex nonlinear programming problems. Starting from an idea introduced several years ago by Polak, we develop a class of implementable methods which, under suitable assumptions, produce a sequence of points converging to a strong local minimum for the problem, regardless of the location of the initial guess. In addition, for sequential minimization type multiplier methods, we make use of a rate of convergence result due to Bertsekas and Polyak, to develop a test for limiting the growth of the penalty parameter and thereby prevent ill-conditioning in the resulting sequence of unconstrained optimization problems.Research sponsored by the National Science Foundation (RANN) Grant ENV76-04264 and the Joint Services Electronics Research Program Contract F44620-76-C-0100.     

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.