At those currents the +/-15V supplies can come from discrete charge pumps from a switching power supply. Each one would need a post regulator - unless the rails are not used as a reference, and merely power for the opamps, then you could get away with simple filters. You would need to 'do the math' and simulate them so they can provide sufficient current.
4 AA's, at the end of their charge will only provide ~4V, so an LDO for the 5V rail is out.
Perhaps a boost converter that has an operating voltage higher than 5.5V (often the max for many converters designed for batteries) set to 6-6.5V. This means the converter will have a very low duty cycle because its only 'boosting' 0.5V or so. Then an LDO for the 5V rail, and then 3 stage charge pumps for ~ +/- 17.7V. - thats (6.5 x 3) minus two diode drops for each stage. A Bat54 will be about 0.3V, so losing 0.6V per stage. With 3 stages, and 6.5V in, thats 19.5V out - (3*2*0.3) = 17.7V.
This gives you enough room for a linear regulator.
Of course you could just boost to 9V, and use a voltage doubler for 18V, then an inverter after that to get ~-17V from that. Many integrated (and easy to use!) charge pump devices can operate up to 250kHz, some up into the MHz, but I have always used a linear regulator after these