We discuss the general method for obtaining full positivity bounds on multifield effective field theories (EFTs). While the leading order forward positivity bounds are commonly derived from the ...elastic scattering of two (superposed) external states, we show that, for a generic EFT containing three or more low-energy modes, this approach only gives incomplete bounds. We then identify the allowed parameter space as the dual to a spectrahedron, constructed from crossing symmetries of the amplitude, and show that finding the optimal bounds for a given number of modes is equivalent to a geometric problem: finding the extremal rays of a spectrahedron. We show how this is done analytically for simple cases and numerically formulated as semidefinite programming (SDP) problems for more complicated cases. We demonstrate this approach with a number of well-motivated examples in particle physics and cosmology, including EFTs of scalars, vectors, fermions, and gravitons. In all these cases, we find that the SDP approach leads to results that either improve the previous ones or are completely new. We also find that the SDP approach is numerically much more efficient.
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Designing unimodular waveforms with a desired beampattern, spectral occupancy, and orthogonality level is of vital importance in the next-generation multiple-input multiple-output (MIMO) radar ...systems. Motivated by this fact, in this article, we propose a framework for shaping the beampattern in MIMO radar systems under the constraints simultaneously ensuring the unimodularity, desired spectral occupancy, and orthogonality of the designed waveform. In this manner, the proposed framework is the most comprehensive approach for MIMO radar waveform design focusing on beampattern shaping. The problem formulation leads to a nonconvex quadratic fractional programming. We propose an effective iterative to solve the problem, where each iteration is composed of a semidefinite programming followed by eigenvalue decomposition. Some numerical simulations are provided to illustrate the superior performance of our proposed over the state of the art.
Research on received signal strength (RSS) based localization has received a lot of attention from both academia and industry because of its low complexity and high efficiency. In this letter, the ...RSS based localization problem when the transmit power is unknown, is addressed based on the least squared relative error (LSRE) estimation. By taking exponential transformation, the original log-normal RSS measurement model is transformed into a multiplicative model, which is used to formulate a non-convex LSRE estimation problem with the source location and the transmit power as variables. We then apply semidefinite relaxation (SDR) to the non-convex LSRE problem to obtain a convex semidefinite programming (SDP) problem. To facilitate SDR, we introduce two compound variables constructed by the source location and the transmit power, and then estimate the two compound variables instead of directly estimating the source location and the transmit power. The source location estimate is recovered according to its relation with the compound variable. Both simulations and real field test demonstrate the superior performance of the proposed method over several existing methods.
The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products ...{α,−α}, α∈0,1), are called equiangular. The problem of determining the maximum size of s-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an s-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in Rn, n≥7, is n(n+1)2 with possible exceptions for some n=(2k+1)2−3, k∈N. We also prove the universal upper bound ∼23na2 for equiangular sets with α=1a and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.
Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we ...extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush–Kuhn–Tucker (AKKT), well known in nonlinear optimization. The second one, called Trace-AKKT, is more natural in the context of semidefinite programming as the computation of eigenvalues is avoided. We propose an augmented Lagrangian algorithm that generates these types of sequences and new constraint qualifications are proposed, weaker than previously considered ones, which are sufficient for the global convergence of the algorithm to a stationary point.
In a smart grid integrated with vehicle-to-grid (V2G) technique, electric vehicles (EVs) fleets under effective coordination can be considered as a massive aggregated power storage to provide ...frequency regulation service. In this article, we propose a hierarchical system model to jointly optimize power flow routing and V2G scheduling for providing regulation service. First of all, by installing power flow routers (PFRs) inside the power grid, we formulate the problem of optimal power flow (OPF) routing at the grid level. Through the utilization of the semidefinite programming (SDP) relaxation, we can transform the original non-deterministic polynomial-time hard (NP-hard) problem into a convex problem. The tree decomposition method is then used to further reduce the complexity of the system network. After solving the grid-level OPF routing problem, a forecast-based scheduling problem is formulated at the EV level to coordinate EVs by providing the V2G regulation service. To cope with the forecast uncertainties, an online scheduling problem is in turn formulated. In order to solve these problems in a scalable manner, decentralized algorithms are then devised to control the EV schedules. The simulation results show that the devised online scheduling algorithm can outperform the existing algorithms, which is able to flatten the power fluctuations at the buses with EVs attached. Additionally, we show that grid stability issue can be alleviated through the proposed model. Finally, for different power systems, the uses of PFRs can reduce the system power loss in a great manner while providing voltage regulation.
The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce first-order and second-order T-derivatives for the multi-variable real-valued ...function with the tensor T-product. Inspired by an equivalent characterization of a twice continuously T-differentiable multi-variable real-valued function being convex, we present a definition of the T-positive semidefiniteness of third-order symmetric tensors. After that, we extend many properties of positive semidefinite matrices to the case of third-order symmetric tensors. In particular, analogue to the widely used semidefinite programming (SDP for short), we introduce the semidefinite programming over the space of third-order symmetric tensors (T-semidefinite programming or TSDP for short), and provide a way to solve the TSDP problem by converting it into an SDP problem in the complex domain. Furthermore, we give several TSDP examples and especially some preliminary numerical results for two unconstrained polynomial optimization problems. Experiments show that finding the global minimums of polynomials via the TSDP relaxation outperforms the traditional SDP relaxation for the test examples.
In this paper, we investigate the two-way communication between two users assisted by a reconfigurable intelligent surface (RIS). The scheme that two users communicate simultaneously over Rayleigh ...fading channels is considered. The channels between the two users and RIS can either be reciprocal or non-reciprocal. For reciprocal channels, we determine the optimal phases at the RIS to maximize the signal-to-interference-plus-noise ratio (SINR). We then derive exact closed-form expressions for the outage probability and spectral efficiency for single-element RIS. By capitalizing the insights obtained from the single-element analysis, we introduce a gamma approximation to model the product of Rayleigh random variables which is useful for the evaluation of the performance metrics in multiple-element RIS. Asymptotic analysis shows that the outage decreases at <inline-formula> <tex-math notation="LaTeX">\left ({\log (\rho)/\rho }\right)^{L} </tex-math></inline-formula> rate where <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula> is the number of elements, whereas the spectral efficiency increases at <inline-formula> <tex-math notation="LaTeX">\log (\rho) </tex-math></inline-formula> rate at large average SINR <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula>. For non-reciprocal channels, the minimum user SINR is targeted to be maximized. For single-element RIS, closed-form solution is derived whereas for multiple-element RIS the problem turns out to be non-convex. The latter one is solved through semidefinite programming relaxation and a proposed greedy-iterative method, which can achieve higher performance and lower computational complexity, respectively.
The Landscape of the Spiked Tensor Model Arous, Gérard Ben; Mei, Song; Montanari, Andrea ...
Communications on pure and applied mathematics,
November 2019, 2019-11-00, 20191101, Volume:
72, Issue:
11
Journal Article
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