The projective method for solving linear matrix inequalities

Numerous problems in control and systems theory can be formulated in terms of linear matrix inequalities (LMI). Since solving an LMI amounts to a convex optimization problem, such formulations are known to be numerically tractable. However, the interest in LMI-based design techniques has really surged with the introduction of efficient interior-point methods for solving LMIs with a polynomial-time complexity. This paper describes one particular method called the Projective Method. Simple geometrical arguments are used to clarify the strategy and convergence mechanism of the Projective algorithm. A complexity analysis is provided, and applications to two generic LMI problems (feasibility and linear objective minimization) are discussed.

[1]  A. Berman.  Cones, matrices and mathematical programming , 1973 .

[2]  N. Rau.  Matrices and Mathematical Programming , 1981 .

[3]  Gene H. Golub,et al.  Matrix computations , 1983 .

[4]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[5]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[6]  Franz Rendl,et al.  Applications of parametric programming and eigenvalue maximization to the quadratic assignment problem , 1992, Math. Program..

[7]  Michael L. Overton,et al.  Large-Scale Optimization of Eigenvalues , 1990, SIAM J. Optim..

[8]  F. Jarre.  An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices , 1993 .

[9]  Stephen P. Boyd,et al.  Method of centers for minimizing generalized eigenvalues , 1993, Linear Algebra and its Applications.

[10]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[11]  P. Gahinet,et al.  A linear matrix inequality approach to H∞ control , 1994 .

[12]  Alan J. Laub,et al.  The LMI control toolbox , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[13]  Stephen P. Boyd,et al.  A primal—dual potential reduction method for problems involving matrix inequalities , 1995, Math. Program..

[14]  Yurii Nesterov,et al.  An interior-point method for generalized linear-fractional programming , 1995, Math. Program..

[15]  Shinji Hara,et al.  Interior Point Methods for the Monotone Linear Complementarity Problem in Symmetric Matrices , 1995 .

[16]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..