It's a result that's not just of interest to mathematicians, but also to engineers and anyone else in the business of optimising things. But luckily, the team have also found a way around this problem that works in most real-life situations.
An Algorithmic Theory of Numbers Graphs and Convexity Information
A continuous function is convex if the area above its graph is a convex set, in other words if the straight line that connects any two points on its graph lies above the bit of the graph between these two points. More formally, a function is convex if for all points and and for all with we have. A function is concave if is convex. These definitions also work for functions of several variables, that is functions , as long as f is defined on the points in question. In this case should be understood as a vector made up of the variables involved.
The graph of a convex function of two variables top and the graph of a non-convex function of two variables bottom.
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But why should we care if a function is convex? Many real-life problems involve minimising something. If your function is convex, however, the job becomes a lot easier.
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Convex functions only have one minimal value. For convex functions of one or two variables, this can be seen from their graphs, which are shaped like a trough with the minimum value sitting at the bottom of the trough. A non-convex function, by contrast, can have many local minima, making its graph look like a mountain range.
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For this and other reasons too, knowing whether a given function is convex is extremely useful. The question of how quickly a computer can check if a given function is convex has been deemed one of the seven most important ones in the field of optimisation. Parrilo, and John N. Tsitsiklis in their paper. The team only looked at a class of functions that are relatively easy to deal with, namely polynomials. These functions are sums of several terms, where each term is a product of the variables, each raised to some power, and then multiplied by a constant, for example:.
The degree of each individual term is the sum of the powers of the variables, and the degree of the whole polynomial is the maximum of the degrees of the individual terms.
In their paper the researchers only looked at polynomials with an even degree, since checking for convexity in the odd degree case is easy. Clearly the problem gets harder the more terms there are in the polynomial, so we expect the answer to depend on the number of terms, which in turn is related to the number of variables. In complexity theory the "time" it takes to solve a problem is measured in terms of the number of steps a computer needs to make to complete the task. This depends on how many components the problem has to start with.
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There are steps in this algorithm, so we say that its execution time is proportional to. Other problems may have an execution time proportional to , , or for some other positive number. This of course means that the execution time increases quite rapidly as , the size of the initial problem, gets bigger. ISSN: X. Discrete Applied Mathematics. Editor-in-Chief: Endre Boros. View Editorial Board.
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