The Catalan-Larcombe-French sequence {Pn}n≥0 arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence {P2 n – Pn–1Pn+1}n≥1 are strictly log-con...

If & ^ 1 for each v, then by reasoning analogous to that of the preceding example, it may be shown, for any set (a), that there is no point p such that tp implies that log St(a, £) is concave. Hence Theorem 4 applies to all such functions log St(a, £). However, for this case the conclusion of the general theorem is weaker than the...

We study data structures for providing ε-approximations of convex functions whose slopes are bounded from above and below by n and −n, respectively. The structures we describe have size O((1/ε) log n) and can answer queries in O(log(1/ε) + log log n) time. We also give an informationtheoretic lower-bound, that shows it is impossible to obtain structures of size O(1/ε) for approximating this cla...

Minimizing a convex function of measure with sparsity-inducing penalty is typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this can be solved by discretizing the and running non-convex gradient descent on positions weights particles. For measures d-dimensional manifold under some non-degeneracy assumptions, leads to global optimiz...

We develop stochastic first-order primal-dual algorithms to solve a class of convex-concave saddle-point problems. When the saddle function is strongly convex in primal variable, we first restart scheme for this problem. gradient noises obey sub-Gaussian distributions, oracle complexity our strictly better than any existing methods, even deterministic case. Furthermore, each problem parameter i...

The Alternating Direction Method of Multipliers (ADMM) has received lots of attention recently due to the tremendous demand from large-scale and data-distributed machine learning applications. In this paper, we present a stochastic setting for optimization problems with non-smooth composite objective functions. To solve this problem, we propose a stochastic ADMM algorithm. Our algorithm applies...

We consider the convex optimization problem P : minx{f(x) : x ∈ K} where f is convex continuously differentiable, and K ⊂ R is a compact convex set with representation {x ∈ R : gj(x) ≥ 0, j = 1, . . . ,m} for some continuously differentiable functions (gj). We discuss the case where the gj ’s are not all concave (in contrast with convex programming where they all are). In particular, even if th...

We present a parallel algorithm for nding the convex hull of a sorted point set. The algorithm runs in O(log log n) (doubly logarithmic) time using n= log logn processors on a Common CRCW PRAM. To break the (log n= loglog n) time barrier required to output the convex hull in a contiguous array, we introduce a novel data structure for representing the convex hull. The algorithm is optimal in two...

Considering optimal non-linear income tax problems when the social welfare function only depends on ranks as in Yaari (Econometrica 55(1):95–115, 1987) and weights agreeing with Lorenz quasi-ordering, we extend analysis of Simula Trannoy (Am Econ J Policy, 2021) two directions. First, establish conditions under which bunching does not occur optimum. We find a sufficient condition individual pre...

It is shown that for every subdivision of the d-dimensional Euclidean space, d ≥ 2, into n convex cells, there is a straight line that stabs at least Ω((log n/ log log n)1/(d−1)) cells. In other words, if a convex subdivision of d-space has the property that any line stabs at most k cells, then the subdivision has at most exp(O(kd−1 log k)) cells. This bound is best possible apart from a consta...