Abstract
In this paper, a class of nonlinear constrained optimization problems with both inequality and equality constraints is discussed. Based on a simple and effective penalty parameter and the idea of primal-dual interior point methods, a QP-free algorithm for solving the discussed problems is presented. At each iteration, the algorithm needs to solve two or three reduced systems of linear equations with a common coefficient matrix, where a slightly new working set technique for judging the active set is used to construct the coefficient matrix, and the positive definiteness restriction on the Lagrangian Hessian estimate is relaxed. Under reasonable conditions, the proposed algorithm is globally and superlinearly convergent. During the numerical experiments, by modifying the technique in Section 5 of (SIAM J. Optim. 14(1): 173-199, 2003), we introduce a slightly new computation measure for the Lagrangian Hessian estimate based on second order derivative information, which can satisfy the associated assumptions. Then, the proposed algorithm is tested and compared on 59 typical test problems, which shows that the proposed algorithm is promising.
Highlights
In this paper, we consider nonlinear constrained optimization problems with inequality and equality constraints (P) min f (x), s.t. gi(x) =, i ∈ I ; gj(x) ≤, j ∈ Iı, ( )where I = {, . . . , m }, Iı = {m +, m +, . . . , m + mı}, the functions f and gj : Rn → R
We review briefly the study on the primal-dual interior point (PDIP) quadratic program (QP)-free algorithms associated with our work
By means of problem (Pρ ), we propose a PDIP-type algorithm for problem (P)
Summary
We consider nonlinear constrained optimization problems with inequality and equality constraints (P) min f (x), s.t. gi(x) = , i ∈ I ; gj(x) ≤ , j ∈ Iı, ( ). In , Mayne and Polak [ ] proposed a simple scheme to convert (P) to a sequence of inequality smoothing constrained optimization (Pρ) min fρ(x) := f (x) – ρ gj(x), s.t. gj(x) ≤ , j ∈ I ∪ Iı, j∈I. With the help of inequality constrained non-smoothing optimization min f (x) + cj gj(x) , s.t. gj(x) ≤ , j ∈ I ∪ Iı, j∈I one can design an algorithm for solving the original problem (P), e.g., [ ], where cj > is the penalty parameter that needs to be updated
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