Partially augmented Lagrangian method for matrix inequality constraints

SIAM Journal on Optimization

Volumn 15, Issue 1, 2005, Pages 161-184

  Noll, Dominikus ^a   Torki, Mounir ^b   Apkarian, Pierre ^a,c  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b UNIVERSITY OF AVIGNON   (France)

^c ONERA   (France)

Author keywords

Augmented Lagrangian; Bilinear matrix inequalities; Linear matrix inequalities; Semidefinite programming

Indexed keywords

COMPUTER SIMULATION; FEEDBACK CONTROL; FUNCTIONS; INTEGER PROGRAMMING; MATRIX ALGEBRA; SET THEORY; SPECIFICATIONS;

AUGMENTED LAGRANGIAN; BILINEAR MATRIX INEQUALITIES; LINEAR MATRIX INEQUALITIES; SEMIDEFINITE PROGRAMMING (SDP);

LAGRANGE MULTIPLIERS;

EID: 14944347490     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623402413963     Document Type: Article

References (54)

1. B. ANDERSON AND S. VONGPANITLERD, Network Analysis and Synthesis: A Modern System Theory Approach, Prentice-Hall, Englewood Cliffs, NJ, 1973.
2. ∞ controllers, IEEE Trans. Automat. Control, 40 (1995), pp. 853-864.
3. ∞ control of linear parameter-varying systems: A design example, Automatica J. IFAC, 31 (1995), pp. 1251-1261.
4. P. APKARIAN, D. NOLL, AND H. D. TUAN, A Prototype Primal-Dual LMI-Interior Algorithm for Nonconvex Robust Control Problems, Rapport Interne 01-08, MIP, UMR 5640, Toulouse, France, 2001. Available online at http://mip.ups-tlse.fr/publi/publi.html.
5. ∞ control design via an augmented Lagrangian method, Internat. J. Robust Nonlinear Control, 13 (2003), pp. 1137-1148.
6. P. APKARIAN AND H. D. TUAN, Concave programming in control theory, J. Global Optim., 15 (1999), pp. 343-370.
7. V. I. ARNOLD, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), pp. 29-43.
8. G. J. BALAS, J. C. DOYLE, K. GLOVER, A. PACKARD, AND R. SMITH, μ-Analysis and Synthesis Toolbox: User's Guide, The MathWorks, Inc., Natick, MA, 1991.
9. K. P. BENNETT AND O. L. MANGASARIAN, Bilinear separation of two sets in n-space, Comput. Optim. Appl., 2 (1993), pp. 207-227.
10. A. BEN-TAL AND A. NEMIROVSKI, Lectures on Modern Convex Optimization, MPS/SIAM Ser. Optim., SIAM, Philadelphia, 2001.
11. A. BEN-TAL, Conic and Robust Optimization, Lecture Notes, Università di Roma La Sapienzia, Rome, Italy, 2002.
12. S. BOYD, L. EL GHAOUI, E. FERON, AND V. BALAKRISHNAN, Linear Matrix Inequalities in System and Control Theory, SIAM Stud. Appl. Math. 15, SIAM, Philadelphia, 1994.
13. J. F. BONNANS AND A. SHAPIRO, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
14. H. BUSCHEK AND H. A. J. CALISE, Mu controllers: Mixed and fixed, AIAA J. Guidance Control Dyn., 20 (1995), pp. 43-41.
15. J. BURKE, A. S. LEWIS, AND M. L. OVERTON, Optimizing matrix stability, Proc. Amer. Math. Soc., 129 (2000), pp. 1635-1642.
16. J. BURKE, A. S. LEWIS, AND M. L. OVERTON, Two numerical methods for optimizing matrix stability, Linear Algebra Appl., 351/352 (2002), pp. 117-145.
17. H. A. J. CALISE AND E. V. BYRN, Parameter sensitivity reduction in fixed-order dynamic compensation, AIAA J. Guidance Control Dyn., 15 (1992), pp. 440-447.
18. m-synthesis via generalized Popov multipliers, in Proceedings of the American Control Conference, Chicago, IL, 1992, pp. 2417-2418.
19. F. H. CLARKE, Optimization and Nonsmooth Analysis, SIAM Class. Appl. Math. 5, SIAM, Philadelphia, 1990.
20. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds, SIAM J. Numer. Anal., 28 (1991), pp. 545-572.
21. A. R. CONN, N. I. M. GOULD, A. SARTENAER, AND PH. L. TOINT, Global Convergence of Two Augmented Lagrangian Algorithms for Optimization with a Combination of General Equality and Linear Constraints, Technical report TR/PA/93/26, Toulouse, France, 1993.
22. A. R. CONN, N. I. M. GOULD, A. SARTENAER, AND PH. L. TOINT, Local Convergence Properties of Two Augmented Lagrangian Algorithms for Optimization with a Combination of General Equality and Linear Constraints, Technical report TR/PA/93/27, Toulouse, France, 1993.
23. A. R. CONN, N. I. M. GOULD, A. SARTENAER, AND PH. L. TOINT, Convergence properties of an augmented Lagrangian algorithm for optimization with a combination of general equality and linear constraints, SIAM J. Optim., 6 (1996), pp. 674-703.
24. J. C. DOYLE, A. PACKARD, AND K. ZHOU, Review of LFT's, LMI's and μ, in Proceedings of the IEEE Conference on Decision and Control, Vol. 2, Brighton, UK, 1991, pp. 1227-1232.
25. G. E. DULLERUD AND F. PAGANINI, A Course in Robust Control Theory: A Convex Approach, Springer Texts Appl. Math. 36, Springer-Verlag, New York, 2000.
