Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance problem, this is a bundle of spectrahedral shadows surrounding the given variety. We characterize the algebraic boundary of this region and we derive a formula for its degree.

Recent years have seen major innovations in cross-layer wireless designs. Despite demonstrating significant throughput gains, hardly any of these technologies have made it into real networks. Deploying cross-layer innovations requires adoption from Wi-Fi chip manufacturers. Yet, manufacturers hesitate to undertake major investments without a better understanding of how these designs interact with real networks and applications.This paper presents the first step towards breaking this stalemate, by enabling the adoption of cross-layer designs in today's networks with commodity Wi-Fi cards and actual applications. We present OpenRF, a cross-layer architecture for managing MIMO signal processing. OpenRF enables access points on the same channel to cancel their interference at each other's clients, while beamforming their signal to their own clients. OpenRF is self-configuring, so that network administrators need not understand MIMO or physical layer techniques.We patch the iwlwifi driver to support OpenRF on off-the-shelf Intel cards. We deploy OpenRF on a 20-node network, showing how it manages the complex interaction of cross-layer design with a real network stack, TCP, bursty traffic, and real applications. Our results demonstrate an average gain of 1.6× for TCP traffic and a significant reduction in response time for real-time applications, like remote desktop.

We study sum of squares (SOS) relaxations to optimize polynomial functions over a set V ∩ R n , where V is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents V with a generic set of complex samples. This approach depends only on the geometry of V, avoiding representation issues such as multiplicity and choice of generators. It also takes advantage of the coordinate ring structure to reduce the size of the corresponding semidefinite program (SDP). In addition, the input can be given as a straight-line program. Our methods are particularly appealing for varieties that are easy to sample from but for which the defining equations are complicated, such as SO(n), Grassmannians or rank k tensors. For arbitrary varieties we can obtain the required samples by using the tools of numerical algebraic geometry. In this way we connect the areas of SOS optimization and numerical algebraic geometry.Problem. Given a bound d ∈ N, a polynomial p(x) and a variety V, find a d-SOS(V) certificate (if it exists).

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