A second order directional derivative and optimality conditions in scalar and Pareto optimization

A printing apparatus has a manual control mode. The manual control mode has a proportion indication, a proportion adjustment key, a selected range adjustment key, and region selection keys. In the manual control mode, an operator selectively touches the region selection keys to select a region in which an ink key opening is to be changed. Next, the operator touches the selected range adjustment key to change the ink key opening of the selected region. The ink key opening is increased or decreased by the percentage indicated at the proportion indication the number of times the selected range adjustment key is touched. The settings made in the manual control mode are stored in a storage section, and are used to control the amount of ink supply during automatic control. Thus, the printing apparatus having the function of automatically controlling the printed density can make partial tone changes.