An algorithm for constrained convex optimization

This paper presents a feasible direction algorithm for the minimization of a pseudoconvex function over a smooth, compact, convex set. We establish that each cluster point of the generated sequence is an optimal solution of the problem without introducing anti-jamming procedures. Each iteration of the algorithm involves as subproblems only one line search for a zero of a continuously differentiable convex function and one univariate function minimization on a compact interval.     

An interval algorithm for constrained global optimization

An interval algorithm for bounding the solutions of a constrained global optimization problem is described. The problem functions are assumed only to be continuous. It is shown how the computational cost of bounding a set which satisfies equality constraints can often be reduced if the equality constraint functions are assumed to be continuously differentiable. Numerical results are presented.     

A penalty-interior-point algorithm for nonlinear constrained optimization

Penalty and interior-point methods for nonlinear optimization problems have enjoyed great successes for decades. Penalty methods have proved to be effective for a variety of problem classes due to their regularization effects on the constraints. They have also been shown to allow for rapid infeasibility detection. Interior-point methods have become the workhorse in large-scale optimization due to their Newton-like qualities, both in terms of their scalability and convergence behavior. Each of these two strategies, however, have certain disadvantages that make their use either impractical or inefficient for certain classes of problems. The goal of this paper is to present a penalty-interior-point method that possesses the advantages of penalty and interior-point techniques, but does not suffer from their disadvantages. Numerous attempts have been made along these lines in recent years, each with varying degrees of success. The novel feature of the algorithm in this paper is that our focus is not only on the formulation of the penalty-interior-point subproblem itself, but on the design of updates for the penalty and interior-point parameters. The updates we propose are designed so that rapid convergence to a solution of the nonlinear optimization problem or an infeasible stationary point is attained. We motivate the convergence properties of our algorithm and illustrate its practical performance on large sets of problems, including sets of problems that exhibit degeneracy or are infeasible.     

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 derivative-free algorithm for linearly constrained optimization problems

Based on the NEWUOA algorithm, a new derivative-free algorithm is developed, named LCOBYQA. The main aim of the algorithm is to find a minimizer $x^{*} \in\mathbb{R}^{n}$ of a non-linear function, whose derivatives are unavailable, subject to linear inequality constraints. The algorithm is based on the model of the given function constructed from a set of interpolation points. LCOBYQA is iterative, at each iteration it constructs a quadratic approximation (model) of the objective function that satisfies interpolation conditions, and leaves some freedom in the model. The remaining freedom is resolved by minimizing the Frobenius norm of the change to the second derivative matrix of the model. The model is then minimized by a trust-region subproblem using the conjugate gradient method for a new iterate. At times the new iterate is found from a model iteration, designed to improve the geometry of the interpolation points. Numerical results are presented which show that LCOBYQA works well and is very competing against available model-based derivative-free algorithms.     

A tolerant algorithm for linearly constrained optimization calculations

Two extreme techniques when choosing a search direction in a linearly constrained optimization calculation are to take account of all the constraints or to use an active set method that satisfies selected constraints as equations, the remaining constraints being ignored. We prefer an intermediate method that treats all inequality constraints with small residuals as inequalities with zero right hand sides and that disregards the other inequality conditions. Thus the step along the search direction is not restricted by any constraints with small residuals, which can help efficiency greatly, particularly when some constraints are nearly degenerate. We study the implementation, convergence properties and performance of an algorithm that employs this idea. The implementation considerations include the choice and automatic adjustment of the tolerance that defines the small residuals, the calculation of the search directions, and the updating of second derivative approximations. The main convergence theorem imposes no conditions on the constraints except for boundedness of the feasible region. The numerical results indicate that a Fortran implementation of our algorithm is much more reliable than the software that was tested by Hock and Schittkowski (1981). Therefore the algorithm seems to be very suitable for general use, and it is particularly appropriate for semi-infinite programming calculations that have many linear constraints that come from discretizations of continua.     

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．     

Parallel SSLE algorithm for large scale constrained optimization

In this paper, a parallel SSLE algorithm is proposed for solving large scale constrained optimization with block-separable structure. At each iteration, the PVD sub-problems are solved inexactly by the SSLE algorithm, which successfully overcomes the constraint inconsistency exited in most SQP-type algorithm, and decreases the computation amount as well. Without assuming the convexity of the constraints, the algorithm is proved to be globally convergent to a KKT point of the original problem.     

A penalty derivative-free algorithm for nonlinear constrained optimization

In this paper, we modify a derivative-free line search algorithm (DFL) proposed in the Ref. (Liuzzi et al. SIAM J Optimiz 20(5):2614–2635, 2010) to minimize a continuously differentiable function of box constrained variables or unconstrained variables with nonlinear constraints. The first-order derivatives of the objective function and of the constraints are assumed to be neither calculated nor explicitly approximated. Different line-searches are used for box-constrained variables and unconstrained variables. Accordingly the convergence to stationary points is proved. The computational behavior of the method has been evaluated on a set of test problems. The performance and data profiles are used to compare with DFL.     

