Kybernetika 46 no. 4, 697-721, 2010

Primal interior point method for minimization of generalized minimax functions

Ladislav Lukšan, Ctirad Matonoha and Jan Vlček


In this paper, we propose a primal interior-point method for large sparse generalized minimax optimization. After a short introduction, where the problem is stated, we introduce the basic equations of the Newton method applied to the KKT conditions and propose a primal interior-point method. Next we describe the basic algorithm and give more details concerning its implementation covering numerical differentiation, variable metric updates, and a barrier parameter decrease. Using standard weak assumptions, we prove that this algorithm is globally convergent if a bounded barrier is used. Then, using stronger assumptions, we prove that it is globally convergent also for the logarithmic barrier. Finally, we present results of computational experiments confirming the efficiency of the primal interior point method for special cases of generalized minimax problems.


unconstrained optimization, large-scale optimization, nonsmooth optimization, interior-point methods, modified Newton methods, variable metric methods, computational experiments, generalized minimax optimization, global convergence


49K35, 90C06, 90C47, 90C51