The MM principle is a device for creating optimization algorithms satisfying the ascent or descent property. The current survey emphasizes the role of the MM principle in nonlinear programming. For smooth functions, one can construct an adaptive interior point method based on scaled Bregman barriers. This algorithm does not follow the central path. For convex programming subject to nonsmooth co...

We consider pure bipartite states shared by Alice and Bob. Let A and B be the labels of their respective quantum systems, each with d dimensions. Suppose Alice and Bob want to transform a state |ψ〉 to another state |φ〉, and they each have complete classical descriptions of the states. We focus on exact and deterministic transformation using only Local quantum Operations and Classical Communicat...

Optimization methods have always played a key role in the development of statistical methodology, but have become of critical importance for modern methods to analyze large size high-dimensional data. Lange, Choi and Zhua (henceforth LCZ) are to be commended for providing a comprehensive overview of optimization methods widely used in statistics. The paper discusses classical unconstrained opti...

For graphs on a finite set of vertices with arbitrary probabilities of independently occurring edges, the reliability is defined as the probability that the graph is connected, and the redundancy as the expected number of spanning trees of the graph. Analogous measures of connectivity are defined for random finite directed graphs with arbitrary probabilities of independently occurring directed ...

The positive partial transpose test is one of the main criteria for detecting entanglement, and the set of states with positive partial transpose is considered as an approximation of the set of separable states. However, we do not know to what extent this criterion, as well as the approximation, are efficient. In this paper, we show that the positive partial transpose test gives no bound on the...

* Correspondence: [email protected] Department of Mathematics, University of Karachi, University Road, Karachi, Pakistan Full list of author information is available at the end of the article Abstract In this article, we give more generalized results than in Anwar et al. (2010) and Latif and Pečarić (2010) in new direction by using second-order divided difference. We investigate the expone...

Rényi divergence is related to Rényi entropy much like information divergence (also called Kullback-Leibler divergence or relative entropy) is related to Shannon’s entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as information divergence. We review the most important properties of Rényi divergence, including it...

Recently a majorization method for optimizing partition functions of log-linear models was proposed alongside a novel quadratic variational upper-bound. In the batch setting, it outperformed state-of-the-art firstand second-order optimization methods on various learning tasks. We propose a stochastic version of this bound majorization method as well as a low-rank modification for highdimensiona...

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let Θ(X ,Y) be the vector of principal angles in nondecreasing order between subspace...

This paper considers the problem of constructing a normatively significant multidimensional Gini index of relative inequality. The social evaluation relation (SER) from which the index is derived is required to satisfy a weak version of the Pigou-Dalton Bundle Principle (WPDBP) (rather than Uniform Majorization or similar conditions). It is also desired to satisfy a weak form of the condition o...