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However, the elimination process results in a new system that possibly contains more inequalities than the original. Yet, often some of the inequalities in the reduced system are redundant. Redundancy may be implied by other inequalities or by inequalities in information theory (a.k.a. Shannon type inequalities).
Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 . Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities ...
Azuma's inequality; Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount
In mathematics, Farkas' lemma is a solvability theorem for a finite system of linear inequalities. It was originally proven by the Hungarian mathematician Gyula Farkas . [ 1 ] Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively ...
A great many important inequalities in information theory are actually lower bounds for the Kullback–Leibler divergence.Even the Shannon-type inequalities can be considered part of this category, since the interaction information can be expressed as the Kullback–Leibler divergence of the joint distribution with respect to the product of the marginals, and thus these inequalities can be ...
Each step often involves approximately solving the subproblem (+) where is the current best guess, is a search direction, and is the step length. The inexact line searches provide an efficient way of computing an acceptable step length α {\displaystyle \alpha } that reduces the objective function 'sufficiently', rather than minimizing the ...
In mathematics, an inequation is a statement that an inequality holds between two values. [1] [2] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation.
An n × n matrix P is doubly stochastic precisely if both P and its transpose P T are stochastic matrices.A stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each column is 1.