A classic mathematical problem manifests itself in the real world

A hundred years ago the great mathematician David Hilbert asked a research question of pure mathematics. The recent development of the theory of optimization endure the work of Hilbert in the world of robomobile

Long before robots were able to run, and cars to drive themselves, mathematicians have thought about a simple mathematical question. Finally, they took him and put aside – not being able to know what the object of their mathematical curiosity will manifest itself in the cars of the distant future.

The future has arrived. The result is new work by Amir Ali Ahmadi and Anybody Macumar from Princeton University, the classical problem of pure mathematics, are ready to provide hard proof that automated drones and robomobile will not crash into trees or to steer into the oncoming lane.

"Given complete, 100% provably guarantee that your system will" be able to avoid collisions, said Georgina Hall, a graduate student from Princeton who collaborated with the Ahmadi in this work.

Amir Ali Ahmadi, a Professor from Princeton

Guarantee is taken in an unexpected place – in a mathematical task known as the "sum of squares". Put it in 1900, the year of the great mathematician David Hilbert. He asked whether certain equations can be expressed as the sum of the two individual members, each of which is squared.

Mathematics answered Hilbert's question in a few decades. Then, nearly 90 years later, computer scientists and engineers have discovered that it is a mathematical property – varasemast equations using the sum of squares – helps to solve many real problems that they would like to solve.

"That I do use a lot of classical mathematics of the XIX century, together with a very modern computational mathematics," said Ahmadi.

However, once researchers realized that the sum of squares can help to answer many types of questions they are faced with the problems of this approach. The new work addresses one of the biggest such problems, forcing the old math question to solve some of the most important technological issues of our time.

Positively guaranteed

Which means that a certain value is a sum of squares? Take the number 13. This is the sum of two squares are 22 and 32. The number 34 is equal to the sum of 32 and 52.

Instead of numbers, the question of Hilbert's 17 – th of 23 problemsthat he proposed in the early twentieth century – dealing with polynomials such as 5x2 + 16x + 13. Such polynomials are sometimes also can be represented as a sum of squares. For example, 5x2 + 16x + 13 can be rewritten as (x + 2)2 + (2x + 3)2.

When the expression is a sum of squares, you know that it is always positive (all [rational] numbers squared give a positive number or zero, and the sum of positive numbers is positive). Hilbert wanted to know whether it works in the other direction: can all non-negative polynomials to Express the sum of squares of rational functions. In 1927 the mathematician Emil Artin proved the conjecture of Hilbert was correct.

This relationship turns out to be quite useful. If you are given a complex polynomial, with dozens of variables, raised to the highest degree – is difficult to determine whether it is always non-negative. "Some obviously non-negative polynomials. Difficult to check that they are nonnegative," said Ahmadi.

But once you show that this polynomial can be expressed via the sum of squares, it is nonnegative to become a consequence. "The sum of the squares gives you a beautiful certificate of positivity," said Pablo Parrilo specialist in computer science engineer from mit who participated in the tug-of-war question about the sum of the squares in the applied world.

The knowledge that a certain polynomial is always nonnegative, it may seem a mathematical triviality. But one hundred years after Hilbert asked the question, nonnegative polynomials turned out to be the answer to applied problems that affect all of us.

The best way

The sum of squares meets the real world in the field of optimization. Optimization theory is concerned with finding the best way to do something, while in constraints – for example, to find the best route for the way to work, given the state of the road and stop, which you need to do on the way. Such scenarios often can be reduced to polynomials. In such cases, to solve or optimize the script, finding the minimum value that the polynomial takes.

Finding the minimum of a polynomial with many variables is a difficult task. There is no simple algorithm from the textbook to compute the minimum values of complex polynomials, moreover, they are difficult to build on the chart.

Georgina Hall

Since the minimum value of a polynomial is difficult to compute directly, researchers make assumptions about the value of other methods. It is here that comes into play nonnegative, and the question whether a polynomial a sum of squares. "Guaranteeing the nonnegativity is the essence of all optimization problems," said Rekha Thomas, a mathematician from the University of Washington.

One of the ways to find the minimum value is to ask the question: what is the maximum value you can deduct the of a non-negative polynomial, so it does not become at some point negative? To answer the question you need to check different values – whether it is possible to subtract 3, so it does not become a negative? And 4? And 5? Repeating the procedure at each step you need to know is whether a polynomial is non-negative. Checked it through the possibility of its expression in the form of a sum of squares.

