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A non-convex optimization approach of searching algebraic degree phase-type representations for general phase-type distributions

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ADVANCES IN APPLIED PROBABILITY 2025年

作者： Liu, Yujie Yao, Dacheng Zhang, Hanqin Natl Univ Singapore Dept Ind Syst Engn & Management Singapore 117576 Singapore Chinese Acad Sci Acad Math & Syst Sci Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Natl Univ Singapore Dept Analyt & Operat Singapore 119245 Singapore

For a continuous-time phase-type (PH) distribution, starting with its Laplace-Stieltjes transform, we obtain a necessary and sufficient condition for its minimal PH representation to have the same order as its algebraic degree. To facilitate finding this minimal representation, we transform this condition equivalently into a non-convex optimization problem, which can be effectively addressed using an alternating minimization algorithm. The algorithm convergence is also proved. Moreover, the method we develop for the continuous-time PH distributions can be used directly for the discrete-time PH distributions after establishing an equivalence between the minimal representation problems for continuous-time and discrete-time PH distributions.

关键词： Markov chain non-convex optimization phase-type distribution rational function

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Accelerated algorithms for convex and non-convex optimization on manifolds

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MACHINE LEARNING 2025年第3期114卷 1-24页

作者： Lin, Lizhen Saparbayeva, Bayan Zhang, Michael Minyi Dunson, David B. Univ Maryland Dept Math William E Kirwan Hall College Pk MD 20742 USA Univ Rochester Dept Biostat & Computat Biol 265 Crittenden Blvd Rochester NY 14642 USA Univ Hong Kong Dept Stat & Actuarial Sci Pok Fu Lam Rd Hong Kong Peoples R China Duke Univ Dept Stat Sci 214 Old Chem Durham NC 27708 USA

We propose a general scheme for solving convex and non-convex optimization problems on manifolds. The central idea is that, by adding a multiple of the squared retraction distance to the objective function in question, we "convexify" the objective function and solve a series of convex sub-problems in the optimization procedure. Our proposed algorithm adapts to the level of complexity in the objective function without requiring the knowledge of the convexity of non-convexity of the objective function. We show that when the objective function is convex, the algorithm provably converges to the optimum and leads to accelerated convergence. When the objective function is non-convex, the algorithm will converge to a stationary point. Our proposed method unifies insights from Nesterov's original idea for accelerating gradient descent algorithms with recent developments in optimization algorithms in Euclidean space. We demonstrate the utility of our algorithms on several manifold optimization tasks such as estimating intrinsic and extrinsic Fr & eacute;chet means on spheres and low-rank matrix factorization with Grassmann manifolds applied to the Netflix rating data set.

关键词： Manifolds non-convex optimization Weakly convex functions

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Hessian barrier algorithms for non-convex conic optimization

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MATHEMATICAL PROGRAMMING 2025年第1-2期209卷 171-229页

作者： Dvurechensky, Pavel Staudigl, Mathias Weierstrass Inst Appl Anal & Stochast Mohrenstr 39 D-10117 Berlin Germany Univ Mannheim Sch Business Informat & Math D-68131 Mannheim Germany

A key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions that may be non-differentiable at the relative boundary of the feasible set. This paper proposes a new family of first- and second-order interior-point methods for non-convex optimization problems with linear and conic constraints, combining logarithmically homogeneous barriers with quadratic and cubic regularization respectively. Our approach is based on a potential-reduction mechanism and, under the Lipschitz continuity of the corresponding derivative with respect to the local barrier-induced norm, attains a suitably defined class of approximate first- or second-order KKT points with worst-case iteration complexity O(epsilon-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon <^>{-2})$$\end{document} (first-order) and O(epsilon-3/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon <^>{-3/2})$$\end{document} (second-order), respectively. Based on these findings, we develop new path-following schemes attaining the same complexity, modulo adjusting constants. These complexity bounds are known to be optimal in the unconstrained case, and our work shows that they are upper bounds in the case with complicated constraints as well. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained non-convex optimization problems.

