After almost 20 years, math problem falls

Jul 18, 2011 Larry Hardesty, MIT News Office
A convex function (top) is one whose graph slopes everywhere toward its minimum value, whereas a nonconvex function (bottom) may have many basins, or local minima. Image: Amir Ali Ahmadi

Mathematicians and engineers are often concerned with finding the minimum value of a particular mathematical function. That minimum could represent the optimal trade-off between competing criteria — between the surface area, weight and wind resistance of a car’s body design, for instance. In control theory, a minimum might represent a stable state of an electromechanical system, like an airplane in flight or a bipedal robot trying to keep itself balanced. There, the goal of a control algorithm might be to continuously steer the system back toward the minimum.

For complex functions, finding global minima can be very hard. But it’s a lot easier if you know in advance that the function is convex, meaning that the graph of the function slopes everywhere toward the minimum. Convexity is such a useful property that, in 1992, when a major conference on optimization selected the seven most important outstanding problems in the field, one of them was whether the convexity of an arbitrary polynomial function could be efficiently determined.

Almost 20 years later, researchers in MIT’s Laboratory for Information and Decision Systems have finally answered that question. Unfortunately, the answer, which they reported in May with one paper at the Society for Industrial and Applied Mathematics (SIAM) Conference on Optimization, is no. For an arbitrary polynomial function — that is, a function in which variables are raised to integral exponents, such as 13x4 + 7xy2 + yz — determining whether it’s convex is what’s called NP-hard. That means that the most powerful computers in the world couldn’t provide an answer in a reasonable amount of time.

At the same conference, however, MIT Professor of Electrical Engineering and Computer Science Pablo Parrilo and his graduate student Amir Ali Ahmadi, two of the first paper’s four authors, showed that in many cases, a property that can be determined efficiently, known as sum-of-squares convexity, is a viable substitute for convexity. Moreover, they provide an algorithm for determining whether an arbitrary function has that property.

Downhill from here

On the first paper, Parrilo and Ahmadi were joined by John N. Tsitsiklis, the Clarence J. LeBel Professor of Electrical Engineering, and Alex Olshevsky, a former student of Tsitsiklis’ who’s now a postdoc at Princeton University. According to Etienne de Klerk, a professor in the Department of Econometrics and Operations Research at Tilburg University in the Netherlands, the revelation that determining convexity is NP-hard is “not only interesting but quite surprising.”

"If you take any textbook of optimization that we use to teach undergrads, it will typically start, 'Let the complex optimization be given,'" de Klerk says. "All the functions are convex functions." The MIT researchers' work, he adds, will "make you view the world of optimization in a slightly different way."

To get a sense of why convexity is so useful, imagine an airplane in flight, and suppose that you have a function that relates its altitude and speed to the amount of fuel it will consume — naturally, a quantity you’d want to minimize. If the function is convex, its graph — which is three-dimensional — looks like a big bowl, and at the bottom is the combination of altitude and speed that minimizes fuel consumption. All the plane’s control algorithm has to do is find a new altitude and speed that are down the slope of the bowl, and it knows it’s heading in the right direction. But if the function isn’t convex, the graph might look like a mountain range. The minimum value might lie across several peaks and basins, and locating it could be too time consuming to do on the fly.

Squaring off

Of course, the types of functions that real control algorithms have to deal with are much more complex. But that only heightens the advantage of convexity: It guarantees that you can make an informed decision using only limited, local information. “In control theory, there’s been a big shift in recent years toward doing online optimizing,” Parrilo explains. “Now, because we have such big computational power, you can afford controllers that do things that are a lot more complicated. They actually solve optimization problems on the fly.” But, Parrilo says, “if you want to do this, you need some kind of guarantee that the decision that you’re going to take is going to be optimal, or close to optimal, and also that you can do this in a certain period of time.” Convexity provides that guarantee.

Since the circumstances surrounding the operation of an electromechanical system are constantly changing, so are the functions that describe the system’s optimal state. It would be nice to be able to tell in advance whether those functions are convex, but alas, the MIT researchers have shown that that’s not always possible.

But Parrilo and Ahmadi also proved that, for polynomial functions with few variables or small exponents, convexity is the same thing as sum-of-squares convexity, which is easy to check. (“Sum of squares” just means that a polynomial, like x2 – 2xy + y2 + z2, can be rewritten as the sum of expressions raised to the power of two — in this case, (x – y)2 + z2.)

Moreover, Ahmadi points out, in order to prove that sum-of-squares convexity is not always the same thing as convexity, he and Parrilo had to come up with some bizarre counterexamples that are unlikely to arise in most engineering contexts. “If you have few enough variables, say less than five or six, it’s not easy to find examples where it doesn’t work,” Ahmadi says. “So it’s pretty powerful, at least for reasonable problems.”


This story is republished courtesy of MIT News (web.mit.edu/newsoffice/), a popular site that covers news about MIT research, innovation and teaching.

Explore further: Computer scientists can predict the price of Bitcoin

Related Stories

The math of the Rubik's cube

Jun 29, 2011

Last August, 30 years after the Rubik’s cube first appeared, an international team of researchers proved that no matter how scrambled a cube got, it could be solved in no more than 20 moves. Although ...

Streamlined rules for robots

Jun 13, 2011

With the explosion of the Internet and the commoditization of autonomous robots (such as the Roomba) and small sensors (such as the ones in most cell phones), computer scientists have become more and more ...

The kids are alright

May 26, 2011

Children should be seen and not heard... who says? A Philosophy academic at The University of Nottingham is challenging the adage by teaching primary school children to argue properly.

Recommended for you

Tablets, cars drive AT&T wireless gains—not phones

5 hours ago

AT&T says it gained 2 million wireless subscribers in the latest quarter, but most were from non-phone services such as tablets and Internet-connected cars. The company is facing pricing pressure from smaller rivals T-Mobile ...

Twitter looks to weave into more mobile apps

6 hours ago

Twitter on Wednesday set out to weave itself into mobile applications with a free "Fabric" platform to help developers build better programs and make more money.

Blink, point, solve an equation: Introducing PhotoMath

7 hours ago

"Ma, can I go now? My phone did my homework." PhotoMath, from the software development company MicroBlink, will make the student's phone do math homework. Just point the camera towards the mathematical expression, ...

Google unveils app for managing Gmail inboxes

7 hours ago

Google is introducing an application designed to make it easier for its Gmail users to find and manage important information that can often become buried in their inboxes.

User comments : 3

Adjust slider to filter visible comments by rank

Display comments: newest first

Koen
5 / 5 (4) Jul 18, 2011
Great article, not just from the physics perspective, this solution has meaning for engineering as well.
Tayrantula
5 / 5 (2) Jul 18, 2011
Having done my fair share of optimization problems (which, by the way, are awful) and having dealt with sum of squares in Engineering math courses, I find this extremely interesting. Fascinating how the simplest approximations can solve problems otherwise near-impossible
Sonhouse
5 / 5 (1) Jul 18, 2011
I wonder if or when quantum computers become real, whether this is the kind of problem amenable to solution on quantum computational physics? All may not be lost if that is the case.

Even if so, quantum computers would likely at first anyway, to be bound to labs and universities able to set up the tricky conditions for such computations, probably at 1 degree Kelvin or so, which would not find its way to aircraft or spacecraft for a long time.

Of course, first we have to make a quantum computer.....