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Guesswork With Quantum Side Information

IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第1期68卷 322-338页

作者： Hanson, Eric P. Katariya, Vishal Datta, Nilanjana Wilde, Mark M. Univ Cambridge Dept Appl Math & Theoret Phys Cambridge CB3 0WA England Louisiana State Univ Dept Phys & Astron Hearne Inst Theoret Phys Baton Rouge LA 70803 USA Louisiana State Univ Ctr Computat & Technol Baton Rouge LA 70803 USA

What is the minimum number of guesses needed on average to guess a realization of a random variable correctly? The answer to this question led to the introduction of a quantity called guesswork by Massey in 1994, which can be viewed as an alternate security criterion to entropy. In this paper, we consider the guesswork in the presence of quantum side information, and show that a general sequential guessing strategy is equivalent to performing a single quantum measurement and choosing a guessing strategy based on the outcome. We use this result to deduce entropic one-shot and asymptotic bounds on the guesswork in the presence of quantum side information, and to formulate a semi-definite program (SDP) to calculate the quantity. We evaluate the guesswork for a simple example involving the BB84 states, both numerically and analytically, and we prove a continuity result that certifies the security of slightly imperfect key states when the guesswork is used as the security criterion.

关键词： Entropy Random variables Security Protocols Task analysis Password Complexity theory Guesswork guesswork with quantum side information quantum cryptography quantum key distribution semidefinite program

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On the tightness of SDP relaxations of QCQPs

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MATHEMATICAL programMING 2022年第1期193卷 33-73页

作者： Wang, Alex L. Kilinc-Karzan, Fatma Carnegie Mellon Univ Pittsburgh PA 15213 USA

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

关键词： Quadratically constrained quadratic programming semidefinite program Relaxation Lagrange function Convex hull

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Multivariate Rational Inattention

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ECONOMETRICA 2022年第2期90卷 907-945页

作者： Miao, Jianjun Wu, Jieran Young, Eric R. Boston Univ Dept Econ Boston MA 02215 USA Zhejiang Univ Acad Financial Res Hangzhou Peoples R China Zhejiang Univ Coll Econ Hangzhou Peoples R China Univ Virginia Dept Econ Charlottesville VA USA Fed Reserve Bank Cleveland Res Dept Cleveland OH USA

We study optimal control problems in the multivariate linear-quadratic-Gaussian framework under rational inattention. We propose a three-step procedure to solve this problem using semidefinite programming and derive the optimal signal structure without strong prior restrictions. We analyze both the transition dynamics of the optimal posterior covariance matrix and its steady state. We characterize the optimal information structure for some special cases and develop numerical algorithms for general cases. Applying our methods to solve three multivariate economic models, we obtain some results qualitatively different from the literature.

关键词： Rational inattention endogenous information choice tracking problem optimal control entropy semidefinite program

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Phase Retrieval by Binary Questions: Which Complementary Subspace is Closer?

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CONSTRUCTIVE APPROXIMATION 2022年第1期56卷 1-33页

作者： Domel-White, Dylan Bodmann, Bernhard G. Vanderbilt Univ 221 Kirkland Hall Nashville TN 37235 USA Univ Houston Houston TX 77004 USA

Phase retrieval in real or complex Hilbert spaces is the task of recovering a vector, up to an overall unimodular multiplicative constant, from magnitudes of linear measurements. In this paper, we assume that the vector is normalized, but retain only qualitative, binary information about the measured magnitudes by comparing them with a threshold. In more specific, geometric terms, we choose a sequence of subspaces in a real or complex Hilbert space and only record whether a given vector is closer to the subspace than to the complementary subspace. The subspaces have half the dimension of the Hilbert space and are independent, uniformly distributed with respect to the action of the orthogonal or unitary groups. The main goal of this paper is to find a feasible algorithm for approximate recovery based on the information gained about the vector from these binary questions and to establish error bounds for its approximate recovery. We provide a pointwise bound for fixed input vectors and a uniform bound that controls the worst-case scenario among all inputs. Both bounds hold with high probability with respect to the choice of the subspaces. For real or complex vectors of dimension n, the pointwise bound requires m >= C delta(-2)n log(n) and the uniform bound m >= C delta(-2) n(2) log(delta(-1)n) binary questions in order to achieve an accuracy of delta. The accuracy delta is measured by the operator norm of the difference between the rank-one orthogonal projections corresponding to the normalized input vector and its approximate recovery.

