(untitled) (Journal of Computational Mathematics, 1987)
This paper is an extension of [1]. In this paper the descent and ascent segments are introduced to replace respectively the descent and ascent directions in [1] and are used to extend the concepts of S-basin and basin of a minimizer of a function. Lemmas and theorems similar to those in [1] are proved for the filled function P(x,r,p)= 1r+F(x)exp(-|x-x^*_1|^2/ρ^2), which is the same as that in [1], where x^*_1 is a constrained local minimizer of the problem (0.3) below and F(x)=f(x)+^m'_i=1μ_i|c_i(x)|+ ^m_i=m'+1μ_i max(0, -c_i(x)) is the exact penalty function for the constrained minimization problem min_x f(x), subject to c_i(x) = 0 , i = 1, 2, …, m', c_i(x) 0 , i = m'+1, …, m, where μ_i ＞ 0 (i=1, 2, …, m) are sufficiently large. When x^*_1 has been located, a saddle point or a minimizer x of P(x,r,ρ) can be located by using the nonsmooth minimization method with some special termination principles. The x is proved to be in a basin of a lower minimizer x^*_2 of F(x), provided that the ratio ρ^2/[r+F(x^*_1)] is appropriately small. Thus, starting with x to minimize F(x), one can locate x^*_2. In this way a constrained global minimizer of (0.3) can finally be found and termination will happen.