Filter Line Search Research Articles

A trust-region algorithm combining line search filter technique for nonlinear constrained optimization

In this paper, we propose a trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming. At current iteration, a trial step is formed as the sum of a normal step and a tangential step which is generated by trust-region subproblem and the step size is decided by interior backtracking line search together with filter methods. Then, the next iteration is determined. This is different from general trust-region methods in which the next iteration is determined by the ratio of the actual reduction to the predicted reduction. The global convergence analysis for this algorithm is presented under some reasonable assumptions and the preliminary numerical results are reported.

Projected affine-scaling interior-point Newton’s method with line search filter for box constrained optimization

We present a new projected affine-scaling interior-point Newton’s method with a line search filter algorithm for nonlinear optimization problems with box constraints. The global convergence of the proposed algorithm is established under some reasonable conditions. Furthermore, the method is shown to be locally quadratically convergent under the strong second order sufficiency condition without assuming strict complementarity of the solution. Numerical experiments are presented that confirm the robustness and efficiency of the approach.

A Filter Method with Unified Step Computation for Nonlinear Optimization

We present a filter line search method for solving general nonlinear and nonconvex optimization problems. The method is of the filter variety but uses a robust (always feasible) subproblem based on an exact penalty function to compute a search direction. This contrasts traditional filter methods that use a (separate) restoration phase designed to reduce infeasibility until a feasible subproblem is obtained. Therefore, an advantage of our approach is that every trial step is computed from subproblems that value reducing both the constraint violation and the objective function. Moreover, our step computation involves subproblems that are computationally tractable and utilize second derivative information when it is available. The formulation of each subproblem and the choice of weighting parameter is crucial for obtaining an efficient, robust, and practical method. Our strategy is based on steering methods designed for exact penalty functions but is fortified with a trial step convexification scheme that en...

Convergence analysis of a regularized interior point algorithm for the barrier problems with singular solutions

Wächter and Biegler (2006) [25] presented and implemented an interior point filter line search method (called IPOPT) for large scale nonlinear programming. To make IPOPT efficient and robust, they add some safeguards in their implementation. One crucial safeguard is to apply the inertia correction technique to modify the KKT matrix. We focus on the degenerate problems in which the KKT matrix might be singular. To complement theoretical foundation of this technique in the degenerate case, we propose and analyze a regularized interior point method for the barrier problems with singular solutions which largely relies on IPOPT. Based on the inertia correction technique, the matrix arising in the KKT equation is corrected to be a nonsingular one so that the Newton step can be calculated at each iteration even in the degenerate case. We emphasize that the special inertia correction for the KKT matrix does not impose any negative impact on the fast convergence of our algorithm. Under some local error bound condition, we prove that the rate of quadratic convergence is achieved when the second order condition or some constraint qualification condition fails to hold (or both).

A Globally Convergent Line Search Filter SQP Method for Inequality Constrained Optimization

A line search filter SQP method for inequality constrained optimization is presented. This method makes use of a backtracking line search procedure to generate step size and the efficiency of the filter technique to determine step acceptance. At each iteration, the subproblem is always consistent, and it only needs to solve one QP subproblem. Under some mild conditions, the global convergence property can be guaranteed. In the end, numerical experiments show that the method in this paper is effective.

A line search filter inexact SQP method for nonlinear equality constrained optimization

A line search filter inexact SQP method for nonlinear equality constrained optimization

A nonmonotone line search filter method with reduced Hessian updating for nonlinear optimization

This paper proposes a nonmonotone line search filter method with reduced Hessian updating for solving nonlinear equality constrained optimization. In order to deal with large scale problems, a reduced Hessian matrix is approximated by BFGS updates. The new method assures global convergence without using a merit function. By Lagrangian function in the filter and nonmonotone scheme, the authors prove that the method can overcome Maratos effect without using second order correction step so that the locally superlinear convergence is achieved. The primary numerical experiments are reported to show effectiveness of the proposed algorithm.

