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In convex optimization, a linear matrix inequality (LMI) is an expression of the form ():= + + + + where = [, =, …,] is a real vector,,,, …, are symmetric matrices, is a generalized inequality meaning is a positive semidefinite matrix belonging to the positive semidefinite cone + in the subspace of symmetric matrices .
Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semidefinite program. [4] The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation. [ 3 ]
A linear programming problem is one in which we wish to maximize or minimize a linear objective function of real variables over a polytope.In semidefinite programming, we instead use real-valued vectors and are allowed to take the dot product of vectors; nonnegativity constraints on real variables in LP (linear programming) are replaced by semidefiniteness constraints on matrix variables in ...
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 ...
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope , which is a set defined as the intersection of finitely many half spaces , each of which is defined by a linear inequality.
To establish whether a form h(x) is SOS amounts to solving a convex optimization problem. Indeed, any h(x) can be written as = {} ′ (+ ()) {} where {} is a vector containing a base for the forms of degree m in x (such as all monomials of degree m in x), the prime ′ denotes the transpose, H is any symmetric matrix satisfying = {} ′ {} and () is a linear parameterization of the linear ...
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in.
One may then solve for by inverting or solving the linear equations. To get X {\displaystyle X} , one must just reshape vec ( X ) {\displaystyle \operatorname {vec} (X)} appropriately. Moreover, if A {\displaystyle A} is stable (in the sense of Schur stability , i.e., having eigenvalues with magnitude less than 1), the solution X ...