关键词： Phase retrieval Coarse quantization semidefinite program Measure concentration

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On Minimizing the Energy of a Spherical Graph Representation 31st

On Minimizing the Energy of a Spherical Graph Representation

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31st International Symposium on Graph Drawing and Network Visualization (GD)

作者： DeVos, Matt Rogers, Danielle Wesolek, Alexandra Simon Fraser Univ Burnaby BC V5A 1S6 Canada

ISBN: (纸本)9783031492747;9783031492754

Graph representations are the generalization of geometric graph drawings from the plane to higher dimensions. A method introduced by Tutte to optimize properties of graph drawings is to minimize their energy. We explore this minimization for spherical graph representations, where the vertices lie on a unit sphere such that the origin is their barycentre. We present a primal and dual semidefinite program which can be used to find such a spherical graph representation minimizing the energy. We denote the optimal value of this program by rho(G) for a given graph G. The value turns out to be related to the second largest eigen-value of the adjacency matrix of G, which we denote by lambda(2). We show that for G regular, rho(G) <= lambda(2)/2 center dot v(G), and that equality holds if and only if the lambda(2) eigenspace contains a spherical 1-design. Moreover, if G is a random d-regular graph, rho(G) = (root(d - 1) + o(1)center dot v(G), asymptotically almost surely.

关键词： Graph representation Energy semidefinite program

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High-speed batch processing of semidefinite programs with feedforward neural networks

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NEW JOURNAL OF PHYSICS 2021年第10期23卷

作者： Krivachy, Tamas Cai, Yu Bowles, Joseph Cavalcanti, Daniel Brunner, Nicolas Univ Geneva Dept Appl Phys CH-1211 Geneva Switzerland Barcelona Inst Sci & Technol ICFO Inst Ciencies Foton E-08860 Barcelona Spain

semidefinite programming is an important optimization task, often used in time-sensitive applications. Though they are solvable in polynomial time, in practice they can be too slow to be used in online, i.e. real-time applications. Here we propose to solve feasibility semidefinite programs using artificial neural networks (NNs). Given the optimization constraints as an input, a NN outputs values for the optimization parameters such that the constraints are satisfied, both for the primal and the dual formulations of the task. We train the network without having to exactly solve the semidefinite program even once, thus avoiding the possibly time-consuming task of having to generate many training samples with conventional solvers. The NN method is only inconclusive if both the primal and dual models fail to provide feasible solutions. Otherwise we always obtain a certificate, which guarantees false positives to be excluded. As a proof-of-principle demonstration, we examine the performance of the method on a hierarchy of quantum information tasks, the Navascues-Pironio-Acin hierarchy applied to the Bell scenario. We demonstrate that the trained NN gives decent accuracy, while showing orders of magnitude increase in speed compared to a traditional solver, once trained. The network does not have to be retrained for similarly structured problems, giving the promise for being a fast solver for these types of problems.

关键词： semidefinite program neural network optimization quantum algorithm machine learning Bell nonlocality

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Moment and Polynomial Optimization

Moment and Polynomial Optimization

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丛书名： MOS-SIAM Series on Optimization

2023年

作者： Jiawang Nie

Moment and polynomial optimization is an active research field. It is used to solve difficult questions in many areas, including global optimization, tensor computation, saddle points, Nash equilibrium, and bilevel pr... 详细信息

ISBN: (数字)9781611977608

ISBN: (纸本)9781611977592

Moment and Polynomial Optimization synthesizes current research and applications to provide the reader with

a systematic introduction to theory and methods,
a comprehensive approach for extracting optimizers and solving truncated moment problems, and
a creative methodology for using optimality conditions to construct tight Moment-SOS relaxations.

