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作者： Venkat, Prayaag Harvard University

学位级别：Ph.D., Doctor of Philosophy

The goal of this thesis to contribute towards a computational complexity theory of statistical inference problems. In recent years, researchers have built evidence in favor of an emerging hypothesis that the class of semi-definite programming (SDP) algorithms is optimal among for computationally efficient algorithms for a certain family of estimation problems. In this thesis, we present four main research efforts that refine this hypothesis and initiate preliminary efforts to go beyond it: • Optimal algorithms for private and robust estimation: We give the first polynomial-time algorithms for privately and robustly estimating a Gaussian distribution with optimal dependence on the dimension in the sample complexity. This adds the fundamental problem of private statistical estimation to a growing list of problems for which SDPs are optimal among polynomial-time algorithms. • Limitations of SDPs: Given independent standard Gaussian points in dimension d, for what values of (n, d) does there exist with high probability an origin-symmetric ellipsoid that simultaneously passes through all of the points? Based on strong numerical evidence, it was conjectured that the ellipsoid fitting problem transitions from feasible to infeasible as the number of points n increases, with a sharp threshold at n ∼ d 2/4; we resolve this conjecture up to logarithmic factors. A corollary of this result is that a canonical SDP-based algorithm fails to successfully solve inference problems involving low-rank matrix decompositions, independent component analysis, and principal component analysis. • New algorithms for discrepancy certification: We initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices, which has connections to conjecturally-hard average-case problems such as negatively-spiked PCA, the number-balancing problem and refuting random constraint satisfaction problems. We give the first polynomial-time algorithms with non-trivial

关键词： Computational complexity semi-definite programming Polynomial-time algorithms Practical algorithms Gaussian distribution

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Minimal compact operators, subdifferential of the maximum eigenvalue and semi-definite programming

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LINEAR ALGEBRA AND ITS APPLICATIONS 2025年 716卷 1-31页

作者： Bottazzi, Tamara Varela, Alejandro Univ Nacl Rio Negro CITECCA Sede Andina RA-8400 San Carlos De Bariloche Argentina Consejo Nacl Invest Cient & Tecn RA-1425 Buenos Aires Argentina Inst Argentino Matemat Alberto P Calderon Saavedra 15 3er PisoC1083ACA Buenos Aires Argentina Univ Nacl Gral Sarmiento Inst Ciencias JM Gutierrez 1150B1613GSX RA-B1613GSX Los Polvorines Argentina

We formulate the issue of minimality of self-adjoint operators on a complex Hilbert space as a semi-definite problem, linking the work by Overton in [18] to the characterization of minimal hermitian matrices. This motivates us to investigate the relationship between minimal self-adjoint operators and the subdifferential of the maximum eigenvalue, initially for matrices and subsequently for compact operators. In order to do it we obtain new formulas of subdifferentials of maximum eigenvalues of compact operators that become useful in these optimization problems. Additionally, we provide formulas for the minimizing diagonals of rank one self-adjoint operators, a result that might be applied for numerical large-scale eigenvalue optimization. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Minimal operators Subdifferential of eigenvalues Moment of a subspace semi-definite programming

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Affinely adjustable robust optimization for radiation therapy under evolving data uncertainty via semi-definite programming

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OPTIMIZATION 2024年第6期73卷 1807-1832页

作者： Jeyakumar, V. Li, G. Woolnough, D. Wu, H. Univ New South Wales Dept Appl Math Sydney Australia

Static robust optimization has played an important role in radiotherapy, where the decisions aim to safeguard against all possible realizations of uncertainty. However, it may lead to overly conservative decisions or too expensive treatment plans, such as delivering significantly more dose than necessary. Motivated by the success of adjustable robust optimization in reducing highly conservative decision-making of static robust optimization in applications, in this paper, we present an affinely adjustable robust optimization (AARO) model for hypoxia-based radiation treatment planning in the face of evolving data uncertainty. We establish an exact semi-definite program reformulation of the model under a so-called affine decision rule and evaluate our model and approach on a liver cancer case as a proof-of-concept. Our AARO model incorporates uncertainties both in dose influence matrix and re-oxygenation data as well as inexactness of the revealed (re-oxygenation) data. Our numerical experiments demonstrate that the adjustable model successfully handles uncertainty in both re-oxygenation and the dose matrix. They also show that, by utilizing information halfway through the treatment plan, the adjustable solutions of the AARO model outperform a static method while maintaining a similar total dose.

