A new published paper called “NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems” brings light (or darkness) into a 20 years-old problem related to the whether or not does exist a polynomial time algorithm that can decide if a multivariate polynomial is globally convex. The answer is: NO.
From the news article:
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.