关键词： non-convex optimization Interior-point methods Self-concordant barrier Cubic regularization of Newton method Conic constraints

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non-convex scenario optimization

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MATHEMATICAL PROGRAMMING 2025年第1-2期209卷 557-608页

作者： Garatti, Simone Campi, Marco C. Politecn Milan Dipartimento Elettron Informaz & Bioingn Piazza Leonardo Vinci 32 I-20133 Milan Italy Univ Brescia Dipartimento Ingn Informaz Via Branze 38 I-25123 Brescia Italy

Scenario optimization is an approach to data-driven decision-making that has been introduced some fifteen years ago and has ever since then grown fast. Its most remarkable feature is that it blends the heuristic nature of data-driven methods with a rigorous theory that allows one to gain factual, reliable, insight in the solution. The usability of the scenario theory, however, has been restrained thus far by the obstacle that most results are standing on the assumption of convexity. With this paper, we aim to free the theory from this limitation. Specifically, we focus on the body of results that are known under the name of "wait-and-judge" and show that its fundamental achievements maintain their validity in a non-convex setup. While optimization is a major center of attention, this paper travels beyond it and into data-driven decision making. Adopting such a broad framework opens the door to building a new theory of truly vast applicability.

关键词： Data-driven optimization Scenario approach non-convex optimization Probabilistic constraints Statistical learning

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non-convex optimization of Resource Allocation in Fog Computing Using Successive Approximation

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Journal of Systems Science & Complexity 2024年第2期37卷 805-840页

作者： LI Shiyong LIU Huan LI Wenzhe SUN Wei School of Economics and Management Yanshan UniversityQinhuangdao 066000China

Fog computing can deliver low delay and advanced IT services to end users with substantially reduced energy ***,with soaring demands for resource service and the limited capability of fog nodes,how to allocate and manage fog computing resources properly and stably has become the ***,the paper investigates the utility optimization-based resource allocation problem between fog nodes and end users in fog *** authors first introduce four types of utility functions due to the diverse tasks executed by end users and build the resource allocation model aiming at utility ***,for only the elastic tasks,the convex optimization method is applied to obtain the optimal results;for the elastic and inelastic tasks,with the assistance of Jensen’s inequality,the primal non-convex model is approximated to a sequence of equivalent convex optimization problems using successive approximation ***,a two-layer algorithm is proposed that globally converges to an optimal solution of the original ***,numerical simulation results demonstrate its superior performance and *** with other works,the authors emphasize the analysis for non-convex optimization problems and the diversity of tasks in fog computing resource allocation.

关键词： Fog computing non-convex optimization optimal resource allocation successive approximation method utility function

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non-convex optimization based optimal bone correction for various beam-hardening artifacts in CT imaging

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JOURNAL OF X-RAY SCIENCE AND TECHNOLOGY 2022年第4期30卷 805-822页

作者： Tang, Shaojie Huang, Tonggang Qiao, Zhiwei Li, Baolei Xu, Yuanfei Mou, Xuanqin Fan, Jiulun Xian Univ Posts & Telecommun Sch Automat Xian 710121 Shaanxi Peoples R China Xian Univ Posts & Telecommun Automat Sorting Technol Res Ctr State Post Bur Peoples Republ China Xian Shaanxi Peoples R China Xian Key Lab Adv Control & Intelligent Proc Xian Shaanxi Peoples R China Shanxi Univ Sch Comp & Informat Technol Taiyuan Shanxi Peoples R China Beijing Hangxing Machinery Co Ltd Beijing Peoples R China Xi An Jiao Tong Univ Sch Elect & Informat Engn Xian Shaanxi Peoples R China Xian Univ Posts & Telecommun Sch Commun & Informat Engn Xian Shaanxi Peoples R China

Tube of X-ray computed tomography (CT) system emitting a polychromatic spectrum of photons leads to beam hardening artifacts such as cupping and streaks, while the metal implants in the imaged object results in metal artifacts in the reconstructed images. The simultaneous emergence of various beam-hardening artifacts degrades the diagnostic accuracy of CT images in clinics. Thus, it should be deeply investigated for suppressing such artifacts. In this study, data consistency condition is exploited to construct an objective function. non-convex optimization algorithm is employed to solve the optimal scaling factors. Finally, an optimal bone correction is acquired to simultaneously correct for cupping, streaks and metal artifacts. Experimental result acquired by a realistic computer simulation demonstrates that the proposed method can adaptively determine the optimal scaling factors, and then correct for various beam-hardening artifacts in the reconstructed CT images. Especially, as compared to the nonlinear least squares before variable substitution, the running time of the new CT image reconstruction algorithm decreases 82.36% and residual error reduces 55.95%. As compared to the nonlinear least squares after variable substitution, the running time of the new algorithm decreases 67.54% with the same residual error.

