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Limiting behavior of weighted central paths in linear programming. (English) Zbl 0841.90089
Given the linear programming problem (P): $$\min c^T x$$, s.t. $$Ax= b$$, $$x\geq 0$$, and its dual (D): $$\max b^T y$$, s.t. $$A^T y+ s= c$$, $$s\geq 0$$, with $$A\in \mathbb{R}^{m\times n}$$, $$b\in \mathbb{R}^m$$ and $$c\in \mathbb{R}^n$$, the paper discusses the solutions $$(x(\mu), s(\mu))$$ of the weighted penalized problems (P$$'$$): $$\min c^T x- \mu \sum^n_{i= 1} \omega_i \ln(x_i)$$ and (D$$'$$): $$\min b^T y+ \mu \sum^n_{i= 1} \omega_i \ln(s_i)$$, s.t. to the same restrictions, for $$\mu\in (0, \infty)$$ and $$\omega> 0$$. In particular, the limiting behaviour of the derivatives $$(x^{(k)}(\mu), s^{(k)}(\mu))$$, $$\mu> 0$$, of the $$\omega$$-central path $$(x(\mu), s(\mu))_{\mu> 0}$$ is studied when $$\mu$$ approaches zero and infinity. It is shown that the derivatives converge if $$\mu$$ approaches zero and that the higher order derivatives vanish at infinity. The author concludes that these observations should be helpful in the construction of new locally fast interior-point algorithms.

##### MSC:
 90C05 Linear programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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