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  2. Fourier–Motzkin elimination - Wikipedia

    en.wikipedia.org/wiki/Fourier–Motzkin_elimination

    Since all the inequalities are in the same form (all less-than or all greater-than), we can examine the coefficient signs for each variable. Eliminating x would yield 2*2 = 4 inequalities on the remaining variables, and so would eliminating y. Eliminating z would yield only 3*1 = 3 inequalities so we use that instead.

  3. List of inequalities - Wikipedia

    en.wikipedia.org/wiki/List_of_inequalities

    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 Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution

  4. Inequality (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Inequality_(mathematics)

    For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. 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 ⁠.

  5. Farkas' lemma - Wikipedia

    en.wikipedia.org/wiki/Farkas'_lemma

    Generalizations of the Farkas' lemma are about the solvability theorem for convex inequalities, [4] i.e., infinite system of linear inequalities. Farkas' lemma belongs to a class of statements called "theorems of the alternative": a theorem stating that exactly one of two systems has a solution. [5]

  6. Inequalities in information theory - Wikipedia

    en.wikipedia.org/wiki/Inequalities_in...

    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 ...

  7. Relaxation (iterative method) - Wikipedia

    en.wikipedia.org/wiki/Relaxation_(iterative_method)

    Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...

  8. Variational inequality - Wikipedia

    en.wikipedia.org/wiki/Variational_inequality

    Following Antman (1983, p. 283), the definition of a variational inequality is the following one.. Given a Banach space, a subset of , and a functional : from to the dual space of the space , the variational inequality problem is the problem of solving for the variable belonging to the following inequality:

  9. Constrained least squares - Wikipedia

    en.wikipedia.org/wiki/Constrained_least_squares

    There are often special-purpose algorithms for solving such problems efficiently. Some examples of constraints are given below: Equality constrained least squares: the elements of β {\displaystyle {\boldsymbol {\beta }}} must exactly satisfy L β = d {\displaystyle \mathbf {L} {\boldsymbol {\beta }}=\mathbf {d} } (see Ordinary least squares ).