A new inexact-restoration method for nonlinear programming is introduced. The iteration of the main algorithm has two phases. In Phase 1, feasibility is improved explicitly; in Phase 2, optimality is improved on a tangent approximation of the constraints. Trust regions are used for reducing the step when the trial point is not good enough. The trust region is not centered in the current point, as in many nonlinear programming algorithms, but in the intermediate more feasible point. Therefore, in this semifeasible approach, the more feasible intermediate point is considered to be essentially better than the current point. This is the first method in which intermediate-point-centered trust regions are combined with the decrease of the Lagrangian in the tangent approximation to the constraints. The merit function used in this paper is also new: it consists of a convex combination of the Lagrangian and the nonsquared norm of the constraints. The Euclidean norm is used for simplicity, but other norms for measuring infeasibility are admissible. Global convergence theorems are proved, a theoretically justified algorithm for the first phase is introduced, and some numerical insight is given.11113958