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This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.

In this paper we worked with the relative divergence of type s , s ∈ ℝ, which include Kullback-Leibler divergence and the Hellinger and χ 2 distances as particular cases. We give here a study of the sym- metrized divergences in additive and multiplicative forms. Some ba-sic properties as symmetry, monotonicity and log-convexity are estab-lished. An important result from the Convexity Theory is also proved.

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body

For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L ♮ -convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ℓ ∞ -distance equal to 2 or not smaller than 2, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions . These functions enjoy nice structural properties. They are stable under scaling and addition and satisfy a family of inequalities named parallelogram inequalities . Furthermore, they admit a proximity theorem with the same small proximity bound as that for L ♮ -convex functions. These structural properties allow us to develop an algorithm for the minimization of locally and globally discrete midpoint convex functions based on the proximity-scaling approach and on a novel 2-neighborhood steepest descent algorithm.

This paper aims to study the directional differential stability results for solutions and values of general parameterized optimization problems. The major results of this paper are the characterizations of directional derivatives of optimal solution sets and optimal value functions for specific convex problems.

We connect high-dimensional subset selection and submodular maximization. Our results extend the work of Das and Kempe [In ICML (2011) 1057–1064] from the setting of linear regression to arbitrary objective functions. For greedy feature selection, this connection allows us to obtain strong multiplicative performance bounds on several methods without statistical modeling assumptions. We also derive recovery guarantees of this form under standard assumptions. Our work shows that greedy algorithms perform within a constant factor from the best possible subset-selection solution for a broad class of general objective functions. Our methods allow a direct control over the number of obtained features as opposed to regularization parameters that only implicitly control sparsity. Our proof technique uses the concept of weak submodularity initially defined by Das and Kempe. We draw a connection between convex analysis and submodular set function theory which may be of independent interest for other statistical learning applications that have combinatorial structure.

The local convexity of a PN space is discussed using the idea of probability metric, and the constraint condition of convexity preservation of probabilistic normed space is given. Based on the probability background, it is proved that the probability norm and its linear combination are convex functions. When α ≥ −1, the lower horizontal set is convex. When α ≥ − ½, the lower level set can form a probabilistic normed space.

The focus of this paper is on three types of convexity: generalized uniform convexity, Φ-convexity and superquadracity. The similar structures of these types of convexity are such that the same processes can be applied to each one of them to obtain further refinements of known inequalities.

Developing a suitable and monotonous health index (HI) that can be used to represent a whole degradation process is a key step for continuous machine health monitoring during its life cycle. It is expected that the potential HI is able to inform incipient fault moment and then track machine degradation trajectories effectively and monotonically. Previously, nearest neighbor convex hull classification (NNCHC) has been widely applied for fault classification. In this paper, a HI construction methodology for machine life cycle health monitoring based on NNCHC is proposed. Firstly, a normal convex hull is modeled based on normal vibration data to fully characterize machine health conditions. Afterward, two HIs are constructed based on ℓ 1 norm and ℓ 2 norm distances between the normal convex hull and test points. The superiority of the developed approach in this study lies in the flexible and efficient development of a HI for fault progress tracking. Moreover, the only usage of a normal dataset in the proposed methodology is closer to real application.

In the present note, we have given a new integral identity via Conformable fractional integrals and some further properties. We have proved some integral inequalities for different kinds of convexity via Conformable fractional integrals. We have also showed that special cases of our findings gave some new inequalities involving Riemann-Liouville fractional integrals.