A class of algorithms for large scale nonlinear minimax optimization
The author wishes to solve the nonlinear minimax problem stated as follows: (P) min F(x) subject to \(x\in X=\{x:\) \(Ax=b\), \(x\geq 0\}\), where \(F(x)=\max \{f_ i(x):\) \(i=1,2,...,m\}\) and \(f_ i\) are continuously differentiable functions defined on X. Of particular interest is the large scale case, that is, problems where the number of functions \(f_ i\) is large. Sequential linear programming and sequential quadratic programming based algorithms are often used to solve such nonlinear minimax problems. However, one of the major difficulties with such algorithms is that each iteration requires the solution of a large scale mathematical program with at least as many constraints as there are functions defining the 'max' function. The author attempts to overcome this difficulty and presents a class of sequential linear programming and sequential quadratic programming based algorithms for (P), that require, at each iteration, only a small fraction of the functions \(f_ i\). He provides proof for global convergence of this class of algorithms.
scientific article; zbMATH DE number 3958601