A Nonmonotone Line Search Filter Algorithm for the System of Nonlinear Equations

We present a new iterative method based on the line search filter method with the nonmonotone strategy to solve the system of nonlinear equations. The equations are divided into two groups; some equations are treated as constraints and the others act as the objective function, and the two groups are just updated at the iterations where it is needed indeed. We employ the nonmonotone idea to the sufficient reduction conditions and filter technique which leads to a flexibility and acceptance behavior comparable to monotone methods. The new algorithm is shown to be globally convergent and numerical experiments demonstrate its effectiveness.

An Improved Line Search Filter Method for the System of Nonlinear Equations

An improved line search filter algorithm for the system of nonlinear equations is presented. We divide the equations into two groups, one contains the equations that are treated as equality constraints and the square of other equations is regarded as objective function. Two groups of equations are updated at every iteration in the works by Nie (2004, 2006, and 2006), by Nie et al. (2008), and by Gu (2011), while we just update them at the iterations when it is needed indeed. As a consequence, the scale of the calculation is decreased in a certain degree. Under some suitable conditions the global convergence can be induced. In the end, numerical experiments show that the method in this paper is effective.

A line search filter algorithm with inexact step computations for equality constrained optimization

In this paper, a new line search filter algorithm for equality constrained optimization is presented. The approach belongs to the class of inexact Newton-like methods. It can also be regarded as an inexact version of generic sequential quadratic programming (SQP) methods. The trial step is obtained by truncatedly solving the primal–dual system based on any robust and efficient linear system solver. Practical termination tests for the linear system solver are established to ensure global convergence. Preliminary numerical results demonstrate the approach is potentially useful.

Assessing the potential of interior point barrier filter line search methods: nonmonotone versus monotone approach

In this article, we present a numerical study of three nonmonotone filter line search techniques, as well as a three-dimensional filter approach, when incorporated into the solver IPOPT, a primal-dual barrier method developed by Wächter and Biegler [On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program. 106 (2006), pp. 25–57] for nonlinear programming. Primary assessment of the proposals has been done with sets of small- and medium-scale problems and large-scale problems separately. Results show that the use of nonmonotone globalization strategies improves efficiency.

Interior point filter method for semi-infinite programming problems

Semi-infinite programming (SIP) problems can be efficiently solved by reduction-type methods. Here, we present a new reduction method for SIP, where the multi-local optimization is carried out with a stretched simulated annealing algorithm, the reduced (finite) problem is approximately solved by a Newton's primal–dual interior point method that uses a novel two-dimensional filter line search strategy to guarantee the convergence to a KKT point that is a minimizer, and the global convergence of the overall reduction method is promoted through the implementation of a classical two-dimensional filter line search. Numerical experiments with a set of well-known problems are shown.

A reduced Hessian algorithm with line search filter method for nonlinear programming

This paper proposes a line search filter reduced Hessian method for nonlinear equality constrained optimization. The feature of the presented algorithm is that the reduced Hessian method is used to produce a search direction, a backtracking line search procedure to generate step size, some filtered rules to determine step acceptance, second order correction technique to reduce infeasibility and overcome the Maratos effects. It is shown that this algorithm does not suffer from the Maratos effects by using second order correction step, and under mild assumptions fast convergence to second order sufficient local solutions is achieved. The numerical experiment is reported to show the effectiveness of the proposed algorithm.

Robust extensions for reduced-space barrier NLP algorithms

Reduced-space barrier NLP algorithms are particularly useful for optimization of large structured systems with few degrees of freedom. Such optimization algorithms are often applied on process models developed within equation oriented process simulators. By partitioning the search direction into tangential and normal steps, these methods can exploit the structure of the equality constraints and adjust the remaining degrees of freedom in a lower dimensional space. Moreover, as shown in previous work, the barrier approach extended with a novel filter linear search algorithm has global and fast local convergence properties. However, convergence properties of the reduced-space barrier algorithm require regularity assumptions. In particular, the method may fail in the presence of linearly dependent active constraints. To deal with these questions, we modify the reduced-space barrier method in two ways. First, as the filter line search requires a feasibility restoration step, we develop and analyze an improved algorithm for this step, which is tailored to the reduced-space method. In addition, a dimension change procedure is proposed to address decomposition of problems with linearly dependent constraints. Finally, both approaches are implemented within a reduced-space version of IPOPT and numerical tests demonstrate the performance of the proposed modifications.

