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作者： Joshi, Babhru Rice University

学位级别：Ph.D., Doctor of Philosophy

We consider the bilinear inverse problem of recovering two vectors, x and w, in RL from their entrywise product. In this dissertation, we consider three different prior on these unknown signals, a subspace prior, a sparsity prior and a generative prior. For both subspace prior and sparsity prior case, we assume the signs of x and w are known which admits intuitive convex programs. For the generative prior case, we study a non-convex program, the empirical risk minimization program. For the case where the vectors have known signs and belong to known subspaces, we introduce the convex program BranchHull, which is posed in the natural parameter space that does not require an approximate solution or initialization in order to be stated or solved. Under the structural assumptions that x and w are members of known K and N dimensional random subspaces, we present a recovery guarantee for the noiseless case and a noisy case. In the noiseless case, we prove that the BranchHull recovers x and w up to the inherent scaling ambiguity with high probability when L ≫ 2(K+N). The analysis provides a precise upper bound on the coefficient for the sample complexity. In a noisy case, we show that with high probability the BranchHull is robust to small dense noise when L = Ω(K+N). We reformulate the BranchHull program and introduce the l1-BranchHull program for the case where w and x are sparse with respect to known dictionaries of size K and N, respectively. Here, K and N may be larger than, smaller than, or equal to L. The l1-BranchHull program is also a convex program that is posed in the natural parameter space. We study the case where x and w are S1- and S2-sparse with respect to a random dictionary, with the sparse vectors satisfying an effective sparsity condition, and present a recovery guarantee that depends on the number of measurements as L > Ω(S1+S2)log2(K+N). We also introduce a variants of l1-BranchHull for the purposes of tolerating noise and outliers, and for the purpose

关键词： Blind demodulation convex programming non-convex programming Sparsity convex relaxation

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Modified interactive chebyshev algorithm (MICA) for non-convex multiobjective programming

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OPTIMIZATION LETTERS 2015年第1期9卷 173-187页

作者： Luque, Mariano Univ Malaga Dept Econ Aplicada Matemat E-29071 Malaga Spain

In this paper, I carry out an extension of the MICA method (modified interactive chebyshev algorithm) for non-convex multiobjective programming. This method is based on the Tchebychev method and in the reference point approach. At each iteration, the decision maker (DM) can provide aspiration levels (desirable values for the objective functions) and also, if the DM wishes, reservation levels (level under which the objective function is not considered acceptable). On the basis of this preferential information, a region of the nondominated objective set is defined. In the convex case, considering the aspiration vector as a reference point in an achievement scalarizing function and taking a set of weight vectors, the efficient solutions generated satisfy the reservation levels. In this work, I analyze the non-convex case. The main result of MICA is verified and demonstrated for the non-convex bi-objective case. The MICA method is not verified in general for multiobjective problems with three or more objective functions, which is demonstrated with a counterexample.

关键词： Multiobjective programming Interactive procedures Tchebychev method Reference point methods Aspiration criterion vectors Reservation levels non-convex programming

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Finding Symmetry Groups of Some Quadratic programming Problems

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Numerical Mathematics(Theory,Methods and Applications) 2023年第2期16卷 370-392页

作者： Anton V.Eremeev Alexander S.Yurkov Sobolev Institute of Mathematics SB RAS Omsk DepartmentOmskRussia Omsk Scientific Center SB RAS(Institute of Radiophysics and Physical Electronics) OmskRussia

Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear *** particular,a knowledge of the symmetries may help decrease the problem dimension,reduce the size of the search space by means of linear *** the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space,the present paper considers a larger group of invertible linear *** study a special case of the quadratic programming problem,where the objective function and constraints are given by quadratic *** formulate conditions,which allow us to transform the original problem to a new system of coordinates,such that the symmetries may be sought only among orthogonal *** particular,these conditions are satisfied if the sum of all matrices of quadratic forms,involved in the constraints,is a positive definite *** describe the structure and some useful properties of the group of symmetries of the *** that,the methods of detection of such symmetries are outlined for different special cases as well as for the general case.

关键词： non-convex programming orthogonal transformation symmetry group Lie group

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Minimizing a complex quadratic fractional optimization problem with two second-order cone constraints

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OPTIMIZATION LETTERS 2024年第5期18卷 1201-1215页

作者： Zare, Arezu Semnan Univ Fac Math Stat & Comp Sci Semnan Iran

One of the problematic research areas in optimization is determining a global optimum for non-convex quadratic fractional optimization problems as a hard problem. This study seeks the quadratic fractional optimization problem in the complex field with two second-order cone constraints. An equivalent quadratic reformulation of the problem is given using the well-known Dinkelbach method, which can obtain its global optimum by applying the semidefinite relaxation approach and rank-one decomposition algorithm at each iteration of some classical methods. Finally, the results show the effectiveness of the proposed methods.

