A dual clutch transmission apparatus comprising: a first clutch configured to transmit or interrupt torque from a crankshaft to a first main shaft; a second clutch configured to transmit or interrupt torque from the crankshaft to a second main shaft; an output shaft to which the torque on the first main shaft is transmitted through an odd-numbered transmission gear mechanism set as an odd-numbered transmission gear (81, 83, 85, 711, 712, 731), and to which the torque on the second main shaft is transmitted through an even-numbered transmission gear mechanism set as an even-numbered transmission gear (82, 84, 86, 721, 722, 732); a transmission mechanism configured to change an engagement between gears (81, 82, 83, 84, 85, 86, 711, 712, 721, 722, 731, 732) in the odd-numbered transmission gear mechanism and the even-numbered transmission gear mechanism; and a transmission control unit configured to, at start of a drive operation of a vehicle, shift from a clutch on a higher transmission gear side to a clutch on a lower transmission gear side by controlling the two clutches with predetermined odd-numbered and even-numbered transmission gears (81, 82, 83, 84, 85, 86, 711, 712, 721, 722, 731, 732), respectively, in a set state, through the transmission mechanism, wherein, at the start of the drive operation of the vehicle, with the vehicle in a state where a total torque capacity of the two clutches is set to a value necessary for starting the drive operation of the vehicle, the transmission control unit is configured to shift the two clutches by gradually increasing the torque capacity of the clutch on the lower transmission gear side curvilinearly according to the difference in clutch rotational speed between a drive side and a driven side in the clutch on the lower transmission gear side, while at the same time gradually reducing the torque capacity of the clutch on the higher transmission gear side. a first clutch configured to transmit or interrupt torque from a crankshaft to a first main shaft; a second clutch configured to transmit or interrupt torque from the crankshaft to a second main shaft; an output shaft to which the torque on the first main shaft is transmitted through an odd-numbered transmission gear mechanism set as an odd-numbered transmission gear (81, 83, 85, 711, 712, 731), and to which the torque on the second main shaft is transmitted through an even-numbered transmission gear mechanism set as an even-numbered transmission gear (82, 84, 86, 721, 722, 732); a transmission mechanism configured to change an engagement between gears (81, 82, 83, 84, 85, 86, 711, 712, 721, 722, 731, 732) in the odd-numbered transmission gear mechanism and the even-numbered transmission gear mechanism; and a transmission control unit configured to, at start of a drive operation of a vehicle, shift from a clutch on a higher transmission gear side to a clutch on a lower transmission gear side by controlling the two clutches with predetermined odd-numbered and even-numbered transmission gears (81, 82, 83, 84, 85, 86, 711, 712, 721, 722, 731, 732), respectively, in a set state, through the transmission mechanism, wherein, at the start of the drive operation of the vehicle, with the vehicle in a state where a total torque capacity of the two clutches is set to a value necessary for starting the drive operation of the vehicle, the transmission control unit is configured to shift the two clutches by gradually increasing the torque capacity of the clutch on the lower transmission gear side curvilinearly according to the difference in clutch rotational speed between a drive side and a driven side in the clutch on the lower transmission gear side, while at the same time gradually reducing the torque capacity of the clutch on the higher transmission gear side.