Optimization problems are everywhere in engineering: Balancing design tradeoffs is an optimization problem, as are scheduling and logistical planning. The theory — and sometimes the implementation — of control systems relies heavily on optimization, and so does machine learning, which has been the basis of most recent advances in artificial intelligence.

This week, at the IEEE Symposium on Foundations of Computer Science, a trio of present and past MIT graduate students won a best-student-paper award for a new “cutting-plane” algorithm, a general-purpose algorithm for solving optimization problems. The algorithm improves on the running time of its most efficient predecessor, and the researchers offer some reason to think that they may have reached the theoretical limit.

But they also present a new method for applying their general algorithm to specific problems, which yields huge efficiency gains — several orders of magnitude.

“What we are trying to do is revive people’s interest in the general problem the algorithm solves,” says Yin-Tat Lee, an MIT graduate student in mathematics and one of the paper’s co-authors. “Previously, people needed to devise different algorithms for each problem, and then they needed to optimize them for a long time. Now we are saying, if for many problems, you have one algorithm, then, in practice, we can try to optimize over one algorithm instead of many algorithms, and we may have a better chance to get faster algorithms for many problems.”

Lee is joined on the paper by Aaron Sidford, who was an MIT graduate student in electrical engineering and computer science when the work was done but is now at Microsoft Research New England, and by Sam Wong, who earned bachelor’s and master’s degrees in math and electrical engineering and computer science at MIT before moving to the University of California at Berkeley for his PhD.

**Inner circle**

Optimization problems are generally framed as trying to find the minimum value of a mathematical function, called a “cost function.” In car design, for example, the cost function might impose penalties for weight and drag but reward legroom and visibility; in an algorithm for object detection, the cost function would reward correct classification of various objects and penalize false positives.

At a very general level, finding the minimum of a cost function can be described as trying to find a small cluster of values amid a much larger set of possibilities. Suppose that the total range of possible values for a cost function is represented by the interior of a circle. In a standard optimization problem, the values clustered around the minimum value would then be represented by a much smaller circle inside of the first one. But you don’t know where it is.

Now pick a point at random inside the bigger circle. In standard optimization problems, it’s generally possible to determine whether that point lies within the smaller circle. If it doesn’t, it’s also possible to draw a line that falls between it and the smaller circle.

Drawing that line cuts off a chunk of the larger circle, eliminating a range of possibilities. With each new random point you pick, you chop off another section of the circle, until you converge on the solution.

If you represent the range of possibilities as a sphere rather than a circle, then you use a plane, rather than a line, to cut some of them off. Hence the name for the technique: the cutting-plane method.

In most real optimization problems, you need a higher-dimensional object than either a circle or a sphere: You need a hypersphere, which you cut with a hyperplane. But the principle remains the same.

**A matter of time**

Theoretical computer scientists measure algorithms’ running times not in seconds or hours, but in the number of operations required, relative to the number of elements being manipulated. With cutting-plane methods, the number of elements is the number of variables in the cost function — the weight of the car, the cost of its materials, drag, legroom, and so on. That’s also the dimension of the hypersphere.

With the best general-purpose cutting-plane method, the time required to select each new point to test was proportional to the number of elements raised to the power 3.373. Sidford, Lee, and Wong get that down to 3.

But they also describe a new way to adapt cutting-plane methods to particular types of optimization problems, with names like submodular minimization, submodular flow, matroid intersection, and semidefinite programming. And in many of those cases, they report dramatic improvements in efficiency, from running times that scale with the fifth or sixth power of the number of variables (n^{5} or n^{6}, in computer science parlance) down to the second or third power (n^{2} or n^{3}).

“This is indeed an astonishing paper,” says Satoru Iwata, a professor of mathematical informatics at the University of Tokyo, who has published widely on the problem of submodular minimization. “For this problem,” he says, “the running time bounds derived with the aid of discrete geometry and combinatorial techniques are by far better than what I could imagine.”

## Comments

M S i N Lund

October 25, 2015

"...pick a point at random inside the bigger circle. In standard

optimization problems, it’s generally possible to determine whether that

point lies within the smaller circle. If it doesn’t, it’s also possible

to draw a line that falls between it and the smaller circle.

Drawing that line cuts off a chunk of the circle..."

How does a line between a point and an area inside a circle, cut off a chunk of the circle?

I understand that it does, IF the line is between two points that are on the edge or outside the circle, but...

...and even if it does, how do you know which of the chunks to cut off?

But then, Im just a simple cave-programmer.

And your mathematics scares and confuses me.

Brandon L

October 25, 2015

brilliant

FredInIT

October 25, 2015

While the maths are way over my head (by several orders of magnitude :), I can grasp the concept of what's being presented here. Now, if this could be adapted provide better optimization to graph traversals, which are also in many instances multi-dimensional cost functions, you have a breakthrough squared.

Nicely written (article author)

Nicely done (paper authors)

FredInIT

Flavio

October 25, 2015

hash it. hash it all. there is not that much of content. people think there is. even in nature fields. submodular flow may have many names. but why to search for a solution if you already have the general solution? if you rethink, it always was renaming works, electric current to 1(bin), 1(bin) to 1(dec), 1(dec) to n(dec), n(dec) to n(dec) = function, and so on.

Mrityunjoy

October 27, 2015

this sounds like stochastic gradient descent. please advise.