26. L. EL GHAOUI, F. OUSTRY, AND M. AIT RAMI, An algorithm for static output-feedback and related problems, IEEE Trans. Automat. Control, 42 (1997), pp. 1171-1176.
27. M. K. H. FAN, A. L. TITS, AND J. C. DOYLE, Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics, IEEE Trans. Automat. Control, 36 (1991), pp. 25-38.
28. B. FARES, P. APKARIAN, AND D. NOLL, An augmented Lagrangian method for a class of LMI constrained problems in robust control theory, Internat. J. Control, 74 (2001), pp. 348-360.
29. B. FARES, D. NOLL, AND P. APKARIAN, Robust control via sequential semidefinite programming, SIAM J. Control Optim., 40 (2002), pp. 1791-1820.
30. E. FERON, P. APKARIAN, AND P. GAHINET, Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions, IEEE Trans. Automat. Control, 41 (1996), pp. 1041-1046.
31. ∞ control, Internat. J. Nonlinear Control, 4 (1994), pp. 421-448.
32. M. X. GOEMANS AND D. P. WILLIAMSON, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), pp. 1115-1145.
33. A. HELMERSSON, Methods for Robust Gain-Scheduling, Ph.D. thesis, Linkoping University, Linkoping, Sweden, 1995.
34. M. R. HESTENES, Multiplier and gradient methods, J. Optim. Theory Appl., 4 (1969), pp. 303-320.
35. ∞ control problem: LMI existence conditions and state space formulas, Automatica J. IFAC, 30 (1994), pp. 1307-1317.
36. J. LASALLE AND S. LEPSCHETZ, Stability by Lyapunov's Direct Method, Academic Press, New York, 1961.
37. F. LEIBFRITZ AND E. M. E. MOSTAFA, Trust region methods for solving the optimal output feedback design problem, Int. J. Control, 76 (2003), pp. 501-519.
38. F. LEIBFRITZ AND E. M. E. MOSTAFA, An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems, SIAM J. Optim., 12 (2002), pp. 1048-1074.
39. O. L. MANGASARIAN AND J. S. PANG, The extended linear complementarity problem, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 359-368.
40. A. MEGRETZKI AND A. RANTZER, System analysis via integral quadratic constraints, IEEE Trans. Automat. Control, 42 (1997), pp. 819-830.
41. M. MESBASHI, A semi-definite programming solution of the least order dynamic output feedback synthesis problem, in Proceedings of the IEEE Conference on Decision and Control, Phoenix, AZ, 1999.
42. A. NEMIROVSKII, Several NP-hard problems arising in robust stability analysis, Math. Control Signals Systems, 6 (1994), pp. 99-105.
43. M. C. DE OLIVEIRA AND J. C. GEROMEL, Numerical comparison of output feedback design methods, in Proceedings of the American Control Conference, Albuquerque, NM, 1997, pp. 72-76.
44. A. PACKARD, Gain scheduling via linear fractional transformations, Systems Control Lett., 22 (1994), pp. 79-92.
45. A. PACKARD AND G. BECKER, Quadratic stabilization of parametrically- dependent linear systems using parametrically-dependent linear dynamic feedback, Adv. Robust Nonlinear Control Syst., 43 (1992), pp. 29-36.
46. A. PACKARD, K. ZHOU, P. PANDEY, AND G. BECKER, A collection of robust control problems leading to LMI's, in Proceedings of the IEEE Conference on Decision and Control, Brighton, UK, 1991, pp. 1245-1250.
47. M. J. D. POWELL, A method for nonlinear constraints in minimization problems, in Optimization, R. Fletcher, ed., Academic Press, London, New York, 1969.
48. M. G. SAFONOV AND G. P. PAPAVASSILOPOULOS, The diameter of an intersection of ellipsoids and BMI robust synthesis, in Proceedings of the IFAC Symposium on Robust Control Design, Rio de Janeiro, Brazil, 1994.
49. A. SARTENAER, Développement d'algorithmes en programmation non-linéaire: de l'étude théorique à l'expérimentation numérique, Habilitation à diriger des recherches, Université Paul Sabatier, Toulouse, France, 1999.
50. C. W. SCHERER, Robust mixed control and linear parameter-varying control with full block scaling, in Advances in Linear Matrix Inequality Methods in Control, Adv. Des. Control 2, L. El Ghaoui and S.-I. Niculescu, eds., SIAM, Philadelphia, 1999, pp. 187-207.
51. G. SCORLETTI AND L. EL GHAOUI, Improved linear matrix inequality conditions for gainscheduling, in Proceedings of the IEEE Conference on Decision and Control, New Orleans, 1995, pp. 3626-3631.
52. A. SHAPIRO, First and second order analysis of nonlinear semidtfinite programs, Math. Program., 77 (1997), pp. 301-320.
53. M. TORKI, First- and second-order epi-differentiability in eigenvalue optimization, J. Math. Anal. Appl., 234 (1999), pp. 391-416.
54. K. ZHOU, J. C. DOYLE, AND K. GLOVER, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, 1996.