In this paper we consider a class of structured nonsmooth difference-of-convex (dc) minimization in which the first convex component is the sum of a smooth and a nonsmooth function while the second convex component is...
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In this paper we consider a class of structured nonsmooth difference-of-convex (dc) minimization in which the first convex component is the sum of a smooth and a nonsmooth function while the second convex component is the supremum of finitely many convex smooth functions. The existing methods for this problem usually have weak convergence guarantees or exhibit slow convergence. Due to this, we propose two nonmonotone enhanced proximal dc algorithms for solving this problem. For possible acceleration, one uses a nonmonotone line-search scheme in which the associated Lipschitz constant is adaptively approximated by some local curvature information of the smooth function in the first convex component, and the other employs an extrapolation scheme. It is shown that every accumulation point of the solution sequence generated by them is a D-stationary point of the problem. These methods may, however, become inefficient when the number of convex smooth functions in defining the second convex component is large. To remedy this issue, we propose randomized counterparts for them and show that every accumulation point of the generated solution sequence is a D-stationary point of the problem almost surely. Some preliminary numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms.
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, differe...
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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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