关键词： Fractional programming non-convex programming Quadratic programming Semidefinite programming Second-order cone

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Energy-Aware Joint Route Selection and Resource Allocation in Heterogeneous Satellite Networks

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2024年第8期73卷 12067-12081页

作者： Li, Jinhong Chai, Rong Liu, Chong Liang, Chengchao Chen, Qianbin Yu, F. Richard Chongqing Univ Posts & Telecommun Sch Commun & Informat Engn Chongqing 400065 Peoples R China Carleton Univ Dept Syst & Comp Engn Ottawa ON Canada

In order to ensure efficient data transmission from satellites to ground stations (GSs), resource allocation schemes must be designed. In cases where direct data transmission from satellites to GSs is not possible due to a dynamic network topology and limited contact time, efficient relay selection or route selection schemes should be employed. This paper considers a satellite communication network in which a number of source low earth orbit(SLEO) satellites are attempting to transmit their data flows to the designated GSs. To improve the transmission performance of the data flows, one geosynchronous earth orbit (GEO) satellite and a number of relay LEO (RLEO) satellites in the network are used as relays. To maximize energy efficiency, a joint route selection and resource allocation mechanism is proposed. The energy cost of the system, which is the sum of the energy cost of the SLEO satellites and the RLEO satellites, is used to formulate the joint route selection and resource allocation as a system energy cost minimization problem. Since the original optimization problem is NP hard, it is transformed into three subproblems: inter-satellite power allocation, satellite-ground power allocation, and route selection. These subproblems are solved using a greedy algorithm, the Lagrange dual method, and a matching-based heuristic algorithm, respectively. The numerical results demonstrate the effectiveness of the proposed scheme.

关键词： Satellites Resource management Satellite broadcasting Data communication Low earth orbit satellites Routing Relays Heterogeneous satellite networks energy consumption resource allocation relay selection non-convex programming

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GLOBAL OPTIMIZATION FOR non-convex PROGRAMS VIA convex PROXIMAL POINT METHOD

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2023年第6期19卷 4591-4614页

作者： Zhao, Yuanyi Xing, Wenxun Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China

In this study, a convex proximal point algorithm (CPPA) is considered for solving constrained non-convex problems, and new theoretical results are proposed. It is proved that every cluster point of CPPA is a stationary point, and the initial point of CPPA is key to global optimization. Several sufficient conditions for the initial point selection are provided for CPPA to find the global minimum. Motivated by these results, numerical experiments were conducted on non-convex quadratic programming problems with convex quadratic constraints. The performance of CPPAs was compared, with the initial point randomly selected or obtained through the Lagrangian dual problem. The numerical results demonstrate that the quality of the CPPA with the computed Lagrangian dual initial point is much better than that with the random initial point, in terms of the objective function value.

关键词： Proximal point method non-convex programming global optimization initial point Lagrangian duality

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Variational model with nonstandard growth conditions for restoration of satellite optical images using synthetic aperture radar

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EUROPEAN JOURNAL OF APPLIED MATHEMATICS 2023年第1期34卷 77-105页

作者： D'Apice, C. Kogut, P., I Manzo, R. Uvarov, M., V Univ Salerno Dipartimento Sci Aziendali Management & Innovat S 132 Via Giovanni Paolo 2 Fisciano SA Italy Oles Honchar Dnipro Natl Univ Dept Differential Equat Gagarin Av 72 UA-49010 Dnipro Ukraine EOS Data Analyt Ukraine Gagarin Av 103a Dnipro Ukraine Univ Salerno Dept Informat Engn Elect Engn & Appl Math Via Giovanni Paolo II 132 Fisciano SA Italy

In this paper, the problem of restoration of cloud contaminated optical images is studied in the case when we have no information about brightness of such images in the damage region. We propose a new variational approach for exact restoration of optical multi-band images utilising Synthetic Aperture Radar (EOS - Spatial Data Analytics, GIS Software, Satellite Imagery - is a cloud-based platform to derive remote sensing data and analyse satellite imagery for business and science purposes) images of the same regions. We prove existence of solutions, propose an alternating minimisation method for computing them, prove convergence of this method to weak solutions of the original problem and derive optimality conditions.

