We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a ba...

Sentiment analysis on user reviews helps to keep track of user reactions towards products, and make advices to users about what to buy. State-of-the-art review-level sentiment classification techniques could give pretty good precisions of above 90%. However, current phrase-level sentiment analysis approaches might only give sentiment polarity labelling precisions of around 70% ∼ 80%, which is f...

In the eld of nonlinear programming (in continuous variables) convex analysis [21, 22] plays a pivotal role both in theory and in practice. An analogous theory for discrete optimization (nonlinear integer programming), called \discrete convex analysis" [18, 17], is developed for L-convex and M-convex functions by adapting the ideas in convex analysis and generalizing the results in matroid theo...

In this paper we will show how a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates may be used to obtain extremely wellt near-exact distributions both for this test statistic as well as for the likelihood ratio test statistic for sphericity, based on a decomposition of this latter tes...

A new simple model has recently been proposed for land mobile satellite channels. In this paper we study the secondorder statistics of the new model. First we develop a characteristic function-based approach for calculating the level crossing rate of the new model in diversity systems, where the branches can have different fading statistics. Then we compare the level crossing rate and the avera...

Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for several first-order algorithms on Hadamard manifolds. Specifically, we prove ...

In this paper we introduce a sequential block iterative method and its simultaneous version with op-timal combination of weights (instead of convex combination) for solving convex feasibility problems.When the intersection of the given family of convex sets is nonempty, it is shown that any sequencegenerated by the given algorithms converges to a feasible point. Additionally for linear feasibil...

The rational cubic function with three parameters has been extended to rational bi-cubic function to visualize the shape of regular convex surface data. The rational bi-cubic function involves six parameters in each rectangular patch. Data dependent constraints are derived on four of these parameters to visualize the shape of convex surface data while other two are free to refine the shape of s...

The notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $CAT(0)$ space, where the curvature is bounded from above by zero. In fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. In this paper, w...