关键词： Affinely adjustable robust optimization semi-definite programming radiation therapy optimization evolving uncertainty

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An accelerated primal-dual method for semi-definite programming relaxation of optimal power flow

IET ENERGY SYSTEMS INTEGRATION

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IET ENERGY SYSTEMS INTEGRATION 2023年第4期5卷 477-490页

作者： Shi, Zhan Wang, Xinying Yan, Dong Chen, Sheng Lin, Zhenwei Xia, Jingfan Deng, Qi China Elect Power Res Inst Artificial Intelligence Applicat Res Dept Beijing Peoples R China Cardinal Operat Beijing Co Ltd Res & Dev Dept Beijing Peoples R China Shanghai Univ Finance & Econ Sch Informat Management & Engn Shanghai Peoples R China China Elect Power Res Inst Beijing 100192 Peoples R China

The application of a semi-definite programming (SDP) approach to the Alternating Current Optimal Power Flow problem has attracted significant attention in recent years. However, the SDP relaxation of optimal power flow (OPF) can be computationally intensive and lead to memory issues when dealing with large-scale power systems. To overcome these challenges, we have developed APD-SDP, an optimisation solver based on a first-order primal-dual algorithm. This framework incorporates various acceleration techniques, such as rescaling, step size decay and reset, adaptive line search, and restart, to improve efficiency. To further speed up computations, we have developed a customised eigenvalue decomposition component by exploiting the 3 x 3 block structure in the dual SDP formulation. Experimental results demonstrate that APD-SDP outperforms other commercial and open-source SDP solvers on large-scale and high-dimensional PGLib-OPF datasets.

关键词： convex relaxation optimal power flow power system primal-dual algorithm semi-definite programming

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Universal prediction band via semi-definite programming

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JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY 2022年第4期84卷 1558-1580页

作者： Liang, Tengyuan Univ Chicago Booth Sch Business Chicago IL 60637 USA

We propose a computationally efficient method to construct nonparametric, heteroscedastic prediction bands for uncertainty quantification, with or without any user-specified predictive model. Our approach provides an alternative to the now-standard conformal prediction for uncertainty quantification, with novel theoretical insights and computational advantages. The data-adaptive prediction band is universally applicable with minimal distributional assumptions, has strong non-asymptotic coverage properties, and is easy to implement using standard convex programs. Our approach can be viewed as a novel variance interpolation with confidence and further leverages techniques from semi-definite programming and sum-of-squares optimization. Theoretical and numerical performances for the proposed approach for uncertainty quantification are analysed.

关键词： heteroscedasticity nonparametric prediction band semi-definite programming sum-of-squares uncertainty quantification

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Optimal Power Flow for AC Power System based on Convex Relaxtion with semi-definite programming 8

Optimal Power Flow for AC Power System based on Convex Relax...

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8th International Conference on Power and Renewable Energy, ICPRE 2023

作者： Sang, Da Zhu, Yaoliang Qian, Chen Chen, Yuxuan State Grid Corporation of China East China Branch Shanghai China Economic Research Institute State Grid Shanghai Municipal Electric Power Company Shanghai China Xi'An Jiaotong University School of Electrical Engineering Xi'an China

ISBN: (纸本)9798350328813

In this paper, the classical AC-OPF model is firstly introduced, and then it is relaxed into semi-definite (SD) OPF and second-order cone (SOC) OPF by the convex relaxation technique. Linear, SOC and SD OPF are compared in the example analysis. The high accuracy of SD OPF is verified. Compared with the classical AC OPF, the SD OPF model does not have the rank-one constraint, so the optimal solution may be outside the feasible area of the original OPF problem which is nonlinear and non-convex. This paper introduces two methods to obtain the exact solution, namely the penalty function method and the iterative algorithm based on convex-concave procedure principle, and then applies these two approaches to the case30 test system in the following part, where the feasibility of the two methods is verified. © 2023 IEEE.

关键词： convex relaxation convex relaxation technology power system rank-one constraint semi-definite programming

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Learning with semi-definite programming: statistical bounds based on fixed point analysis and excess risk curvature

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JOURNAL OF MACHINE LEARNING RESEARCH 2021年第1期22卷 1-64页

作者： Chretien, Stephane Cucuringu, Mihai Lecue, Guillaume Neirac, Lucie Univ Lyon 2 Natl Phys Lab Lyon France Alan Turing Inst London England Univ Oxford Oxford England IPParis ENSAE CREST Paris France

Many statistical learning problems have recently been shown to be amenable to semi definite programming (SDP), with community detection and clustering in Gaussian mixture models as the most striking instances Javanmard et al. (2016). Given the growing range of applications of SDP-based techniques to machine learning problems, and the rapid progress in the design of efficient algorithms for solving SDPs, an intriguing question is to understand how the recent advances from empirical process theory and Statistical Learning Theory can be leveraged for providing a precise statistical analysis of SDP estimators. In the present paper, we borrow cutting edge techniques and concepts from the Learning Theory literature, such as fixed point equations and excess risk curvature arguments, which yield general estimation and prediction results for a wide class of SDP estimators. From this perspective, we revisit some classical results in community detection from Gue acute accent don and Vershynin (2016) and Fei and Chen (2019b), and we obtain statistical guarantees for SDP estimators used in signed clustering, angular group synchronization (for both multiplicative and additive models) and MAX-CUT. Our theoretical findings are complemented by numerical experiments for each of the three problems considered, showcasing the competitiveness of the SDP estimators.

