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A primal-dual trust-region algorithm for non-convex nonlinear programming. (English) Zbl 0970.90116
The authors propose a primal-dual algorithm for the problem $$\min f(x)$$ subject to $$Ax=b, c(x) \geq 0$$, where $$f: \mathbb{R}^n \to \mathbb{R}$$ and $$c: \mathbb{R}^n \to \mathbb{R}^m$$ are twice continuously differentiable and an $$m\times n$$- matrix $$A$$ has full rank. The algorithm is basically a sequential minimization of a logarithmic barrier function $$\phi(x,\mu_k)=f(x)-\mu_k \langle e, \log(c(x))\rangle, e=(1, \dots, 1)$$, for a sequence $$\mu_k \to 0$$. Convergence is proved to second-order critical points from arbitrary starting points. Numerical results are presented for general quadratic problems.

##### MSC:
 90C51 Interior-point methods 90C30 Nonlinear programming
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