A New Decomposition Method for Multiuser DC-Programming and Its Applications

Alvarado, Alberth; Scutari, Gesualdo; Pang, Jong-Shi

doi:10.1109/tsp.2014.2315167

Cited by 87 publications

(40 citation statements)

References 30 publications

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“…Difference-of-convex (DC) optimization problems are problems whose objective can be written as the difference of a proper closed convex function and a continuous convex function. They arise in various applications such as digital communication system [2], assignment and power allocation [29] and compressed sensing [35]; we refer the readers to Sections 7.6 to 7.8 of the recent monograph [33] for more applications of DC optimization problems.…”

Section: Introductionmentioning

confidence: 99%

“…This idea appears in numerous work and is also recently adopted in [18], where they proposed the so-called proximal DCA. 2 This algorithm not only majorizes the concave part in the objective by a linear majorant in each iteration, but also majorizes the smooth convex part by a quadratic majorant. When the proximal mapping of the proper closed convex function is easy to compute, the subproblems of the proximal DCA can be solved efficiently.…”

Section: Introductionmentioning

confidence: 99%

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A proximal difference-of-convex algorithm with extrapolation

2017

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We consider a class of difference-of-convex (DC) optimization problems whose objective is levelbounded and is the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous concave function. While this kind of problems can be solved by the classical difference-of-convex algorithm (DCA) [26], the difficulty of the subproblems of this algorithm depends heavily on the choice of DC decomposition. Simpler subproblems can be obtained by using a specific DC decomposition described in [27]. This decomposition has been proposed in numerous work such as [18], and we refer to the resulting DCA as the proximal DCA. Although the subproblems are simpler, the proximal DCA is the same as the proximal gradient algorithm when the concave part of the objective is void, and hence is potentially slow in practice. In this paper, motivated by the extrapolation techniques for accelerating the proximal gradient algorithm in the convex settings, we consider a proximal difference-of-convex algorithm with extrapolation to possibly accelerate the proximal DCA. We show that any cluster point of the sequence generated by our algorithm is a stationary point of the DC optimization problem for a fairly general choice of extrapolation parameters: in particular, the parameters can be chosen as in FISTA with fixed restart [15]. In addition, by assuming the Kurdyka-Lojasiewicz property of the objective and the differentiability of the concave part, we establish global convergence of the sequence generated by our algorithm and analyze its convergence rate. Our numerical experiments on two difference-of-convex regularized least squares models show that our algorithm usually outperforms the proximal DCA and the general iterative shrinkage and thresholding algorithm proposed in [17].

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A proximal difference-of-convex algorithm with extrapolation

2017

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“…Our interest in this class of nonconvex optimization problems stemmed initially from a particular application pertaining to physical layer based security in a digital communication system [1,2] and a related one of joint base-station assignment and power allocation [38]. A first glance at their formulations does not immediately reveal that the resulting nonsmooth maximization problem (see (3)) is of the dc type.…”

Section: Introductionmentioning

confidence: 99%

Computing B-Stationary Points of Nonsmooth DC Programs

2017

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Motivated by a class of applied problems arising from physical layer based security in a digital communication system, in particular, by a secrecy sum-rate maximization problem, this paper studies a nonsmooth, difference-of-convex (dc) minimization problem. The contributions of this paper are: (i) clarify several kinds of stationary solutions and their relations; (ii) develop and establish the convergence of a novel algorithm for computing a d-stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions; (iii) extend the algorithm in several directions including: a randomized choice of the subproblems that could help the practical convergence of the algorithm, a distributed penalty approach for problems whose objective functions are sums of dc functions, and problems with a specially structured (nonconvex) dc constraint. For the latter class of problems, a pointwise Slater constraint qualification is introduced that facilitates the verification and computation of a B(ouligand)-stationary point.

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“…Our own interest in dc functions stemmed from the optimization of some physical layer problems in signal processing and communication [1,3,33]. Most worthy of note in these references are the following.…”

Section: Introductionmentioning

confidence: 99%

On the pervasiveness of difference-convexity in optimization and statistics

2018

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With the increasing interest in applying the methodology of difference-of-convex (dc) optimization to diverse problems in engineering and statistics, this paper establishes the dc property of many functions in various areas of applications not previously known to be of this class. Motivated by a quadratic programming based recourse function in two-stage stochastic programming, we show that the (optimal) value function of a copositive (thus not necessarily convex) quadratic program is dc on the domain of finiteness of the program when the matrix in the objective function's quadratic term and the constraint matrix are fixed. The proof of this result is based on a dc decomposition of a piecewise LC 1 function (i.e., functions with Lipschitz gradients). Armed with these new results and known properties of dc functions existed in the literature, we show that many composite statistical functions in risk analysis, including the value-at-risk (VaR), conditional value-at-risk (CVaR), Optimized Certainty Equivalent (OCE), and the expectation-based, VaR-based, and CVaR-based random deviation functionals are all dc. Adding the known class of dc surrogate sparsity functions that are employed as approximations of the 0 function in statistical learning, our work significantly expands the classes of dc functions and positions them for fruitful applications. * All authors are affiliated with the Daniel J. Epstein

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A New Decomposition Method for Multiuser DC-Programming and Its Applications

Cited by 87 publications

References 30 publications

A proximal difference-of-convex algorithm with extrapolation

A proximal difference-of-convex algorithm with extrapolation

Computing B-Stationary Points of Nonsmooth DC Programs

On the pervasiveness of difference-convexity in optimization and statistics

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