Is it possible to power on/off a monitor using the computer?

I want my computer to power off my monitors (not standby mode) when I don't use it (no keyboard/mouse input) for more than 15 minutes, and power them back when such input is received.

What you want is exactly what you are trying to avoid.

If the monitor is actually, fully off, then it can’t turn back on from a signal on the video cable. To allow it to turn on like that, there needs to be some sort of circuit in the monitor that remains on and active to watch for the signal. Monitors already have such a circuit, but turning them off turns that circuit off as well.

To use that circuit, you need to leave some electricity in the monitor, and that is exactly what standby does: it turns the display (and speakers, and everything else) off while leaving that one small circuit active.

With most modern monitors, there is essentially no difference between standby mode and fully off other than a tiny low-voltage trickle in that circuit and the LED on the front.

I have to pay for electricity, so we always avoid using electricity as much as is humanly possible, yet, I leave it in standby when I am using the computer and need to step away for a while (I turn both off when I am done for the day).

Instead of letting the monitor remain on for 15 minutes for nothing, your best best is to do what I do and simply get into the habit of either turning the monitor off whenever you get up to step away, or to manually put it into standby mode. What I do is to use the AutoHotkey script below (can be compiled to an executble that runs in the background if desired) to let me press ⊞ Win+M to sleep the monitor whenever I get up. Other options include using a shortcut or program, using a script or program to do it with a mouse-cursor hot-corner, or even just reducing the timeout from 15 minutes to five or so.

;Monitor Standby Hotkey
;⊞ Win + M puts monitor in standby
#m::
  Sleep 1000 ; Pause for 1sec to prevent un-sleeping when key released

  SendMessage, 0x112, 0xF170, 2,, Program Manager
  ; 0x112 is WM_SYSCOMMAND, 0xF170 is SC_MONITORPOWER
  ; Use  1 in place of 2 to activate the monitor's low-power mode
  ; Use -1 in place of 2 to turn the monitor on
return

The HDMI-CEC specification, which allows HDMI-connected devices to act as a limited remote control any other connected CEC-enabled device, defines the command System Standby which would allow the "remote" to switch the target into standby mode.

This is the closest to what you have in mind that I know of. It requires that the target HDMI device (your monitor) be wired for CEC (which is mandatory under the spec, so it is) and that the device implement CEC in general and that command specifically (which is totally optional and not very common -- yet). It's pretty unlikely that your monitor implements CEC but may be possible to confirm from the manual or published spec, or perhaps easier to simply test by connecting a device via HDMI that sends CEC commands and seeing if your monitor responds.

Yes, but you shouldn't. I think Synetech sums it up well, but if you must:

Exactly how depends on the screen, but if it has an IR receiver you can connect an IR blaster to your PC to signal shut off (though as Synetech pointed out, if the IR is working, power is still being drawn).

Alternatively you could use an Arduino or similar device to actually switch off the power to your monitor. This would take some work, and some DIY skills - but it is certainly a possibility.