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Nonmonotone algorithm for minimax optimization problems. (English) Zbl 1215.65114

The paper deals with finite minimax optimization problems \[ \min_{x\in\mathbb{R}^n}\;\max_{1\leq i\leq m}\, f_i(x), \] where all functions \(f_i\) are twice continuously differentiable. Equivalently, the function \(\Phi(x)= \max f_i(x)\) is to minimize. The problem is not necessarily differentiable, but it is possible to transform the task to a differentiable programming problem with side conditions (inequalities). The authors introduce a nonmonotone algorithm for the construction of a minimizing sequence. The algorithm takes not only the advantage of nonmonotone strategy and second-order step but also the advantages of trust-region methods and line search methods. The new algorithm is of strongly global convergence and superlinear convergence. Numerical experiments (10 examples) indicate the efficiency.

MSC:

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C47 Minimax problems in mathematical programming
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