"The question that we must ask is "whether a polynomial nonnegative?" The problem is that with a lot of variables to this question difficult to answer, said Ahmadi. – So we use the sum of squares as a replacement for the nonnegativity of".

As soon as researchers become well-known minimum is the optimal value of the polynomial they can use other methods to determine input parameters that lead to this value. However, for nonnegative helped to solve optimization problems, it is necessary to come up with a way to quickly calculate whether a polynomial is expressed as a sum of squares. And the fact that the researchers were able to do it, gone 100 years since the formulation of the question by Hilbert.

Breaking problem

17th Hilbert's problem has moved from the world of pure mathematics to the application plane at approximately 2000-m to year. That's when several researchers came up with an algorithm verifying, imagine if the polynomial using the sum of squares. They got that way, solving the issue with the squares using "polupriznanie programming"by which computers are able to cope with this task. This made it possible for researchers from areas such as computer science and engineering, to use the possibilities of non-negativity for directing their search towards optimal solutions to problems.

Anirudha Majumdar

But poliproteinov programming has a big limitation: it is slow on large tasks and unable to handle some of the more complex polynomials that are particularly interested in researchers. Polupriznanie programming can be used to decompose the sum of squares of such polynomials, which consist of not more than a dozen variables raised to the power of not more than 6. In the polynomials that describe complex engineering problems – such as a guarantee that the robot will remain standing on my feet, can contain 50 variables or even more.The program can chew a polynomial until the end of time and never to give the sum of squares.

In the work, published online last June, Ahmadi and Majumdar explain how to get around the slow poliproteinov programming. Instead of trying to find the decomposition of the sum of squares in a single, slow programs, and they show how you can do it with solution a sequence of more simple problems, calculate which will be much faster.

The problems of such kind are called "linear" and they were developed in the 1940s for solving optimization problems related to military matters. Linear programs are now well understood and fast. In the new work of Ahmadi and Majumdar show that it is possible to solve multiple related linear programs (or, in some cases, a problem of a different type, the program cone of the second order), and combine the results in order to obtain something almost as good as the answer that could give the program for poliproteinov programming.In the end, the engineers have a new, practical tool that they can use to check for nonnegative and quickly find the decomposition of the sum of squares.

"We have studied several problems of robotics and control theory, and showed that the quality of the obtained solutions is useful for practical use, and that they are much faster to calculate," said Majumdar.

Proof of security

Solution speed is everything, if you are in robomobile. In this situation, the polynomial can be a mathematical barrier installed around obstacles that you don't want to crash – if it can be fast enough to calculate.

Imagine a simple example: robomobile a giant Parking lot. The Parking is nothing but the gate at the far end. Your task is to program a machine so that it never ran into this booth.

In this case, you can pull on the Parking grid. Then make a polynomial as the input of the receiving point on the grid. To ensure that the values of the polynomial at the location of the machine is negative, and the value at the location of the gate is positive.

At some set of points between your vehicle and the box, the polynomial will go from minus to plus. Since your machine can only be where the polynomial is negative, these points will play the role of the wall.

"If you start with a certain place, the robot won't cross the line, behind which is the obstacle. It gives you a formal proof of collision avoidance," said Ahmadi.

Of course, it will be inconvenient if the wall will be half way between the car and the booth. Need to construct a polynomial so that the wall surrounding the obstacle as tightly as possible. This option will protect the booth, and give the car plenty of room for movement.

In practice, it is necessary to minimize the value of the distance between the wall and the booth – so you need to shift the graph of the polynomial and see how far it can push until, until he will move from minus to plus. This is achieved by verifying if the polynomial a sum of squares.

Almost empty Parking lot is one thing. But in realistic scenarios the machine control, its sensors continuously determine a new moving obstacles – cars, bicycles, children. Every time a new obstacle or movement is known, the machine must build a new complex polynomials, enclosing these obstacles. That's a lot of checks on the sum of squares that should be done on the fly.

Seven years ago, another pair of researchers have imagined that these techniques for working with polynomials, it may be possible to use for separation of robomobile and the places where they are not supposed to get. But while the power of computers made this idea just a dream.

A new approach of Ahmadi and Majumdar opens a new way of carrying out such instantaneous calculations. So, if and when robomobile will be able to safely move around the world, we can thank Google and Tesla – as well as Gilbert.