Systems and techniques are described for optimizing a circuit design by using a numerical solver. Some embodiments construct a set of lower bound expressions for a parameter that is used in an approximation of an objective function. Next, the embodiments evaluate the set of lower bound expressions to obtain a set of lower bound values. The embodiments then determine a maximum lower bound value from the set of lower bound values. Next, while solving a gate sizing problem using the numerical solver, the embodiments evaluate the approximate objective function and the partial derivatives of the approximate objective function by using the maximum lower bound value of the parameter. The maximum lower bound value of this parameter determines the accuracy of the approximation of the objective function.