Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.

Principle of smooth fit and diffusions with angles

Abstract: We show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problem fails to satisfy the smooth fit condition at the optimal stopping point b. On the other hand, if the scale function S of X is differentiable at b, then the smooth fit condition holds (whenever X is regular and G is differentiable at b). We give an example showing that the latter can happen even when at b.

Norm and anti-norm inequalities for positive semi-definite matrices

Abstract: Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms, To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten q-norms for q ∈ (0, 1] and q < 0. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let f : [0, ∞) → [0, ∞) be concave and p ∈(1, ∞). If fp(t) is superadditive, then for all positive m × m matrix A = [aij]. Furthermore, for the normalized trace τ, we consider functions φ(t) and f(t) for which the functional A ↦ φ ◦ τ ◦ f(A) is convex or concave, and obtain a simple analytic criterion.

The Piecewise Concave Function

Abstract: An important function which is neither concave nor convex often arises in production and inventory models. This function is herein called piecewise concave and can be considered to be a generalization of the concave function. Essentially it is the maximum of a collection of concave functions. Various properties of piecewise concave functions are explored in this paper. In particular, a theory is developed to locate the point at which such functions are minimized. A short discussion of applications is included.

Successive Concave Sparsity Approximation for Compressed Sensing

Abstract: In this paper, based on a successively accuracy-increasing approximation of the $\ell _0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $\ell _0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $\ell _1$ and $\ell _0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed that leads to a number of weighted $\ell _1$ minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.