An interior-point trust-region algorithm to solve a finite nonlinear minimax problem

Abstract

In this paper, we introduce an algorithm to solve a nonlinear finite minimax problem. This algorithm is based on converting minimax problems to differentiable optimization problems with constraints. This allows using a slack variable with a penalty method to convert the differentiable optimization problems with constraints to an unconstrained optimization problem with bound-on variables. To solve the unconstrained optimization problem with bound-on variables, Newton’s interior-point method is used. But Newton’s method is local and so it may not converge if the starting point is far away from a stationary point. To guarantee convergence from any starting point to the stationary point, a trust-region globalization strategy is used.

A global convergence theory for the proposed algorithm is introduced under three standard assumptions.

Finally, numerical experiments are reported to indicate that the proposed algorithm performs efficiently in practice