We define a class of L-convex-concave subsets of RP , where L is a projective line in RP . These are sets whose sections by any plane containing L are convex and concavely depend on this plane. We prove a version of Arnold’s conjecture for these sets, namely we prove that each such set contains a line.

In [K–S 1] it was shown that Ave π ( n ∑

We define a class of L-convex-concave subsets of RP 3 , where L is a projective line in RP 3. These are sets whose sections by any plane containing L are convex and concavely depend on this plane. We prove a version of Arnold hypothesis for these sets, namely we prove that each such set contains a line.

A positive real sequence (ak) n k=0 is said to be unimodal if there exist integers k0, k1, 0 ≤ k0 ≤ k1 ≤ n such that a0 ≤ a1 ≤ · · · ≤ ak0 = ak0+1 = · · · = ak1 ≥ ak1+1 ≥ · · · ≥ an. The integers l, k0 ≤ l ≤ k1 are called the modes of the sequence. If k0 < k1 then (ak)k=0 is said to have a plateau of k1 − k0 + 1 elements; if k0 = k1 then it is said to have a peak. A real sequence is said to be ...

Variational Gaussian (VG) inference methods that optimize a lower bound to the marginal likelihood are a popular approach for Bayesian inference. A difficulty remains in computation of the lower bound when the latent dimensionality L is large. Even though the lower bound is concave for many models, its computation requires optimization over O(L) variational parameters. Efficient reparameterizat...

In this paper, we present some operator and eigenvalue inequalities involving monotone, doubly concave convex functions. These provide variants of Acz\'{e}l inequality its reverse via generalized Kantorovich constant.

In this paper we consider Newton’s problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in R. We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized prob...

We consider convex-concave saddle point problems with a separable structure and non-strongly convex functions. We propose an efficient stochastic block coordinate descent method using adaptive primal-dual updates, which enables flexible parallel optimization for large-scale problems. Our method shares the efficiency and flexibility of block coordinate descent methods with the simplicity of prim...

The amplituhedron \(\mathcal{A}_{n,k,m}\) was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes planar \(\mathcal{N}=4\) supersymmetric Yang-Mills theory. It is projection inside the Grassmannian \(\text{Gr}_{k,k+m}\) of totally nonnegative part \(\text{Gr}_{k,n}\). Karp Williams (2019) studied \(m=1\) \(\mathcal{A}_{n,k,1}\), g...

We study the one-warehouse multi-retailer (OWMR) problem under deterministic dynamic demand and concave batch order costs, where order batches have an identical capacity and the order cost function for each facility is concave within the batch. Under appropriate assumptions on holding cost structure, we obtain lower bounds via a decomposition that splits the two-echelon problem into single-faci...