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An unconstrained convex programming view of linear programming. (English) Zbl 0759.90064
For a linear program in Karmarkar standard form (P) $\min\{c\sp Tx; Ax=0, e\sp T x=1, x\ge 0\}$ the author considers a program with entropic barrier function $(\text{P}\sb \mu)$ $\min\{c\sp T x+\mu\sum\sb j x\sb j\log x\sb j; Ax=0, e\sp T x=1, x\ge 0\}$ and shows by using a geometric programming technique that the dual problem $(\text{D}\sb \mu)$ to $(\text{P}\sb \mu)$ is an unconstrained convex programming problem. Explicit formulae are derived for the computation of $\varepsilon$- optimal solutions to (P) and to its dual (D) from the optimal solution to $(\text{D}\sb \mu)$. Solving $(\text{D}\sb \mu)$ via unconstrained minimization techniques is discussed briefly.
Reviewer: J.Rohn (Praha)

90C05Linear programming
90C25Convex programming
90-08Computational methods (optimization)
90C30Nonlinear programming
Full Text: DOI
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