Abstract
We propose a class of inexact secant methods in association with the line search filter technique for solving nonlinear equality constrained optimization. Compared with other filter methods that combine the line search method applied in most large-scale optimization problems, the inexact line search filter algorithm is more flexible and realizable. In this paper, we focus on the analysis of the local superlinear convergence rate of the algorithms, while their global convergence properties can be obtained by making an analogy with our previous work. These methods have been implemented in a Matlab code, and detailed numerical results indicate that the proposed algorithms are efficient for 43 problems from the CUTEr test set.
Highlights
After more than 20 years of development, filter algorithm has become an important method to solve constrained optimization problems and has been successfully applied to various fields of optimization
When solving large-scale nonlinear programming problems or system of nonlinear equations, it would take too much time to get an exact solution in each iteration
Mathematical Problems in Engineering assumptions, the global convergence of inexact SQP steps was proved. ese research results show that the use of inexact algorithm can save a lot of computing time. Inspired by these ideas above, we propose a class of inexact secant methods in association with the line search filter technique, which has both global convergence and qsuperlinear local convergence rate. ese methods are globalized by line search and filter methods
Summary
(2h) where ‖rk‖ ≤ ηk‖ck‖ with ηk ∈ [0, t] and t ∈ (0, 1), L(x, λ) f(x) + λTc(x) for λ ∈ Rm, and B is the BFGS or the DFP secant update formula generating matrices Wk approximating the Hessian of the Lagrangian at each xk. ere are many different multiplier updates; λk+1 in (2a) is chosen from the most commonly used multiplier updates: projection update, null-space update, and Newton update. A†k in (2d) is the pseudo-inverse of ATk. With different choices for λk+1 in (2a), P(x) in (2c), and A†k in (2d), we get the following three algorithms:. E class of complete algorithms can be stated as follows. 3. Convergence Analysis e global convergence analysis of the algorithms in Section 3 of [10] still holds. Is lemma proves that the search direction is a sufficient descent direction for Lat the points that are sufficiently close to feasible and nonoptimal. Assume that xk and λk generated by the class of algorithms converge to local solution x∗ and λ∗, respectively. They are contained in a convex set Ω we give the following assumptions
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