Abstract
In this study, a new smoothing nonlinear penalty function for constrained optimization problems is presented. It is proved that the optimal solution of the smoothed penalty problem is an approximate optimal solution of the original problem. Based on the smoothed penalty function, we develop an algorithm for finding an optimal solution of the optimization problems with inequality constraints. We further discuss the convergence of this algorithm and test this algorithm with three numerical examples. The numerical examples show that the proposed algorithm is feasible and effective for solving some nonlinear constrained optimization problems.
Highlights
IntroductionThe exact penalty function methods have been widely used for solving constrained optimization problems (see, e.g., [2,3,4,5,6,7,8,9])
Consider the following constrained optimization problem:(P) min f (x) s.t. gi(x) ≤ 0, i = 1, 2, . . . , m, x ∈ Rn, where the functions f, gi : Rn → R, i ∈ I = {1, 2, . . . , m}, are continuously differentiable functions
To obtain an optimal solution for the original problem, the conventional quadratic penalty function method usually requires that the penalty parameter tends to infinity, which is undesirable in practical computation
Summary
The exact penalty function methods have been widely used for solving constrained optimization problems (see, e.g., [2,3,4,5,6,7,8,9]). In order to use existing gradient-based algorithms, such as a Newton method, it is necessary to smooth the exact penalty function. Pinar and Zenios [21] and Wu et al [22] discussed a quadratic smoothing approximation to nondifferentiable exact penalty functions for constrained optimization.
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