Globally convergent algorithm for nonlinearly constrained optimization problems

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 first solving a linear program and using the information gained thereby to define a quadratic approximation, with a guaranteed solution, to the original problem; the solution of the quadratic problem is the desired search direction. The algorithm incorporates a new method for choosing the penalty parameter. Numerical results illustrate the performance of the algorithm.The author wishes to thank Professor D. Q. Mayne and Dr. F. A. Pantoja for critically reviewing the first draft of this paper, for their suggestions, criticism, and contributions to some of the proofs. Support of the UK Science Research and Engineering Council is gratefully acknowledged.     

A trust region algorithm for equality constrained optimization

A trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function. Under certain conditions global convergence and local superlinear convergence results are proved.     

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.This work is supported by the National Natural Science Foundation (No. 10261001) and Guangxi Science Foundation (No. 0236001 and 0249003) of China. Acknowledgement.We would like to thank one anonymous referee for his valuable comments and suggestions, which greatly improved the quality of this paper.     

An algorithm for solving linearly constrained optimization problems

An algorithm for minimization of functions of many variables, subject possibly to linear constraints on the variables, is described. In it a subproblem is solved in which a quadratic approximation is made to the object function and minimized over a region in which the approximation is valid. A strategy for deciding when this region should be expanded or contracted is given. The quadratic approximation involves estimating the hessian of the object function by a matrix which is updated at each iteration by a formula recently reported by Powell [6]. This formula enables convergence of the algorithm from any feasible point to be proved. Use of such an approximation, as against using exact second derivatives, also enables a reduction of about 60% to be made in the number of operations to solve the subproblem. Numerical evidence is reported showing that the algorithm is efficient in the number of function evaluations required to solve well known test problems.This paper was presented at the 7th International Mathematical Programming Symposium 1970, The Hague, The Netherlands.     

A two-stage feasible directions algorithm for nonlinear constrained optimization

We present a feasible directions algorithm, based on Lagrangian concepts, for the solution of the nonlinear programming problem with equality and inequality constraints. At each iteration a descent direction is defined; by modifying it, we obtain a feasible descent direction. The line search procedure assures the global convergence of the method and the feasibility of all the iterates. We prove the global convergence of the algorithm and apply it to the solution of some test problems. Although the present version of the algorithm does not include any second-order information, like quasi-Newton methods, these numerical results exhibit a behavior comparable to that of the best methods known at present for nonlinear programming. Research performed while the author was on a two years appointment at INRIA, Rocquencourt, France, and partially supported by the Brazilian Research Council (CNPq).     

A hybrid glowworm swarm optimization algorithm to solve constrained multimodal functions optimization

In this paper, a novel hybrid glowworm swarm optimization (HGSO) algorithm is proposed. The HGSO algorithm embeds predatory behaviour of artificial fish swarm algorithm (AFSA) into glowworm swarm optimization (GSO) algorithm and combines the GSO with differential evolution on the basis of a two-population co-evolution mechanism. In addition, to overcome the premature convergence, the local search strategy based on simulated annealing is applied to make the search of GSO approach the true optimum solution gradually. Finally, several benchmark functions show that HGSO has faster convergence efficiency and higher computational precision, and is more effective for solving constrained multi-modal function optimization problems.     

A progressive barrier derivative-free trust-region algorithm for constrained optimization

We study derivative-free constrained optimization problems and propose a trust-region method that builds linear or quadratic models around the best feasible and around the best infeasible solutions found so far. These models are optimized within a trust region, and the progressive barrier methodology handles the constraints by progressively pushing the infeasible solutions toward the feasible domain. Computational experiments on 40 smooth constrained problems indicate that the proposed method is competitive with COBYLA, and experiments on two nonsmooth multidisciplinary optimization problems from mechanical engineering show that it can be competitive with the NOMAD software.     

An improved feasible QP-free algorithm for inequality constrained optimization

In this paper, an improved feasible QP-free method is proposed to solve nonlinear inequality constrained optimization problems. Here, a new modified method is presented to obtain the revised feasible descent direction. In view of the computational cost, the most attractive feature of the new algorithm is that only one system of linear equations is required to obtain the revised feasible descent direction. Thereby, per single iteration, it is only necessary to solve three systems of linear equations with the same coefficient matrix. In particular, without the positive definiteness assumption on the Hessian estimate, the proposed algorithm is still global convergence. Under some suitable conditions, the superlinear convergence rate is obtained.     

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.     

Projected quasi-Newton algorithm with trust region for constrained optimization

In Ref. 1, Nocedal and Overton proposed a two-sided projected Hessian updating technique for equality constrained optimization problems. Although local two-step Q-superlinear rate was proved, its global convergence is not assured. In this paper, we suggest a trust-region-type, two-sided, projected quasi-Newton method, which preserves the local two-step superlinear convergence of the original algorithm and also ensures global convergence. The subproblem that we propose is as simple as the one often used when solving unconstrained optimization problems by trust-region strategies and therefore is easy to implement.This research was supported in part by the National Natural Science Foundation of China.     

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.