Sparse Semidefinite Programs with Near-Linear Time Complexity

Zhang, Richard Y.; Lavaei, Javad

doi:10.1109/cdc.2018.8619478

Cited by 18 publications

(16 citation statements)

References 76 publications

(143 reference statements)

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“…Inspired by the Goemans-Williamson algorithm for MAX-CUT [19], we prove that projecting the SDP solution onto a random hyperplane recovers a solution to the QCQP with an approximation ratio of π/4. We solve the SDP relaxation using the sparsityexploiting chordal conversion technique of Fukuda et al [20] and the dualization technique recently developed by Zhang and Lavaei [21], [22]. Directly solving the SDP in the complex domain yields significant improvement on runtime, compared to our previous results in the conference version of this paper [1].…”

Section: B Main Resultsmentioning

confidence: 99%

“…. , W j can be found in [21]. The linear operator R k,j : C |Ij |×|Ij | → C |I k |×|I k | is defined to output the overlapping elements of two principal submatrices indexed by I k and I j , given the latter as the argument:…”

Section: B Clique Tree Conversion and Recoverymentioning

confidence: 99%

“…Here, q is the total number of equality constraints in the original problem (21), and {0} q+1 denotes the so-called "equality-constraint cone", whose dual cone is a free variable of dimension ν + 1.…”

Section: Dualizationmentioning

confidence: 99%

“…Combined, we have the following complexity result. Proof: The proof in [21], [22] for real-valued SDPs can be adopted to prove this theorem. First, note that a general-purpose interior-point method solves an order-θ linear conic program posed in the canonical form to -accuracy in O( √ θ log −1 ) iterations.…”

Section: Dualizationmentioning

confidence: 99%

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Large-Scale Traffic Signal Offset Optimization

Ouyang¹,

Zhang

Lavaei

et al. 2020

IEEE Trans. Control Netw. Syst.

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The offset optimization problem seeks to coordinate and synchronize the timing of traffic signals throughout a network in order to enhance traffic flow and reduce stops and delays. Recently, offset optimization was formulated into a continuous optimization problem without integer variables by modeling traffic flow as sinusoidal. In this paper, we present a novel algorithm to solve this new formulation to near-global optimality on a large-scale. Specifically, we solve a convex relaxation of the nonconvex problem using a tree decomposition reduction, and use randomized rounding to recover a near-global solution. We prove that the algorithm always delivers solutions of expected value at least 0.785 times the globally optimal value. Moreover, assuming that the topology of the traffic network is "tree-like", we prove that the algorithm has near-linear time complexity with respect to the number of intersections. These theoretical guarantees are experimentally validated on the Berkeley, Manhattan, and Los Angeles traffic networks. In our numerical results, the empirical time complexity of the algorithm is linear, and the solutions have objectives within 0.99 times the globally optimal value.

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Section: B Main Resultsmentioning

confidence: 99%

Section: B Clique Tree Conversion and Recoverymentioning

confidence: 99%

Section: Dualizationmentioning

confidence: 99%

Section: Dualizationmentioning

confidence: 99%

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Large-Scale Traffic Signal Offset Optimization

Ouyang¹,

Zhang

Lavaei

et al. 2020

IEEE Trans. Control Netw. Syst.

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“…This approach has enabled parallel implementations that can solve larger instances with the use of supercomputers [68]. In a series of works, Zhang and Lavaei have presented SDP algorithms that can properly take advantage of the problem's sparsity [69,70]. Primal-dual operator splitting for SDP…”

Section: Sdp Solution Methodsmentioning

confidence: 99%

Semidefinite Programming via Generalized Proximal Point Algorithm

NETO¹

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I am grateful to my advisor Álvaro Veiga and the professors Alexandre Street and Carlos Kubrusly for their excellent technical assistance and inspiring lectures. I would like to express my gratitude to all members of LAMPS (Laboratory of Applied Mathematical Programming and Statistics) for the daily support and fruitful discussions. I would like to acknowledge the collaboration with Joaquim Garcia, which was essential to the development of this work. A special thanks to Thuener Silva for the support regarding the software development of the SDP solver. Since all code used in this work was developed in the Julia language, I would like to thank all developers involved in this open source project. In particular, I would like to acknowledge the developers of the package JuMP. The use of the domain-specific modeling language for mathematical optimization, JuMP, was fundamental for the development of this work. A special thanks to Benoît Legat for helping with ProxSDP's MOI interface. Finally, I would also like to thank the Brazilian agency CNPq and PUC-Rio university for financial support.

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Robustness Verification of Quantum Classifiers

Guan

Wang

Ying

2021

Lecture Notes in Computer Science

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Several important models of machine learning algorithms have been successfully generalized to the quantum world, with potential speedup to training classical classifiers and applications to data analytics in quantum physics that can be implemented on the near future quantum computers. However, quantum noise is a major obstacle to the practical implementation of quantum machine learning. In this work, we define a formal framework for the robustness verification and analysis of quantum machine learning algorithms against noises. A robust bound is derived and an algorithm is developed to check whether or not a quantum machine learning algorithm is robust with respect to quantum training data. In particular, this algorithm can find adversarial examples during checking. Our approach is implemented on Google’s TensorFlow Quantum and can verify the robustness of quantum machine learning algorithms with respect to a small disturbance of noises, derived from the surrounding environment. The effectiveness of our robust bound and algorithm is confirmed by the experimental results, including quantum bits classification as the “Hello World” example, quantum phase recognition and cluster excitation detection from real world intractable physical problems, and the classification of MNIST from the classical world.

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Sparse Semidefinite Programs with Near-Linear Time Complexity

Cited by 18 publications

References 76 publications

Large-Scale Traffic Signal Offset Optimization

Large-Scale Traffic Signal Offset Optimization

Semidefinite Programming via Generalized Proximal Point Algorithm

Robustness Verification of Quantum Classifiers

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