Optimization problems arising from engineering design problems often involve the solution of one or several constrained minimax optimization problems. It is sometimes crucial that all iterates constructed when solving such problems satisfy a given set of 'hard' inequality constraints, and generally desirable that the (maximum) objective function value improve at each iteration. In this paper, we propose an algorithm of the sequential quadratic programming (SQP) type that enjoys such properties. This algorithm is inspired from an algorithm recently proposed for the solution of single objective constrained optimization problems. Preliminary numerical results are very promising.