关键词： X-ray CT beam hardening consistency condition non-convex optimization

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non-convex optimization and rate control for multi-class services in the Internet

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IEEE-ACM TRANSACTIONS ON NETWORKING 2005年第4期13卷 827-840页

作者： Lee, JW Mazumdar, RR Shroff, NB Princeton Univ Dept Elect Engn Princeton NJ 08544 USA Univ Waterloo Dept Elect & Comp Engn Waterloo ON N2L 3G1 Canada Purdue Univ Sch Elect & Comp Engn W Lafayette IN 47907 USA

In this paper, we investigate the problem of distributively allocating transmission data rates to users in the Internet. We allow users to have concave as well as sigmoidal utility functions as appropriate for different applications. In the literature, for simplicity, most works have dealt only with the concave utility function. However, we show that applying rate control algorithms developed for concave utility functions in a more realistic setting (with both concave and sigmoidal types of utility functions) could lead to instability and high network congestion. We show that a pricing-based mechanism that solves the dual formulation can be developed based on the theory of subdifferentials with the property that the prices "self-regulate" the users to access the resources based on the net utility. We discuss convergence issues and show that an algorithm can be developed that is efficient in the sense of achieving the global optimum when there are many users.

关键词： rate control Internet non-concave utility function non-convex optimization

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non-convex optimization with Certificates and Fast Rates Through Kernel Sums of Squares 35

Non-Convex Optimization with Certificates and Fast Rates Thr...

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35th Conference on Learning Theory (COLT)

作者： Woodworth, Blake Bach, Francis Rudi, Alessandro PSL Res Univ Ecole Normale Super Inria Paris France

We consider potentially non-convex optimization problems, for which optimal rates of approximation depend on the dimension of the parameter space and the smoothness of the function to be optimized. In this paper, we propose an algorithm that achieves close to optimal a priori computational guarantees, while also providing a posteriori certificates of optimality. Our general formulation builds on infinite-dimensional sums-of-squares and Fourier analysis, and is instantiated on the minimization of periodic functions.

关键词： non-convex optimization sum of squares kernel methods

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non-convex optimization and Applications to Bilinear Programming and Super-Resolution Imaging

Non-Convex Optimization and Applications to Bilinear Program...

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作者： Gronski, Jessica University of Colorado Boulder

学位级别：博士

Bilinear programs and Phase Retrieval are two instances of nonconvex problems that arise in engineering and physical applications, and both occur with their fundamental difficulties. In this thesis, we consider various methods and algorithms for tackling these challenging problems and discuss their effectiveness. Bilinear programs (BLPs) are ubiquitous in engineering applications, economics, and operations research, and have a natural encoding to quadratic programs. They appear in the study of Lyapunov functions used to deduce the stability of solutions to differential equations describing dynamical systems. For multivariate dynamical systems, the problem formulation for computing an appropriate Lyapunov function is a BLP. In electric power systems engineering, one of the most practically important and well-researched subfields of constrained nonlinear optimization is Optimal Power Flow wherein one attempts to optimize an electric power system subject to physical constraints imposed by electrical laws and engineering limits, which can be naturally formulated as a quadratic program. In a recent publication, we studied the relationship between data flow constraints for numerical domains such as polyhedra and bilinear constraints. The problem of recovering an image from its Fourier modulus, or intensity, measurements emerges in many physical and engineering applications. The problem is known as Fourier phase retrieval wherein one attempts to recover the phase information of a signal in order to accurately reconstruct it from estimated intensity measurements by applying the inverse Fourier transform. The problem of recovering phase information from a set of measurements can be formulated as a quadratic program. This problem is well-studied but still presents many challenges. The resolution of an optical device is defined as the smallest distance between two objects such that the two objects can still be recognized as separate entities. Due to the physics of diffraction

关键词： non-convex optimization super-resolution imaging bilinear programming quadratic programming

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non-convex optimization FOR THE DESIGN OF SPARSE FIR FILTERS

NON-CONVEX OPTIMIZATION FOR THE DESIGN OF SPARSE FIR FILTERS

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15th IEEE/SP Workshop on Statistical Signal Processing

作者： Wei, Dennis MIT Dept Elect Engn & Comp Sci Cambridge MA 02139 USA

ISBN: (纸本)9781424427093

This paper presents a method for designing sparse FIR filters by means of a sequence of p-norm minimization problems with p gradually decreasing from 1 toward 0. The lack of convexity for p < 1 is partially overcome by appropriately initializing each subproblem. A necessary condition of optimality is derived for the subproblem of p-norm minimization, forming the basis for an efficient local search algorithm. Examples demonstrate that the method is capable of producing filters approaching the optimal level of sparsity for a given set of specifications.

关键词： Sparse filters non-convex optimization FIR digital filters

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