Going by UK prices as that was the subject of the article and what I'm familiar with, you can get 1 kg of long grain rice for £0.45 (~$0.60), which contains 3,650 calories. 1kg of lentils is £1.40 ($1.83) and contains 3,530 calories and a healthy amount of proteins, minerals and vitamins. A female needs 2000 calories per day, so for £0.67p ($0.88) you can get 1000 calories from rice and 1000 calories from lentils and meet your minimum requirements, and have some money left over to buy salt, spices and veg.
Other staples such as oats, beans, pasta are at similar prices (1kg of oats can be found as low as £0.75 and contains 3650 calories) and can provide additional nutritional diversity, but there is no question that you can meet your minimum calorie requirements with $1/day, at least in the UK.
This may not be the most appealing diet to some (although spices go a long way into making cheap palatable dishes), but if the alternative is going so hungry you are unable to move without grabbing on to furniture, you cannot afford to complain about the taste of a bland bowl of a rice.
You buy off whatever veg is on-sale (e.g. the Aldi Super 6) - citrus can be less than a pound per kg, so 12 cents can actually get you a whole medium-sized orange (you buy a pack for 50p and eat it over the next few days). Cabbage, broccoli and misc frozen veg are generally more cost-effective, however.
Based upon that diet you'd get 0% of your Vitamin D, 5 % of your vitamin A, and 15% of your calcium. You'd get 4 grams of fat, no sodium, and barely any potassium. You'd be getting 180% of your recommended carbs for the day.
I doubt that would be sustainable over the long term.
You can take supplements or research the other low cost foods that compensate for the deficiencies (spinach, eggs, fish oils, etc.). Do note that the primary goal is to obtain enough calories to avoid suffering from hunger and dizziness, not achieving the most optimal and sustainable diet (although a rice, beans, oat & veg diet is doubtlessly a far better base than the typical low-income western pattern diet of fast food and sugary drinks).
Obviously you'd obviously want to find ways to increase your income in order to have more than a $1/day to spend on food, but you'll never get there if you're starving.
If your budget is $1/day I doubt that supplements are in the cards, and it's obviously not sustainable, as part B of this nutritional plan is "get a better job".
You can do $2 a day pretty easily in my opinion. Basically you want rice (or pasta) and beans to be the bulk of your calories, and anything else just adds flavor. Helps that in America you generally have access to $2/lb boneless chicken breast (and I've gotten drum sticks for 69c a pound before, but those are more of a pain) at one of your grocery stores, and rice and beans are almost free (25c-50c a day), even in smaller bulk sizes such as 5lb bags. Potatoes, carrots, bananas, and apples are examples of fruits and vegetables that are generally incredibly cheap.
Once you get up to $5 a day per person it really feels like you can eat any thing, once you start following a budget. I've met people with budgets around $400 a month for one person, though that usually includes a lot of eating out.
Seasoning is something people seem to miss a lot. Just garlic powder, paprika, chili powder, and of course salt and pepper can bring a dish up to something you could see eating at a restaurant.
Right? It's not even that hard. In Indian cooking the 'tadka' method is used for lentils and beans (also called chonk, chaunce, vaghar, etc.). It is basically heated oil with some spices thrown in and gently fried (not burnt), this tempering is then mixed in with the rest of food. Here is a good example:
If you can't prepare lentils in a way that you'd describe as tasty, then you owe it to yourself to look up any of the thousands of youtube videos or recipes that prepare it a thousand different ways.
You can make some extremely delicious meals with lentils, especially yellow and red split lentils, given the right spices, which are also pretty cheap.
How?