This article considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first consider a distributed f...
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This article considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first consider a distributed first-order primal-dual algorithm. We show that it converges sublinearly to a stationary point if each local cost function is smooth and linearly to a global optimum under an additional condition that the global cost function satisfies the Polyak-Lojasiewicz condition. This condition is weaker than strong convexity, which is a standard condition for proving linear convergence of distributed optimization algorithms, and the global minimizer is not necessarily unique. Motivated by the situations where the gradients are unavailable, we then propose a distributed zeroth-orderalgorithm, derived from the considered first-order algorithm by using a deterministic gradient estimator, and show that it has the same convergence properties as the considered first-order algorithm under the same conditions. The theoretical results are illustrated by numerical simulations.
This paper studies the communication complexity of convex risk-averse optimization over a network. The problem generalizes the well-studied risk-neutral finite-sum distributed optimization problem, and its importance ...
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This paper studies the communication complexity of convex risk-averse optimization over a network. The problem generalizes the well-studied risk-neutral finite-sum distributed optimization problem, and its importance stems from the need to handle risk in an uncertain environment. For algorithms in the literature, a gap exists in communication complexities for solving risk-averse and risk-neutral problems. We propose two distributed algorithms, namely the distributed risk-averse optimization (DRAO) method and the distributed risk-averse optimization with sliding (DRAO-S) method, to close the gap. Specifically, the DRAO method achieves optimal communication complexity by assuming a certain saddle point subproblem can be easily solved in the server node. The DRAO-S method removes the strong assumption by introducing a novel saddle point sliding subroutine which only requires the projection over the ambiguity set P. We observe that the number of P-projections performed by DRAO-S is optimal. Moreover, we develop matching lower complexity bounds to show the communication complexities of both DRAO and DRAO-S to be unimprovable. Numerical experiments are conducted to demonstrate the encouraging empirical performance of the DRAO-S method.
This work studies multigroup multicasting transmission in cloud radio access networks (C-RANs) with simultaneous wireless information and power transfer where densely packed remote radio heads (RRHs) cooperatively pro...
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This work studies multigroup multicasting transmission in cloud radio access networks (C-RANs) with simultaneous wireless information and power transfer where densely packed remote radio heads (RRHs) cooperatively provide information and energy services for information users (IUs) and energy users (EUs) respectively. To maximize the weighted sum rate (WSR) of information services while satisfying the energy harvesting levels at EUs an optimization of joint beamforming design for the fronthaul and access links is formulated which is however neither smooth nor convex and is indeed NP-hard. To tackle this difficulty the smooth and successive convex approximations are used to transform the original problem into a sequence of convex problems and two first-order algorithms are developed to find the initial feasible point and the nearly optimal solution respectively. Moreover an accelerated algorithm is designed to improve the convergence speed by exploiting both Nesterov and heavy-ball momentums. Numerical results demonstrate that the proposed first-order algorithms achieve almost the same WSR as that of traditional second-order approaches yet with much lower computational complexity and the proposed scheme outperforms state-of-the-art competing schemes in terms of WSR.
We present a new variant of the Chambolle-Pock primal-dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method...
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We present a new variant of the Chambolle-Pock primal-dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.
In this paper, we propose a fixed-point augmented Lagrangian method (FPALM) for general convex problems arising in image processing. We can easily obtain the alternating minimization algorithm (AMA) referred to [1] fr...
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In this paper, we propose a fixed-point augmented Lagrangian method (FPALM) for general convex problems arising in image processing. We can easily obtain the alternating minimization algorithm (AMA) referred to [1] from the proposed FPALM. The proof for the convergence of the FPALM is provided under some mild assumptions. We present two kinds of first-order augmented Lagrangian schemes and show their connections to first-order primal-dual algorithms [2]. Furthermore, we apply an acceleration rule to both the FPALM and AMA to achieve better convergence rates. Numerical examples on different image denosing models including the ROF model, the vectorial TVmodel, high order models and the TV-L-1 model are provided to demonstrate the efficiency of the proposed algorithms. (C) 2013 Elsevier Inc. All rights reserved.
A continuous compressed sensing method for 2D radar imaging is adopted. Atomic norm minimisation is used for sparse signal recovery and the property of the dual optimal solution is utilised to generate a super-resolut...
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A continuous compressed sensing method for 2D radar imaging is adopted. Atomic norm minimisation is used for sparse signal recovery and the property of the dual optimal solution is utilised to generate a super-resolution radar image. A feasible first-order algorithm based on the alternating direction method of multipliers is presented for problem solving where the primal and dual optimal solution can be obtained simultaneously. A fast implementation is also developed by exploiting the low rank structure of the subproblem. Experimental results on real data validate the effectiveness of the proposed algorithm.
In this paper,we propose an image denoising algorithm for compressed sensing based on alternating direction method of multipliers(ADMM).We prove that the objective func-tion of the iterates approaches the optimal *** ...
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In this paper,we propose an image denoising algorithm for compressed sensing based on alternating direction method of multipliers(ADMM).We prove that the objective func-tion of the iterates approaches the optimal *** also prove the O(1/N)convergence rate of our algorithm in the ergodic *** the same time,simulation results show that our algorithm is more efficient in image denoising compared with existing methods.
We consider the design of experiments when estimation is to be performed using locally weighted regression methods. We adopt criteria that consider both estimation error (variance) and error resulting from model missp...
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We consider the design of experiments when estimation is to be performed using locally weighted regression methods. We adopt criteria that consider both estimation error (variance) and error resulting from model misspecification (bias). Working with continuous designs, we use the ideas developed in convex design theory to analyze properties of the corresponding optimal designs. Numerical procedures for constructing optimal designs are developed and applied to a variety of design scenarios in one and two dimensions. Among the interesting properties of the constructed designs are the following: (1) Design points tend to be more spread throughout the design space than in the classical case. (2) The optimal designs appear to be less model and criterion dependent than their classical counterparts. (3) While the optimal designs are relatively insensitive to the specification of the design space boundaries, the allocation of supporting points is strongly governed by the points of interest and the selected weight function, if the latter is concentrated in areas significantly smaller than the design region. Some singular and unstable situations occur in the case of saturated designs. The corresponding phenomenon is discussed using a univariate linear regression example. (C) 1999 Elsevier Science B.V. All rights reserved.
An optimization approach to solving a convex finite-dimensional variational inequality with nonpotential operator is examined. It is shown how to construct an optimization problem equivalent to the variational inequal...
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An optimization approach to solving a convex finite-dimensional variational inequality with nonpotential operator is examined. It is shown how to construct an optimization problem equivalent to the variational inequality in the space of the original variables. A first-order algorithm to solve variational inequalitieis is formulated based on this optimization problem.
A new class of optimal design problems that incorporates environmental uncertainty is formulated and related to worst-case design, minimax objective design, and game theory. A numerical solution technique is developed...
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A new class of optimal design problems that incorporates environmental uncertainty is formulated and related to worst-case design, minimax objective design, and game theory. A numerical solution technique is developed and applied to a weapon allocation problem, a structural design problem with an infinite family of load conditions, and a vibration isolator design problem with a band of excitation frequencies.
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