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A fast dual gradient method for separable convex optimization via smoothing
Li, J.,Wu, Z.,Wu, C.,Long, Q.,Wang, X.,Lee, J.-M.,Jung, K.-H. YOKOHAMA PUBLISHERS 2016 Pacific journal of optimization Vol.12 No.2
<P>This paper considers a class of separable convex optimization problems with linear coupled constraints arising in many applications. Based on a novel smoothing technique, a simple fast dual gradient method is presented to solve this class of problems. Then the convergence of the proposed method is proved and the computational complexity bound of the method for achieving an approximately optimal solution is given explicitly. An improved iteration complexity bound is obtained when the objective function of the problem is strongly convex. Our algorithm is simple and fast, which can be implemented in a parallel fashion. Numerical experiments on a network utility maximization problem are presented to illustrate the effectiveness of the proposed algorithm.</P>
Sums of squares characterizations of containment of convex semialgebraic sets
Jeyakumar, V.,Lee, G. M.,Lee, J. H. YOKOHAMA PUBLISHERS 2016 Pacific journal of optimization Vol.12 No.1
<P>In this paper, we establish numerically checkable sums of squares characterizations of containment of a convex semialgebraic set in another reverse convex semialgebraic set, described by SOS-convex polynomials. The significance of these characterizations is that they hold without any qualifications. In particular, when the semialgebraic sets are described by convex quadratic functions, we obtain a simple linear matrix inequality characterization for the containment. We also present robust set containment characterizations for convex semialgebraic sets in the face of data uncertainty of the SOS-convex polynomials that define the convex semialgebraic sets.</P>