关键词： semi-definite programming Statistical Learning Group Synchronization Signed Clustering

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Simultaneous and Sequential Control Design for Discrete-Time Switched Linear Systems Using semi-definite programming

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IEEE CONTROL SYSTEMS LETTERS 2021年第4期5卷 1393-1398页

作者： Guicherd, Romain Mills, Andrew R. Trodden, Paul A. Kadirkamanathan, Visakan Univ Sheffield Dept Automat Control & Syst Engn Sheffield S1 3JD S Yorkshire England

The control of switched linear discrete-time systems occurs in multiple engineering fields, where it has been used to deal with complex and non-linear systems. This letter presents two strategies to design control laws for discrete-time switched linear systems, whilst guaranteeing asymptotic stability of the closed loop. Firstly, an arbitrary switching signal is considered. In this scenario a common quadratic Lyapunov function is used for stability, but subsystem Lyapunov functions are employed to improve local subsystem performance. Secondly, a constrained switching signal, associated with subsystem lower dwell time bounds is studied. In this case, a decrease in Lyapunov cost is achieved by design, based on dwell time constraints only, thus removing the need for both a common quadratic Lyapunov function or direct stable switches. It is shown in both cases that the control design problems can be formulated as one or a sequence of semi-definite programming problems, and therefore can be solved efficiently. Finally, two examples are provided in order to illustrate the different techniques presented.

关键词： Switched systems optimal control optimization semi-definite programming

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An Optimal Polarization SAR Three-Component Target Decomposition Based on semi-definite programming

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REMOTE SENSING 2023年第22期15卷 5292-5292页

作者： Wang, Tingting Suo, Zhiyong Jiang, Penghui Ti, Jingjing Ding, Zhiquan Qin, Tianqi Xidian Univ Natl Key Lab Radar Signal Proc Xian 710071 Peoples R China Sichuan Inst Aerosp Elect Equipment Chengdu 610100 Peoples R China

The model-based polarimetric synthetic aperture radar (PolSAR) target decomposition decodes the scattering mechanism of the target by analyzing the essential scattering components. This paper presents a new general three-component scattering power decomposition method by establishing optimization problems. It is known that the existing three-component decomposition method prioritizes the contribution of volume scattering, which often leads to volume scattering energy overestimation and may make double-bounce scattering and odd-bounce scattering component power negative. In this paper, a full parameter optimization method based on the remainder matrix is proposed, where all the elements of the coherency matrix will be taken into account including the remaining T13 component. The optimization is achieved with no priority order by solving the problem using semi-definite programming (SDP) based on the Schur complement theory. By doing so, the problem of volume scattering energy overestimation and negative powers will be avoided. The performance of the proposed approach is demonstrated and evaluated with AIRSAR and GF-3 PolSAR data sets. The experimental results show that by using the proposed method, the power contributions of volume scattering in two sets of data were reduced by at least 2.6% and 3.7% respectively, compared to traditional methods. And the appearance of negative power of double-bounce scattering and odd-bounce scattering are also avoided compared with those of the existing three-component decomposition.

关键词： model-based decomposition polarimetric SAR full parameter optimization semi-definite programming Schur complement

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Localisation algorithm based on weighted semi-definite programming

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INTERNATIONAL JOURNAL OF EMBEDDED SYSTEMS 2020年第3期13卷 283-291页

作者： Lu, Jianfeng Yang, Xuanyuan Taizhou Polytech Coll Inst Informat Technol Taizhou 225300 Peoples R China

In order to improve the performance of reduced complexity positive semi-definite programming (RCSDP) algorithm based on time difference of arrival (TDOA), a weighted positive semi-definite programming (WSDP) scheme is proposed in this paper. Based on the squared distance differences between the target node to one anchor node and to the other anchor node, the location of the target node is described as the optimal solution of a non-convex optimisation problem. The semi-definite relaxation technique is used to transform the original non-convex problem into a weighted convex problem, which takes the measurement noise into consideration, and then the estimated location of the target node is obtained. The simulation results show that the localisation performance of WSDP algorithm is better than that of RCSDP algorithm, regardless of whether the target node is located inside or outside the area surrounded by anchor nodes.

关键词： localisation semi-definite programming time difference of arrival TDOA weight factor

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