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Systems of linear inequalities can be simplified by Fourier–Motzkin elimination. [ 17 ] The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them.
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).
A linear programming problem seeks to optimize (find a maximum or minimum value) a function (called the objective function) subject to a number of constraints on the variables which, in general, are linear inequalities. [6] The list of constraints is a system of linear inequalities.
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 ...
Many traditional polyhedral forms are n-dimensional polyhedra. Other examples include: A half-space is a polyhedron defined by a single linear inequality, a 1 T x ≤ b 1.; A hyperplane is a polyhedron defined by two inequalities, a 1 T x ≤ b 1 and a 1 T x ≥ b 1 (which is equivalent to -a 1 T x ≤ -b 1).
Bernstein inequalities (probability theory) Boole's inequality; Borell–TIS inequality; BRS-inequality; Burkholder's inequality; Burkholder–Davis–Gundy inequalities; Cantelli's inequality; Chebyshev's inequality; Chernoff's inequality; Chung–Erdős inequality; Concentration inequality; Cramér–Rao inequality; Doob's martingale inequality
In mathematics, a differential variational inequality (DVI) is a dynamical system that incorporates ordinary differential equations and variational inequalities or complementarity problems. DVIs are useful for representing models involving both dynamics and inequality constraints.
Muirhead's inequality states that [a] ≤ [b] for all x such that x i > 0 for every i ∈ { 1, ..., n} if and only if there is some doubly stochastic matrix P for which a = Pb.
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