关键词： polynomial moment optimization semidefinite program sum-of-squares convex relaxation flat truncation variety ideal quadratic module preordering Positivsellensatz Lagrange multiplier expression nonnegative polynomial positive semidefinite

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Optimizing power system wide-area damping controllers under communication time delays and model reduction errors

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Smart Power & Energy Security 2025年第1期1卷 36-48页

作者： Chao Duan Chongtao Li Wanliang Fang Wei Yao Chuanke Zhang Lin Jiang Department of Electrical Engineering Xi'an Jiaotong UniversityXi'an710049China School of Electrical and Electronic Engineering Huazhong University of Science and TechnologyWuhan430074China School of Automation China University of GeosciencesWuhan430074China Department of Electrical Engineering and Electronics University of LiverpoolLiverpoolL693BXMerseysideUK

This paper tackles two pivotal challenges within Lyapunov-functional-based approaches to analyze and design power system damping control amidst communication time *** first challenge addresses the inadequacy of the reduced-order system model commonly employed in time-domain methods to accurately capture the stability of the original full-order *** harnessing dissipativity theory,we introduce a model reduction technique coupled with a condition guaranteeing e-exponential stability solely through the employment of the reduced closed-loop system *** not only validates the use of model reduction strategies but also furnishes a theoretical underpinning for their *** second challenge pertains to the computational complexity of nonlinear semidefinite programming problems encountered during the coordinated design of *** than pursuing the global optimum,we advocate for a path-following methodology to systematically explore local *** iterative algorithm enhances system damping and H 2 performance by orchestrating automatic and co-ordinated adjustments to control *** experiments conducted on various benchmark systems underscore the efficacy of the proposed approach.

关键词： Power system damping control Model reduction Path-following method Dissipativity theory semidefinite program

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Deep Reinforcement Learning for Demand Response in Distribution Networks

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IEEE TRANSACTIONS ON SMART GRID 2021年第2期12卷 1496-1506页

作者： Bahrami, Shahab Chen, Yu Christine Wong, Vincent W. S. Univ British Columbia Dept Elect & Comp Engn Vancouver BC V6T 1Z4 Canada

Load aggregators can use demand response programs to motivate residential users toward reducing electricity demand during peak time periods. This article proposes a demand response algorithm for residential users, while accounting for uncertainties in the load demand and electricity price, users' privacy concerns, and power flow constraints imposed by the distribution network. To address the uncertainty issues, we develop a deep reinforcement learning (DRL) algorithm using an actor-critic method. We apply federated learning to enable users to determine the neural network parameters in a decentralized fashion without sharing private information (e.g., load demand, users' potential discomfort due to load scheduling). To tackle the nonconvex power flow constraints, we apply convex relaxation and transform the problem of updating the neural network parameters into a sequence of semidefinite programs (SDPs). Simulations on an IEEE 33-bus test feeder with 32 households show that the proposed demand response algorithm can reduce the peak load by 33% and the expected cost of each user by 13%. Also, we demonstrate the scalability of the proposed algorithm in 330-bus and 1650-bus feeders with real-time pricing scheme.

关键词： Load management Load flow control Distribution networks Pricing Load modeling Reinforcement learning Uncertainty Demand response deep reinforcement learning federated learning power flow semidefinite program

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Convex Relaxation Methods for Community Detection

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STATISTICAL SCIENCE 2021年第1期36卷 2-15页

作者： Li, Xiaodong Chen, Yudong Xu, Jiaming Univ Calif Davis Dept Stat 4232 Math Sci Bldg399 Crocker Lane Davis CA 95616 USA Cornell Univ Sch Operat Res & Informat Engn Ithaca NY 14853 USA Duke Univ Fuqua Sch Business Durham NC 27708 USA

This paper surveys recent theoretical advances in convex optimization approaches for community detection. We introduce some important theoretical techniques and results for establishing the consistency of convex community detection under various statistical models. In particular, we discuss the basic techniques based on the primal and dual analysis. We also present results that demonstrate several distinctive advantages of convex community detection, including robustness against outlier nodes, consistency under weak assortativity, and adaptivity to heterogeneous degrees. This survey is not intended to be a complete overview of the vast literature on this fast-growing topic. Instead, we aim to provide a big picture of the remarkable recent development in this area and to make the survey accessible to a broad audience. We hope that this expository article can serve as an introductory guide for readers who are interested in using, designing, and analyzing convex relaxation methods in network analysis.

关键词： Community detection semidefinite program strong consistency weak consistency assortativity degree correction robustness

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