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We address the problem of finding a local solution to a nonconvex-nonconcaveminmax optimization using Newton type methods, including interior-point ones.We modify the Hessian matrix of these methods such that, at each step, themodified Newton update direction can be seen as the solution to a quadraticprogram that locally approximates the minmax problem. Moreover, we show that byselecting the modification in an appropriate way, the only stable equilibriumpoints of the algorithm's iterations are local minmax points. As a consequence,the algorithm can only converge towards an equilibrium point if such point is alocal minmax, and it will escape if the point is not a local minmax. Usingnumerical examples, we show that the computation time of our algorithm scalesroughly linearly with the number of nonzero elements in the Hessian.

Newton and interior-point methods for (constrained) nonconvex-nonconcave minmax optimization with stability and instability guarantees