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Optimization of cooperative Beamforming for Sc-FdMA Multi-User Multi-Relay Networks by Tractable d.c. programming

IEEE TRANSAcTIONS ON SIGNAL PROcESSING 2013年第2期61卷 467-479页

作者： Ha Hoang Kha Hoang duong Tuan Nguyen, Ha H. Pham, Tung T. Univ Technol Sydney Fac Engn & Informat Technol Sydney NSW 2007 Australia Univ Saskatchewan Dept Elect & Comp Engn Saskatoon SK Canada

This paper addresses the optimal cooperative beamforming design for multi-user multi-relay wireless networks in which the single-carrier frequency division multiple access (Sc-FdMA) technique is employed at the terminals. The problem of interest is to find the beamforming weights across relays to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among source users subject to individual power constraints at each relay. Such a beamforming design is shown to be a hard non-convex optimization problem and therefore it is mathematically challenging to find the optimal solution. By exploring its partial convex structures, we recast the design problem as minimization of a d.c. (difference of two convex) objective function subject to convex constraints and develop an effective iterative algorithm of low complexity to solve it. Simulation results show that our optimal cooperative beamforming scheme realizes the inherent diversity order of the relay network and it performs significantly better than the equal-power beamforming weights.

关键词： cooperative beamforming d.c. programming frequency-domain equalization maximin optimization power allocation Sc-FdMA wireless relay networks

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A general double-proximal gradient algorithm for d.c. programming

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MATHEMATIcAL programming 2019年第1-2期178卷 301-326页

作者： Banert, Sebastian Bot, Radu Ioan KTH Royal Inst Technol Dept Math Lindstedtsvagen 25 S-10044 Stockholm Sweden Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria

The possibilities of exploiting the special structure of d.c. programs, which consist of optimising the difference of convex functions, are currently more or less limited to variants of the dcA proposed by Pham dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are evaluated by one of their subgradients. In this paper we propose an algorithm which allows the evaluation of both the concave and the convex part by their proximal points. Additionally, we allow a smooth part, which is evaluated via its gradient. In the spirit of primal-dual splitting algorithms, the concave part might be the composition of a concave function with a linear operator, which are, however, evaluated separately. For this algorithm we show that every cluster point is a solution of the optimisation problem. Furthermore, we show the connection to the Toland dual problem and prove a descent property for the objective function values of a primal-dual formulation of the problem. convergence of the iterates is shown if this objective function satisfies the Kurdyka-Lojasiewicz property. In the last part, we apply the algorithm to an image processing model.

关键词： d.c. programming Toland dual Proximal-gradient algorithm Kurdyka-Lojasiewicz property convergence analysis

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Solving an inverse problem for an elliptic equation by d.c. programming

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JOURNAL OF GLOBAL OPTIMIZATION 2003年第4期25卷 407-423页

作者： An, LTH Tao, Pd Hào, dN Inst Natl Sci Appl Rouen Lab Modelisat & Optimisat Rech Operat F-76131 Mont St Aignan France Inst Math Hanoi 10000 Vietnam

An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (dcA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.

关键词： branch-and-bound technique dcA d.c. programming ill-posed problem inverse problem Tikhonov regularization

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Optimizing radial basis functions by d.c. programming and its use in direct search for global derivative-free optimization

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TOP 2012年第1期20卷 190-214页

作者： Le Thi, H. A. Vaz, A. I. F. Vicente, L. N. Univ Minho Dept Prod & Syst P-4710057 Braga Portugal Paul Verlaine Univ Lab Theoret & Appl Comp Sci LITA EA 3097 F-57045 Metz France

In this paper, we address the global optimization of functions subject to bound and linear constraints without using derivatives of the objective function. We investigate the use of derivative-free models based on radial basis functions (RBFs) in the search step of direct-search methods of directional type. We also study the application of algorithms based on difference of convex (d.c.) functions programming to solve the resulting subproblems which consist of the minimization of the RBF models subject to simple bounds on the variables. Extensive numerical results are reported with a test set of bound and linearly constrained problems.

关键词： Global optimization derivative-free optimization direct-search methods Search step Radial basis functions d.c. programming dcA

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On solving a d.c. programming problem by a sequence of linear programs

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Journal of Global Optimization 1991年第2期1卷 183-203页

作者： Horst, R. Phong, T.Q. Thoai, Ng.V. de Vries, J. Fachbereich IV-Mathematik Universität Trier Trier Germany Institute of Technology Danang Viet Nam Institute of Mathematics Hanoi Viet Nam

We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed.c.nvex set in ℝn. This algorithm combines a new prismatic branch and bo... 详细信息

We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed.c.nvex set in ℝⁿ. This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved. © 1991 Kluwer Academic Publishers.

关键词： branch and bound d.c. programming global optimization Nonlinear programming outer approximation prismatic partition

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d.c. programming for cooperative Beamforming in Sc-FdMA Multi-User Multi-Relay Networks

D.C. Programming for Cooperative Beamforming in SC-FDMA Mult...

