A certificate of non-negativity is a way to write a given function so that its non-negativity becomes evident. Certificates of non-negativity are fundamental tools in optimization, and they underlie powerful algorithmic techniques for various types of optimization problems. We propose certificates of non-negativity of polynomials based on copositive polynomials. The certificates we obtain are valid for generic semialgebraic sets and have a fixed small degree, while commonly used sums-of-squares (SOS) certificates are guaranteed to be valid only for compact semialgebraic sets and could have large degree. Optimization over the cone of copositive polynomials is not tractable, but this cone has been well studied. The main benefit of our copositive certificates of non-negativity is their ability to translate results known exclusively for copositive polynomials to more general semialgebraic sets. In particular, we show how to use copositive polynomials to construct structured (e.g., sparse) certificates of non-negativity, even for unstructured semialgebraic sets. Last but not least, copositive certificates can be used to obtain not only hierarchies of tractable lower bounds, but also hierarchies of tractable upper bounds for polynomial optimization problems.

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Recently, non-SOS Positivstellensätze for polynomials on compact semialgebraic sets, following the general form of Schmüdgen's Positivstellensatz, have been derived by appropriately replacing the SOS polynomials with other classes of polynomials. An open question in the literature is how to obtain similar results following the general form of Putinar's Positivstellensatz. Extrapolating the algebraic geometry tools used to obtain this type of result in the SOS case fails to answer this question, because algebraic geometry methods strongly use hallmark properties of the class of SOS polynomials, such as closure under multiplication and closure under composition with other polynomials. In this article, using a new approach, we show the existence of Putinar-type Positivstellensätze that are constructed using non-SOS classes of non-negative polynomials, such as SONC, SDSOS and DSOS polynomials. Even not necessarily non-negative classes of polynomials such as sums of arithmetic-mean/geometric-mean polynomials could be used. Furthermore, we show that these certificates can be written with inherent sparsity characteristics. Such characteristics can be further exploited when the sparsity structure of both the polynomial whose non-negativity is being certified and the polynomials defining the semialgebraic set of interest are known. In contrast with related literature focused on exploiting sparsity in SOS Positivstellensätze, these latter results show how to exploit sparsity in a more general setting in which non-SOS polynomials are used to construct the Positivstellensätze.

We provide new insight into the convergence properties of the Douglas-Rachford algorithm for the problem min x {f (x) + g(x)}, where f and g are convex functions. Our approach relies on and highlights the natural primal-dual symmetry between the above problem and its Fenchel dual min u {f * (u) + g * (u)} where g * (u) := g * (−u). Our main development is to show the linear convergence of the algorithm when a natural error bound condition on the Douglas-Rachford operator holds. We leverage our error bound condition approach to show and estimate the algorithm's linear rate of convergence for three special classes of problems. The first one is when f or g and f * or g * are strongly convex relative to the primal and dual optimal sets respectively. The second one is when f and g are piecewise linear-quadratic functions. The third one is when f and g are the indicator functions of closed convex cones. In all three cases the rate of convergence is determined by a suitable measure of well-posedness of the problem. In the conic case, if the two closed convex cones are a linear subspace L and R n + , we establish the following stronger finite termination result: the Douglas-Rachford algorithm identifies the maximum support sets for L ∩ R n + and L ⊥ ∩ R n + in finitely many steps. Our developments have straightforward extensions to the more general linearly constrained problem min x,y {f (x)+g(y) : Ax+By = b} thereby highlighting a direct and straightforward relationship between the Douglas-Rachford algorithm and the alternating direction method of multipliers (ADMM).