This paper presents a new methodology for analyzing multiresponse experiments. The methodology consists of computing confidence regions for the stationary points of quadratic responses and confidence cones for the direction of maximum improvement for linear responses. The stationary points are constrained to lie within the region of experimentation. It is shown that the confidence regions depend on the value of the Lagrange multiplier of the region's constraint. The value of the Lagrange multiplier is found by solving the Karush-Kuhn-Tucker optimality conditions. Then, nonlinear optimization problems are set up and solved for obtaining experimental points that lie inside all the confidence regions, cones and constraints. Robust process design examples illustrate the methods proposed. The examples address the“target is best”and“larger the better”cases.