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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.
The Klee–Minty cube was originally specified with a parameterized system of linear inequalities, with the dimension as the parameter. The cube in two-dimensional space is a squashed square, and the "cube" in three-dimensional space is a squashed cube. Illustrations of the "cube" have appeared besides algebraic descriptions. [3]
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.
A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5] A linear system may behave in any one of three possible ways: The system has infinitely many solutions.
A system of linear inequalities defines a polytope as a feasible region. The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimal solution. Polyhedron of simplex algorithm in 3D. The simplex algorithm operates on linear programs in the canonical form
Given a linear constraints system, if the -th inequality is satisfied for any solution of all other inequalities, then it is redundant. Similarly, STIs refers to inequalities that are implied by the non-negativity of information theoretic measures and basic identities they satisfy.
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
Finsler's lemma can be used to give novel linear matrix inequality (LMI) characterizations to stability and control problems. [4] The set of LMIs stemmed from this procedure yields less conservative results when applied to control problems where the system matrices has dependence on a parameter, such as robust control problems and control of ...
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