We develop a computationally efficient approximation of the maximum likelihood (ML) detector for 16 quadrature amplitude modulation (16-QAM) in multiple-input multiple-output (MIMO) systems. The detector is based on solving a convex relaxation of the ML problem by a box constrained optimization scheme. Simulation results in a random MIMO system show that this proposed approach outperforms the c...

Semidefinite relaxation methods transform a variety of non-convex optimization problems into convex problems, but square the number of variables. We study a new type of convex relaxation for phase retrieval problems, called PhaseMax, that convexifies the underlying problem without lifting. The resulting problem formulation can be solved using standard convex optimization routines, while still w...

In this paper, a detection strategy based on variable neighborhood search (VNS) and semidefinite relaxation of the CDMA maximum likelihood (ML) is investigated. The VNS method provides a good method for solving the ML problem while keeping the integer constraints. A SDP relaxation is used as an efficient way to generate an initial solution in a limited amount of time, in particular using early ...

We propose an enhanced semidefinite program (SDP) relaxation to enable the tight and efficient verification of neural networks (NNs). The tightness improvement is achieved by introducing a nonlinear constraint existing SDP relaxations previously proposed for NN verification. efficiency proposal stems from iterative nature algorithm in that it solves resulting non-convex recursively solving auxi...

A sensor network localization problem can be formulated as a quadratic optimization problem (QOP). For quadratic optimization problems, semidefinite programming (SDP) relaxation by Lasserre with relaxation order 1 for general polynomial optimization problems (POPs) is known to be equivalent to the sparse SDP relaxation by Waki et al. with relaxation order 1, except the size and sparsity of the ...

Abstract The angular synchronization problem of estimating a set unknown angles from their known noisy pairwise differences arises in various applications. It can be reformulated as an optimization on graphs involving the graph Laplacian matrix. We consider general, weighted version this problem, where impact noise differs between different pairs entries and some are erased completely; for exam...

Determination of a protein’s structure can facilitate an understanding of how the structure changes when that protein combines with other proteins or smaller molecules. In this paper we study a semidefinite programming (SDP) relaxation of the (NP-hard) side chain positioning problem (SCP) presented in Chazelle et al. [4]. We show that the Slater constraint qualification (SCQ) fails for the SDP ...

In the rest of this course we use the proof of Marcus, Spielman and Srivastava to prove an upper bound of polyloglog(n) on the integrality gap of the Held-Karp relaxation for ATSP. The materials will be based on the work of Oveis Gharan and Anari [AO14]. We start by introducing the sparsest cut problem and Cheeger’s inequalities. The ideas that we develop here will be crucially used later. In p...

In this paper, we show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractable semidefinite programming (SDP) relaxation that is tighter than the current SDP relaxation schemes in the literature. In contrast to the existing schemes, our proposed SDP formula...

This report presents a convex relaxation method that globally solves for the camera position and orientation from a set of image pixel measurements associated with a scene of reference points of known 3D positions. The pose optimisation is formulated as a semidefinite positive relaxation program and this approach shows superior performance over existing methods. RELEASE LIMITATION Approved for ...