关键词： nonlinear programming non-convex programming applications of mathematical programming image processing (reconstruction) approximation methods in mathematical programming

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A MODIFICATION PIECEWISE convexIFICATION METHOD WITH A CLASSIFICATION STRATEGY FOR BOX-CONSTRAINED non-convex

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF nonLINEAR AND VARIATIONAL ANALYSIS 2024年第1期8卷 125-142页

作者： Zhu, Qiao Tang, Liping Yang, Xinmin Chongqing Normal Univ Natl Ctr Appl Math Chongqing Chongqing 401331 Peoples R China Sichuan Univ Coll Math Chengdu 610065 Peoples R China

This paper presents a piecewise convexification method with a box classification strategy to approximate the entire globally optimal solution set of non-convex optimization problems with box con-straints. First, the box classification strategy is proposed based on the convexity of the objective function on the sub-boxes, which helps to reduce the number of box divisions and improve the computational efficiency. At the same time, we construct the piecewise convexification problem of the original non-convex optimization problem by applying the a-based Branch-and-Bound (aBB) method, and we define the (approximate) solution set of the piecewise convexification problem based on the result of classifying the sub-boxes. Then, it is deduced that the globally optimal solution set can be approximated by the (ap-proximate) solution set of the piecewise convexification problem. Finally, a piecewise convexification algorithm is proposed that includes a new subset selection technique for division and two new termina-tion tests. The results of our experiments demonstrate the effectiveness and general superiority of our approach over the competition.

关键词： . a-based Branch-and-Bound Global optimization non-convex programming Optimal solution set Piecewise convexification

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Optimal EV Charger Level Specification for Residential Buildings with Renewable Energy 19

Optimal EV Charger Level Specification for Residential Build...

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19th International Conference on Smart Technologies (IEEE EUROCON)

作者： Knowles, Kaleb Faye, Bilal Orrson, Alex Abdeltawab, Hussein Bayrakci-Boz, Mesude Anwar, Sohail Penn State Univ Altoona Engn Altoona PA 16601 USA Penn State Univ Hazleton Engn Hazleton PA USA Penn State Univ Behrend Sch Engn Erie PA USA

ISBN: (纸本)9781665432993

Electric Vehicles (EVs) have witnessed high interest lately due to different environmental, socio-political, and economic factors. The chargers of EVs are categorized into three levels;L1 and L2. These chargers have different power ratings and prices. On the other hand, many houses started to add distributed energy resources such as photovoltaic (PV) to their existing power service. This paper investigates the optimal level of an EV charger for a residential building with renewable energy. The optimization aims to reduce the energy cost during the lifetime of the EV battery. The optimization constraints include the EV state of charge, the charging/discharging power limits of the EV, the EV expected availability, the house service size, and the EV maximum number of cycles. The study also investigates the feasibility of conducting a power service upgrade while keeping a power reserve for future load uncertainty. The problem is formulated as a non-convex optimization problem utilizing the Gurobi solver. The optimization problem is solved for different scenarios presenting different loads, PV power, and EV availability scenarios. Case studies are conducted using household data in Los Angeles, CA. The studies evaluated the energy cost saving for different levels of the chargers in the case of various PV sizes and energy tariffs. This research is a collaboration result between three commonwealth campuses in Pennsylvania State University.

关键词： Energy Storage System Energy Management System non-convex programming Electric Vehicle Photovoltaic Systems

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The global optimization method of homotopy continuation for nonlinear optimization problems

The global optimization method of homotopy continuation for ...

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作者： L. Martens Eindhoven University of Technology

学位级别：硕士

In this master thesis project, a global optimization method for nonlinear problems in relation to pressurized pipe flow equations is further developed. The solution methods used in this research are the homotopy continuation method and interior-point method. Furthermore, the performance and suitability of the optimization method for implementation in real-life water networks is investigated. In these real-life water networks, one would like to calculate the flow rate Q, total head H and demands d at certain points in the network. By doing this, potential leakage problems can be detected. To calculate these values, an optimization problem including different pressurized pipe flow equations, like the Hazen-Williams equation and Darcy-Weisbach approximation, is stated. We can conclude from this research that the performance of the solution methods is significant, resulting in both accurate and fast solving of the nonlinear problems in water networks. This is supported by analysing multiple different existing water networks while validating certain outputs to the software application EPANET. The computational effort of mere seconds is of utmost importance for real-time global optimization because software continuously updates the needed values in order to display decisions. Additionally, the mathematical proof is present to show that the used solution methods guarantee convergence, and thus, a globally optimal solution is always reached when using the Hazen-Williams equation in the optimization problem.

关键词： Optimal control homotopy continuation bifurcation analysis non-convex programming interiorpoint methods global optimization pipe networks leak detection water distribution networks

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