In
convex optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems ...
, a linear matrix inequality (LMI) is an expression of the form
:
where
*
symmetric matrices
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...
,
*
is a generalized inequality meaning
is a
positive semidefinite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf.
M ...
belonging to the positive semidefinite cone
in the subspace of symmetric matrices
.
This linear matrix inequality specifies a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
constraint on
.
Applications
There are efficient numerical methods to determine whether an LMI is feasible (''e.g.'', whether there exists a vector ''y'' such that LMI(''y'') ≥ 0), or to solve a
convex optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems ...
problem with LMI constraints.
Many optimization problems in
control theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
,
system identification
The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design#System identification and stochastic approximation, optimal de ...
and
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
can be formulated using LMIs. Also LMIs find application in
Polynomial Sum-Of-Squares. The prototypical primal and dual
semidefinite program is a minimization of a real linear function respectively subject to the primal and
dual convex cone
In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, C is a cone if x\in C implies sx\in C for e ...
s governing this LMI.
Solving LMIs
A major breakthrough in convex optimization was the introduction of
interior-point methods. These methods were developed in a series of papers and became of true interest in the context of LMI problems in the work of
Yurii Nesterov and
Arkadi Nemirovski.
See also
*
Semidefinite programming
*
Spectrahedron
*
Finsler's lemma
References
* Y. Nesterov and A. Nemirovsky, ''Interior Point Polynomial Methods in Convex Programming.'' SIAM, 1994.
External links
* S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan
Linear Matrix Inequalities in System and Control Theory (book in pdf)
* C. Scherer and S. Weiland
Linear Matrix Inequalities in Control
Convex optimization