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IEEE Global communications conference

作者： Ha H. Kha H. d. Tuan Ha H. Nguyen Tung T. Pham Faculty of Engineering and Information Technology University of Technology Department of Electrical and Computer Engineering University of Saskatchewan

ISBN: (纸本)9781467309202

We are concerned with a cooperative beamforming design for multi-user multi-relay wireless networks in which the single-carrier frequency division multiple access (Sc-FdMA) technique is employed at the terminals. The problem of interest is to find the beamforming weights across relays to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among users subject to individual power constraints at each relay. Such a beamforming design is shown to be a hard nonconvex program and therefore it is mathematically challenging to find the optimal solution. By exploring its partial convex structures, we recast the design problem as minimization of a d.c. (difference of two convex) objective function subject to convex constraints and develop an effective iterative algorithm of low complexity to solve it. Simulation results show that our optimal cooperative beamforming scheme realizes the inherent diversity order of the relay network and it performs significantly better than the equal-power beamforming weights.

关键词： cooperative beamforming Power allocation Sc-FdMA Frequency-domain equalization Wireless relay networks d.c. programming Maximin optimization Frequency domain equalization Beamforming power splitting Relay networks wireless relay networks design problems Frequency division multiaccess Individual Power convexity

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dc programming for cooperative Beamforming in Sc-FdMA Multi-User Multi-Relay Networks

DC Programming for Cooperative Beamforming in SC-FDMA Multi-...

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IEEE Global communications conference (GLOBEcOM)

作者： Kha, Ha H. Tuan, H. d. Nguyen, Ha H. Pham, Tung T. Univ Technol Sydney Fac Engn & Informat Technol Sydney NSW 2007 Australia

ISBN: (纸本)9781467309219;9781467309202

关键词： cooperative beamforming power allocation Sc-FdMA frequency-domain equalization wireless relay networks d.c. programming maximin optimization

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THE dESIGN cENTERING PROBLEM AS A dc programming PROBLEM

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MATHEMATIcAL programming 1988年第2期41卷 229-248页

作者： THAcH, PT 1.Institute of Mathematics P.O. Box 631 Bo Ho Hanoi Vietnam

The following problem is studied: Given a compact setS inR n and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ R n satisfyingp(y?x) ≤ r is... 详细信息

The following problem is studied: Given a compact setS inR ^class="a-plus-plus"> n and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ R ^{class="a-plus-plus"> n} satisfyingp(y?x) ≤ r is contained inS. It is shown that whenS is the intersection of a closed.c.nvex set and several complementary convex sets (sets whose complements are open convex) this “design centering problem” can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR ^{class="a-plus-plus"> n}. In the case where, moreover,p(x) = (x ^{class="a-plus-plus"> T} Ax)^{class="a-plus-plus">1/2}, withA being a symmetric positive definite matrix, a solution method is developed which is based on the red.c.ion of the problem to the global minimization of a concave function over a compact convex set.

关键词： design centering problem d.c. programming difference of two convex functions complementary convex sets reverse convex constraints global minimization of concave functions outer approximation algorithm

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The hill detouring method for minimizing hinging hyperplanes functions

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cOMPUTERS & OPERATIONS RESEARcH 2012年第7期39卷 1763-1770页

作者： Huang, Xiaolin Xu, Jun Mu, Xiaomu Wang, Shuning Tsinghua Univ Tsinghua Natl Lab Informat Sci & Technol TNList Dept Automat Beijing 100084 Peoples R China

This paper studies the problem of minimizing hinging hyperplanes (HH) which is a widely applied nonlinear model. To deal with HH minimization, we transform it into a d.c. (difference of convex functions) programming and a concave minimization on a polyhedron, then some mature techniques are applicable. More importantly, HH is a continuous piecewise linear function and for concave HH, the super-level sets are polyhedra. Inspired by this property, we establish a method which searches on the counter map in order to escape a local optimum. Intuitively, this method bypasses the super-level set and is hence called hill detouring method, following the name of hill climbing. In numerical experiments, the proposed algorithm is compared with cPLEX and a heuristic algorithm showing its effectiveness. (c) 2011 Elsevier Ltd. All rights reserved.

关键词： Hinging hyperplanes continuous piecewise linear Global optimum d.c. programming concave minimization

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Joint Optimization of cooperative Beamforming and Relay Assignment in Multi-User Wireless Relay Networks

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IEEE TRANSAcTIONS ON WIRELESS cOMMUNIcATIONS 2014年第10期13卷 5481-5495页

作者： che, Enlong Hoang duong Tuan Nguyen, Ha H. Univ Technol Sydney Fac Engn & Informat Technol Sydney NSW 2007 Australia Univ Saskatchewan Dept Elect & Comp Engn Saskatoon SK S7N 5A9 Canada

This paper considers joint optimization of cooperative beamforming and relay assignment for multi-user multi-relay wireless networks to maximize the minimum of the received signal-to-interference-plus-noise ratios (SINR). Separated.c.ntinuous optimization of beamforming and binary optimization of relay assignment already pose very challenging programs. certainly, their joint optimization, which involves nonconvex objectives and.c.upled.c.nstraints in continuous and binary variables, is among the most challenging optimization problems. Even the conventional relaxation of binary constraints by continuous box constraints is still computationally intractable because the relaxed program is still highly nonconvex. However, it is shown in this paper that the joint programs fit well in the d.c. (difference of two convex functions/sets) optimization framework. Efficient optimization algorithms are then developed for both cases of orthogonal and nonorthogonal transmission by multiple users. Simulation results show that the jointly optimized beamforming and relay assignment not only save transmission bandwidth but can also maintain well the network SINRs.

关键词： cooperative beamforming power allocation relay selection relay assignment wireless relay networks d.c. programming

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