Obesity, insulin resistance, diabetes are associated with this PCOS. We also know that in these conditions insulin (a hormone) levels are elevated. By this logic, whatever lowers insulin levels, will also restore fertility in these women.
With nutrition, it is, in fact, possible to lower insulin levels: by avoiding refined carbohydrates and sugar. Biologists around the world know that the availability of food is a requirement for fertility and reproduction. However, if the food type is carbohydrate and sugar, insulin levels go up, which will incite the body to store the calories as fat. The brain can't see this stored energy in the fat tissue, it can only see the available energy in the bloodstream. As there is little energy left in the bloodstream, the body goes into "infertility" mode and stops to cycle (in order to save more energy), until calories are readily available again in the bloodstream.
"This was suggested in the late 1980s, when the reproductive biologists George Wade and Jill Schneider described their research on hamsters [...]. These animals will go into heat whether they are fat or lean, and they will continue to cycle, as long as they can eat as much food as they want. If both fatty acid and glucose oxidation are inhibited, however, and they are not allowed to increase their food intake in response, their estrous cycles stop. They'll remain infertile, whether they are gaining or losing weight at the time."
- Gary Taubes (Good Calories, Bad Calories)
In the case of obese, insulin resistant and diabetic patients, insulin plays the role of starving the organs of fuel, thus putting the body on a slow motion. As a result, the natural reaction of the body is to restrict it's normal operation to a bare minimum: less exercise, more rest and less ovulation. This mechanism is very similar to starvation, our body's reaction to a lack of food is the same.
This is the reason why, we find easily emaciated (especially children) and obese patients in the same population. Obesity is a form of malnutrition and not a form of over-nutrition.