Fly nonstop from Los Angeles to Fiji on Fiji Airways for just $795 round trip across a wide range of dates throughout the remainder of 2017.
Better yet, I believe that these tickets can be purchased as Qantas tickets even though they are operated by Fiji Airways, which I believe will allow you to credit them to Alaska Airlines for 100% mileage credit. A round trip could then possibly earn ~11,000 Alaska miles if I am correct.
I stand corrected! As reader Jose points out, these Qantas flight numbers will NOT accrue Alaska miles. That means you would want to book them as Fiji Airways flight numbers and credit to Alaska in order to receive 50% of the flown miles for ~5,500 Alaska miles earned.
At any rate, $795 round trip direct to Fiji is a solid deal and I am finding some availability from March through December (except in July), with particularly wide open availability in September and October:
Pricing out the same flights on the Qantas website came up around $1400 for me, so I suggest going with your favorite OTA.