To shorten the luminous flux rise time and prevent melting/deformation of the projections in the initial ignition period of high pressure discharge lamps that have a sealed mercury volume of no less than 0.20 mg/mm.sup.3, a power supply device feeds a current that exceeds the maximum current value of the constant power control period to the high pressure discharge lamp during a first control period from the initial discharge until the lamp voltage reaches a specified lamp voltage and causes a switch to constant power control, and then, during the following period until a change to steady power control, the power supply device feeds it a current that is less than the maximum current during regular operation.