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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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A globally convergent interior point algorithm for non-convex nonlinear programming

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2009年第2期224卷 622-627页

作者： Fan, Xiaona Nanjing Univ Posts & Telecommun Coll Math & Phys Nanjing 210046 Jiangsu Peoples R China

in this paper, a new algorithm for tracing the combined homotopy path of the non-convex nonlinear programming problem is proposed. The algorithm is based on the techniques of beta-cone neighborhood and a combined homotopy interior point method. The residual control criteria, which ensures that the obtained iterative points are interior points, is given by the condition that ensures the beta-cone neighborhood to be included in the interior part of the feasible region. The global convergence and polynomial complexity are established under some hypotheses. (c) 2008 Elsevier B.V. All rights reserved.

关键词： non-convex programming Combined interior homotopy Path following algorithm Global convergence

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Homotopy Method for non-convex programming in Unbonded Set

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Northeastern Mathematical Journal 2005年第1期21卷 25-31页

作者：徐庆于波 Management School Fudan University Shanghai 200433 Department of Applied Mathematics Dalian University of Technology Dalian 116024

In the past few years, much and much attention has been paid to the method for solving non-convex programming. Many convergence results are obtained for bounded sets. In this paper, we get global convergence results f... 详细信息

关键词： non-convex programming unbounded set interior homotopy global convergence

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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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Homotopy Method for non-convex programming in Unbonded Set

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Communications in Mathematical Research 2005年第1期000卷

作者： XU Qing YU Bo Management School Fudan UniversityShanghai200433 Department of Applied Mathematics Dalian University of TechnologyDalian116024

关键词： non-convex programming unbounded set interior homotopy global convergence

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Sequential convex programming for nonlinear optimal control problems in UAV path planning

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AEROSPACE SCIENCE AND TECHNOLOGY 2018年第May期76卷 280-290页

作者： Zhang, Zhe Li, Jianxun Wang, Jun Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Dept Automat 800 Dong Chuan Rd Shanghai Peoples R China Luoyang Elect Equipment Test Ctr China Luoyang Peoples R China

Usually, an UAV (Unmanned Aerial Vehicle) path planning problem can be modeled as a nonlinear optimal control problem with non-convex constraints in practical applications. However, it is quite difficult to obtain stable solutions quickly for this kind of non-convex optimization with certain convergence and optimality. In this paper, an algorithm is proposed to solve the problem through approximating the non-convex parts by a series of sequential convex programming problems. Under mild conditions, the sequence generated by the proposed algorithm is globally convergent to a KKT (Karush-Kuhn-Tucker) point of the original nonlinear problem, which is verified by a rigorous theoretical proof. Compared with other methods, the convergence and effectiveness of the proposed algorithm is demonstrated by trajectory planning applications. (C) 2018 Elsevier Masson SAS. All rights reserved.

关键词： UAV path planning Sequential convex programming nonlinear optimal control Globally convergent algorithm non-convex programming

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Globally convergent homotopy algorithm for solving the KKT systems to the principal-agent bilevel programming

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OPTIMIZATION METHODS & SOFTWARE 2017年第1期32卷 69-85页

作者： Zhu, Zhichuan Yu, Bo Northeast Normal Univ Sch Math & Stat Changchun 130024 Jilin Peoples R China Jilin Univ Finance & Econ Fac Stat Changchun 130117 Jilin Peoples R China Dalian Univ Technol Sch Math Sci Dalian 116024 Liaoning Peoples R China

In this paper, a constraint set swelling homotopy (CSSH) algorithm for solving the single-level non-convex programming problem with designing piecewise linear contractual function which is equivalent to the principal-agent model with integral operator is proposed, and the existence and global convergence is proven under some mild conditions. As a comparison, a piecewise constant contract is also designed for solving the single-level non-convex programming problem with the corresponding discrete distributions. And some numerical tests are done by the proposed homotopy algorithm as well as by using fmincon in Matlab, LOQO and MINOS. The numerical results show that the CSSH algorithm is robust, feasible and effective.

关键词： principal-agent model homotopy algorithm bilevel programming non-convex programming

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Sum-Rate Maximization in Two-Way AF MIMO Relaying: Polynomial Time Solutions to a Class of DC programming Problems

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2012年第10期60卷 5478-5493页

作者： Khabbazibasmenj, Arash Roemer, Florian Vorobyov, Sergiy A. Haardt, Martin Univ Alberta Dept Elect & Comp Engn Edmonton AB T6G 2V4 Canada Ilmenau Univ Technol Commun Res Lab D-98693 Ilmenau Germany

Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relaying belongs to the class of difference-of-convex functions (DC) programming problems. DC programming problems occur also in other signal processing applications and are typically solved using different modifications of the branch-and-bound method which, however, does not have any polynomial time complexity guarantees. In this paper, we develop two efficient polynomial time algorithms for the sum-rate maximization in two-way AF MIMO relaying. The first algorithm guarantees to find at least a Karush-Kuhn-Tucker (KKT) solution. There is a strong evidence, however, that such a solution is actually globally optimal. The second algorithm that is based on the generalized eigenvectors shows the same performance as the first one with reduced computational complexity. The objective function of the problem is represented as a product of quadratic fractional ratios and parameterized so that its convex part (versus the concave part) contains only one (or two) optimization variables. One of the algorithms is called POlynomial Time DC (POTDC) and is based on semi-definite programming (SDP) relaxation, linearization, and an iterative Newton-type search over a single parameter. The other algorithm is called RAte-maximization via Generalized EigenvectorS (RAGES) and is based on the generalized eigenvectors method and an iterative search over two (or one, in its approximate version) optimization variables. We derive an upper-bound for the optimal value of the corresponding optimization problem and show by simulations that this upper-bound is achieved by both algorithms. It provides an evidence that the algorithms find a global optimum. The proposed methods are also superior to other state-of-the-art algorithms.

关键词： Difference-of-convex functions (DC) programming non-convex programming semi-definite programming relaxation sum-rate maximization two-way relaying

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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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SUCCESSIVE LINEAR-programming FOR RATIO GOAL PROBLEMS

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1987年第3期32卷 426-434页

作者： ARMSTRONG, R CHARNES, A HAKSEVER, C UNIV TEXAS CTR CYBERNET STUDIESAUSTINTX 78712 BOSTON COLL SCH MANAGEMENTCHESTNUT HILLMA 02167

This paper develops two effective methods for solution of the nonlinear and non-convex programming problems of multiple ratio goal models. They are based on Charnes and Cooper's convergence theorem for an associated sequence class of linear programs. The Charnes-Cooper results are extended to develop two alternate algorithms and implementations effectively employing new information en route to solution to achieve significant savings in computation time over methods not taking advantage of such information. Possible uses for other applications than the computed examples or for other model classes are also indicated. [ABSTRACT FROM AUTHOR]

关键词： Multiobjective programming non-convex programming ratio goals programming

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