A row relaxation method for large minimax problems

AbstractThis paper presents a row relaxation method for solving the doubly regularized minimax problem $$minimize \tfrac{1}{2}\varepsilon (\parallel x\parallel _2^2 + \parallel Ax - b\parallel _\infty ^2 ) + \parallel Ax - b\parallel _\infty $$ whereε is a given positive constant. It is shown that the dual of this problem can be brought to the form $$minimize \tfrac{1}{2}\parallel A^T y\parallel _2^2 - \varepsilon b^T y + \tfrac{1}{2}(max\{ 0,\parallel y\parallel _1 - 1\} )^2 $$ and ify solves the dual thenATy/ε solves the primal. The dual problem is solved via a row relaxation method that resembles Kaczmarz's method. This feature makes the new method suitable for problems in whichA is large, sparse, and unstructured. Another advantage is that the method is easily adapted to handle linear constraints. Numerical results are included.