We consider the problem of minimizing a high-dimensional objective function, which may include regularization term, using only (possibly noisy) evaluations function. Such optimization is also called derivative-free, zeroth-order, or black-box optimization. propose new zeroth-order regularized method, dubbed ZORO. When underlying gradient approximately sparse at an iterate, ZORO needs very few f...

Recently, some combinatorial properties for the the Catalan-Larcombe-French numbers have been proved by Sun and Wu, and Zhao. Recently, Z. W. Sun conjectured that the root of the Catalan-Larcombe-French numbers is log-concave. In this paper, we confirm Sun's conjecture by establishing the lower and upper bound for the ratios of the Catalan-Larcombe-French numbers.

Very recently, Ehrenborg and Steingrı́msson [7] studied enumerative properties of the excedance statistic. Let Sn denote the permutation group on the set {1, 2, . . . , n} and π = π1 π2 · · ·πn ∈ Sn . An excedance in π is an index i such that πi > i . Following [7], we encode the excedance set of a permutation as a word in the letters a and b. The excedance word w(π) of π is the ab-word w1w2 · ·...

Log-concave and Log-convex sequences arise often in combinatorics, algebra, probability and statistics. There has been a considerable amount of research devoted to this topic in recent years. Let {xi}i≥0 be a sequence of non-negative real numbers. We say that {xi} is Log-concave ( Log-convex resp.) if and only if xi−1xi+1 ≤ xi (xi−1xi+1 ≥ xi resp.) for all i ≥ 1 (relevant results can see [2] an...

E. Mossel and J. Neeman recently provided a heat flow monotonicity proof of Borell’s noise stability theorem. In this note, we develop the argument to include in a common framework noise stability, Brascamp-Lieb inequalities (including hypercontractivity), and even a weak form of Slepian inequalities. The scheme applies furthermore to families of measures with are more log-concave than the Gaus...

For a real-valued nonnegative and log-concave function f defined in R, we introduce a notion of difference function ∆f ; the difference function represents a functional analog on the difference body K + (−K) of a convex body K. We prove a sharp inequality which bounds the integral of ∆f from above, in terms of the integral of f and we characterize equality conditions. The investigation is exten...

Let R+ = (0,∞) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 ∈ M, we say that a function f : R+ → R+ is (m1, m2)-convex if f(m1(x, y)) ≤ m2(f(x), f(y)) for all x, y ∈ R+. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1, m2)-convexit...

We consider the problem of optimizing a high-dimensional convex function using stochastic zeroth-order queries. Under sparsity assumptions on the gradients or function values, we present two algorithms: a successive component/feature selection algorithm and a noisy mirror descent algorithm using Lasso gradient estimates, and show that both algorithms have convergence rates that depend only loga...

In the article, a notion “logarithmically absolutely monotonic function” is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function ` 1+ α x ́x+β are proved, where α and β are given real parameters, a new proof for the inclusion t...