An Affine Scaling Interior Point Filter Line-Search Algorithm for Linear Inequality Constrained Minimization

We propose and analyze a new affine scaling interior point method in association with line-search filter technique for solving the linear inequality constrained optimization. We extend an existing filter of Wächter and Biegler [9] for nonlinear equality constrained problems to linear inequality constraint problems. The two main ingredients of the method are a filter-line-search algorithm which is used to determine step size and an affine-scaling interior point Newton method which is used to generate a search direction. The global convergence of the proposed algorithm is established under some reasonable conditions. Further, the method is shown to be locally Q-superlinearly convergent under the strong second order sufficiency condition. Numerical tests are presented that confirm the robustness and efficiency of the approach.

A secant algorithm with line search filter method for nonlinear optimization

Filter methods were initially designed for nonlinear programming problems by Fletcher and Leyffer. In this paper we propose a secant algorithm with line search filter method for nonlinear equality constrained optimization. The algorithm yields the global convergence under some reasonable conditions. By using the Lagrangian function value in the filter we establish that the proposed algorithm can overcome the Maratos effect without using second order correction step, so that fast local superlinear convergence to second order sufficient local solution is achieved. The primary numerical results are presented to confirm the robustness and efficiency of our approach.

A line search filter secant method for nonlinear equality constrained optimization

This paper formulates and analyzes a line search method for general nonlinear equality constrained optimization based on filter methods for step acceptance and secant methods for search direction. The feature of the new algorithm is that the secant algorithm is used to produce a search direction, a backtracking line search procedure is used to generate step size, some filtered rules are used to determine step acceptance, second order correction technique is used to reduce infeasibility and overcome the Maratos effect. Global convergence properties of this method are analyzed: under mild assumptions it is showed that every limit point of the sequence of iterates generated by the algorithm is feasible, and that there exists at least one limit point that is a stationary point for the problem. Moreover, it is also established that the Maratos effect can be overcome in our new approach by adding second order correction steps so that fast local superlinear convergence to a second order sufficient local solution is achieved. Finally, the results of numerical experiments are reported to show the effectiveness of the line search filter secant method.

A filter interior-point algorithm with projected Hessian updating for nonlinear optimization

Filter approaches, initially presented by Fletcher and Leyffer in 2002, are attractive methods for nonlinear programming. In this paper, we propose an interior-point barrier projected Hessian updating algorithm with line search filter method for nonlinear optimization. The Lagrangian function value instead of the objective function value is used in the filter. The damped BFGS updating is employed to maintain the positive definiteness of the matrices in projected Hessian updating algorithm. The numerical experiments are reported to show the effectiveness of the proposed algorithm.

Practical implementation of an interior point nonmonotone line search filter method

Here we present a primal-dual interior point nonmonotone line search filter method for nonlinear programming. The filter relies on three measures, the feasibility, the centrality and the optimality presented in the optimality conditions, and considers relaxed acceptability criteria for the step size and includes a feasibility restoration phase. Evaluation of the method has, until now, been made on small problems and a comparison is provided with a merit function approach.

Global convergence of a tri-dimensional filter SQP algorithm based on the line search method

In this paper, we propose a new filter line search SQP method in which the violations of equality and inequality constraints are considered separately. Thus the filter in our algorithm is composed by three components: objective function value, equality and inequality constraints violations. The filter with three components accepts reasonable steps flexibly, comparing that with two components. The new filter shares some features with the Chin and Fletcher's approach, namely the “slanting envelope” and the “inclusion property”. Under mild conditions, the filter line search SQP method is proven to be globally convergent. Numerical experiments also show the efficiency of our method.