Research Reveals the Optimal Way to Optimize

The archetypal version of this story appeared in Quanta Magazine.

In 1939, upon arriving precocious to his statistic people astatine UC Berkeley, George Dantzig—a first-year postgraduate student—copied 2 problems disconnected the blackboard, reasoning they were a homework assignment. He recovered the homework “harder to bash than usual,” helium would aboriginal recount, and apologized to the prof for taking immoderate other days to implicit it. A fewer weeks later, his prof told him that helium had solved 2 celebrated unfastened problems successful statistics. Dantzig’s enactment would supply the ground for his doctoral dissertation and, decades later, inspiration for the movie Good Will Hunting.

Dantzig received his doctorate successful 1946, conscionable aft World War II, and helium soon became a mathematical advisor to the recently formed US Air Force. As with each modern wars, World War II’s result depended connected the prudent allocation of constricted resources. But dissimilar erstwhile wars, this struggle was genuinely planetary successful scale, and it was won successful ample portion done sheer concern might. The US could simply nutrient much tanks, craft carriers, and bombers than its enemies. Knowing this, the subject was intensely funny successful optimization problems—that is, however to strategically allocate constricted resources successful situations that could impact hundreds oregon thousands of variables.

The Air Force tasked Dantzig with figuring retired caller ways to lick optimization problems specified arsenic these. In response, helium invented the simplex method, an algorithm that drew connected immoderate of the mathematical techniques helium had developed portion solving his blackboard problems astir a decennary before.

Nearly 80 years later, the simplex method is inactive among the astir wide utilized tools erstwhile a logistical oregon supply-chain determination needs to beryllium made nether analyzable constraints. It’s businesslike and it works. “It has ever tally fast, and nobody’s seen it not beryllium fast,” said Sophie Huiberts of the French National Center for Scientific Research (CNRS).

At the aforesaid time, there’s a funny spot that has agelong formed a shadiness implicit Dantzig’s method. In 1972, mathematicians proved that the clip it takes to implicit a task could emergence exponentially with the fig of constraints. So, nary substance however accelerated the method whitethorn beryllium successful practice, theoretical analyses person consistently offered worst-case scenarios that connote it could instrumentality exponentially longer. For the simplex method, “our accepted tools for studying algorithms don’t work,” Huiberts said.

But successful a caller insubstantial that volition beryllium presented successful December astatine the Foundations of Computer Science conference, Huiberts and Eleon Bach, a doctoral pupil astatine the Technical University of Munich, look to person flooded this issue. They’ve made the algorithm faster, and besides provided theoretical reasons wherefore the exponential runtimes that person agelong been feared bash not materialize successful practice. The work, which builds connected a landmark effect from 2001 by Daniel Spielman and Shang-Hua Teng, is “brilliant [and] beautiful,” according to Teng.

“It’s precise awesome method work, which masterfully combines galore of the ideas developed successful erstwhile lines of research, [while adding] immoderate genuinely bully caller method ideas,” said László Végh, a mathematician astatine the University of Bonn who was not progressive successful this effort.

Optimal Geometry

The simplex method was designed to code a people of problems similar this: Suppose a furnishings institution makes armoires, beds, and chairs. Coincidentally, each armoire is 3 times arsenic profitable arsenic each chair, portion each furniture is doubly arsenic profitable. If we wanted to constitute this arsenic an expression, utilizing a, b, and c to correspond the magnitude of furnishings produced, we would accidental that the full nett is proportional to 3a + 2b + c.

To maximize profits, however galore of each point should the institution make? The reply depends connected the constraints it faces. Let’s accidental that the institution tin crook out, astatine most, 50 items per month, truthful a + b + c is little than oregon adjacent to 50. Armoires are harder to make—no much than 20 tin beryllium produced—so a is little than oregon adjacent to 20. Chairs necessitate peculiar wood, and it’s successful constricted supply, truthful c indispensable beryllium little than 24.

The simplex method turns situations similar this—though often involving galore much variables—into a geometry problem. Imagine graphing our constraints for a, b and c successful 3 dimensions. If a is little than oregon adjacent to 20, we tin ideate a level connected a three-dimensional graph that is perpendicular to the a axis, cutting done it astatine a = 20. We would stipulate that our solution indispensable prevarication determination connected oregon beneath that plane. Likewise, we tin make boundaries associated with the different constraints. Combined, these boundaries tin disagreement abstraction into a analyzable three-dimensional signifier called a polyhedron.