Abstract
In this study, we deal with the nonlinear constrained global optimization problems. First, we introduce a new smooth exact penalty function for constrained optimization problems. We combine the exact penalty function with the auxiliary function in regard to constrained global optimization. We present a new auxiliary function approach and the adapted algorithm for solving non-linear inequality constrained global optimization problems. Finally, we illustrate the efficiency of the algorithm on some numerical examples.
Highlights
Introduction are usedThe penalized objective function is defined asWe consider the following continuous constrained m optimization problemF (x, ρ) = f (x) + ρ b(gj(x)), (1) min f (x) (P )x∈Rn s.t. gj(x) ≤ 0, j = 1, 2, ..., m, where f : Rn → R and gj(x) : Rn → R, j ∈ J = {1, 2, ..., m} are continuously differentiable functions
The penalty function method has been proposed in order to transform a constrained optimization problem to an unconstrained optimization problem
The main idea in exact penalty function approach is to construct a barrier at the boundary of D0 such that any local solver can not find a point outside the set D0
Summary
We assume that the set D0 is closed and bounded and the function f has a finite number of local minimizers in D0. Throughout the paper, we use x∗k to denote the k−th local minimizer of f whereas by x∗ we mean the global minimizer. X2k denotes the Euclidean norm in Definition 1. [13] Let f : Rn → R be a continuous function. The function f : Rn × R+ → R is called a smoothing function of f (x), if f(·, β) is continuously differentiable in Rn for any fixed β, and for any x ∈ Rn, lim f(z, β) = f (x). [19] Let ε > 0, a point xε is called ε−feasible solution for the problem (P ), if gj(x) ≤ ε, j = 1, 2, .
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