A global and quadratically convergent method for linear l ∞ problems
SIAM Journal on Numerical Analysis
Journal of Optimization Theory and Applications
The shortest network under a given topology
Journal of Algorithms
A primal-dual algorithm for minimizing a sum of Euclidean norms
Journal of Computational and Applied Mathematics
An Efficient Primal-Dual Interior-Point Method for Minimizing a Sum of Euclidean Norms
SIAM Journal on Scientific Computing
A Smoothing Newton Method for Minimizing a Sum of Euclidean Norms
SIAM Journal on Optimization
An Efficient Algorithm for Minimizing a Sum of Euclidean Norms with Applications
SIAM Journal on Optimization
An Efficient Algorithm for Minimizing a Sum of p-Norms
SIAM Journal on Optimization
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
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In this paper, we consider the problem of minimizing a sum of Tchebycheff norms @F(x)=@?"i"="1^m@?b"i-A"i^Tx@?"~, where A"i@?R^n^x^d and b"i@?R^d. We derive a smooth approximation of @F(x) by the entropy regularization technique, and convert the problem into a parametric family of strictly convex minimization. It turns out that the minimizers of these problems generate a trajectory that will go to the primal-dual solution set of the original problem as the parameter tends to zero. By this, we propose a smoothing algorithm to compute an @e-optimal primal-dual solution pair. The algorithm is globally convergent and has a quadratic rate of convergence. Numerical results are reported for a path-following version of the algorithm and made comparisons with those yielded by the primal-dual path-following interior point algorithm, which indicate that the proposed algorithm can yield the solutions with favorable accuracy and is comparable with the interior point method in terms of CPU time for those problems with m@?max{n,d}.