WebConcave maximisation(!), but two complications: I The optimum must be found overfeasiblemeans. Interdependance of the sufficient ... exponentially many hyperplanes connecting the discrete configuration stats: called themarginal polytope. I Even when restricted to the marginal polytope, evaluating ( ) can be challenging. WebSep 30, 2024 · In this paper, we define and introduce some new concepts of the exponentially m -convex functions involving a fixed constant m ∈ (0, 1]. We investigate several properties of the exponentially m -convex functions and discuss their relations with convex functions. Optimality conditions are characterized by a class of variational …

Concave function - Wikipedia

WebSep 1, 2024 · For K ∈ R and N > 0, a function φ: R n → [− ∞, ∞) is said to be (K, N) exponentially concave if Φ ≔ exp N − 1 φ is a concave function on R n satisfying 1 Φ Hess Φ ≤ − K N. The right hand side above represents the scalar − K ∕ N multiplied with the identity matrix and the inequality is in the sense of two nonpositive ... WebThe exponential function is a mathematical function denoted by () = ⁡ or (where the argument x is written as an exponent).Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. rophnane

Exponential families: convexity, duality and free …

Webthe cumulant generating function is α-exponentially concave/convex (Propositions 2, 3), and the corresponding L(±α)-divergence is the Rényi divergence (Theorem 13). Moreover, the dual function is the Rényi entropy. Our results thus provide a new approach to the geometry of Rényi and α-divergences. When α → 0 we recover the WebExponential growth occurs when b > 1, and y-values increase to the right. Exponential decay occurs when 0 b 1, and y-values decrease to the right. Both graphs are concave-up. When a 0, the graphs of exponential equations become concave down, and increasing x-values yield increasingly negative y-values. Webthe negative exponential distributions are such cases. In fact, since linear functions are the only functions which are both concave and convex, the only distributions which are both log-concave and log-convex are exponential or truncated exponential. There are distributions which are neither log-concave nor log-convex over the entire support. rop hits