Tractable fitting with convex polynomials via sum-of-squares

Abstract

Abstract-We consider the problem of fitting given data (u1, y1), . . . , (um, ym) where ui ∈ R n and yi ∈ R with a convex polynomial f . A technique to solve this problem using sum of squares polynomials is presented. This technique is extended to enforce convexity of f only on a specified region. Also, an algorithm to fit the convex hull of a set of points with a convex sub-level set of a polynomial is presented. This problem is a natural extension of the problem of finding the minimum volume ellipsoid covering a set. The algorithm, like that for the minimum volume ellipsoid problem, has the property of being invariant to affine coordinate transformations. We generalize this technique to fit arbitrary unions and intersections of polynomial sub-level sets.

NP-hardness of deciding convexity of quartic polynomials and related problems

“…Related to convexity of polynomials, a concept that has attracted recent attention is the algebraic notion of sos-convexity (see Definition 2.4) [22], [24], [25], [3], [29], [10]. This is a powerful sufficient condition for convexity that relies on an appropriately defined sum of squares decomposition of the Hessian matrix, and can be efficiently checked by solving a single semidefinite program.…”

“…In several other problems of practical relevance, we might not just be interested in checking whether a given polynomial is convex, but to parameterize a family of convex polynomials and perhaps search or optimize over them. For example we might be interested in approximating the convex envelope of a complicated nonconvex function with a convex polynomial, or in fitting a convex polynomial to a set of data points with minimum error [29]. Not surprisingly, if testing membership to the set of convex polynomials is hard, searching and optimizing over that set also turns out to be a hard problem.…”

“…In several statistics and clustering problems, we are interested in finding minimum volume convex sets that contain a set of data points in space. This problem can be tackled by searching over the set of quasiconvex polynomials [29]. In economics, quasiconcave functions are prevalent as desirable utility functions [27], [4].…”

A convex polynomial force-motion model for planar sliding: Identification and application

“…Meanwhile, for a polynomial of degree greater than 2 whose hessian matrix ∇ 2 H(F; a) is a function of both F and a, certification of positive semi-definiteness is NP-hard. However, recent progress [17], [18] in sum-ofsquares programming has given powerful semi-definite relaxations of global positiveness certification of polynomials. Specifically, let z be an arbitrary non-zero vector in R 3 and…”

Semidefinite Characterization of Sum-of-Squares Cones in Algebras

“…Magnani et. al [27] consider this problem in the special case when f is a multivariate polynomial, and suggest the following approach: H(x) 0 for every x ∈ S if and only if h(x, y) = y ⊤ H(x)y ≥ 0 for every x ∈ ∆ and y ∈ R n . Since h is also a polynomial, the problem is reduced to POP.…”