Research Reveals the Optimal Way to Optimize

The original version to This story Featured in Quanta Magazine.

In 1939, arriving late to a statistics course at UC Berkeley, George Dantzig—a first-year graduate student—copied two problems from the blackboard, thinking they were homework. He found the homework “harder than usual,” he later recounted, and apologized to the professor for taking a few extra days to complete it. A few weeks later, his professor told him that he had solved two famous open problems in statistics. Dantzig’s work would provide the basis for his doctoral thesis and, decades later, the inspiration for the film Goodwill fishing.

Dantzig earned his doctorate in 1946, just after World War II, and soon became a sports advisor to the newly formed United States Air Force. As with all modern wars, the outcome of World War II depended on the judicious allocation of limited resources. But unlike previous wars, this conflict was truly global in scope, and was won largely through sheer industrial might. The United States could simply produce more tanks, aircraft carriers, and bombers than its enemies. Recognizing this, the Army has been keenly interested in optimization problems, that is, how to strategically allocate limited resources in situations that can involve hundreds or thousands of variables.

The Air Force tasked Dantzig with discovering new ways to solve optimization problems like this. In response, he invented the simple method, an algorithm based on some mathematical techniques he had developed while solving blackboard problems about a decade earlier.

Nearly 80 years later, the simple method is still among the most widely used tools when a logistics or supply chain decision needs to be made under complex constraints. It’s effective and it works. “It was always going fast, and no one saw that it was not that fast,” said Sophie Huiberts of the French National Center for Scientific Research (CNRS).

At the same time, there is a strange quality that has long cast a shadow over Dantzig’s method. In 1972, mathematicians demonstrated that the time it takes to complete a task can increase dramatically as the number of constraints increases. So, no matter how fast this method seems in practice, theoretical analyzes have consistently presented worst-case scenarios that imply it could take significantly longer. “For a simple approach, our traditional tools for studying algorithms don’t work,” Huiberts said.

But in a new paper to be presented in December at the Foundations of Computer Science conference, Heberts and Elion Bach, a PhD student at the Technical University of Munich, appear to have overcome this problem. They made the algorithm faster, and also provided theoretical reasons why the long-feared exponential runtimes were not achieved in practice. The work, based on a landmark 2001 score by Daniel Spielman and Shang Hua Teng, is considered “outstanding”. [and] Beautiful,” according to Teng.

“It is a very impressive work of art, brilliantly bringing together many of the ideas developed in previous lines of research. [while adding] “We’ve come up with some really cool new technical ideas,” said Laszlo Vig, a mathematician at the University of Bonn, who was not involved in the effort.

Optimal engineering

The simple method is designed to address a class of problems like this: Suppose a furniture company makes dressers, beds, and chairs. Coincidentally, each dresser is three times as profitable as each chair, while each bed is twice as profitable. If we wanted to write this as an expression, using A, forand C To represent the quantity of furniture produced, we can say that the total profit is proportional to 3A + 2for + C.

To maximize profits, how many products of each product should the company produce? The answer depends on the limitations it faces. Let’s assume that the company can produce at most 50 items per month, so A + for + C Less than or equal to 50. The wheel is difficult to make – no more than 20 can be produced – so A Less than or equal to 20. Chairs require special wood, which is in limited supply, so C Must be less than 24.

The simple method turns such situations – although they often involve many variables – into an engineering problem. Imagine a graph of our constraints A, for and C In three dimensions. if A Less than or equal to 20, we can imagine a level on a 3D graph perpendicular to A axle, and cut through it in A = 20. We will stipulate that the solution must lie somewhere at or below this level. Likewise, we can create limits that are linked to other constraints. Combined, these boundaries can divide space into a complex three-dimensional shape called a polyhedron.