Projection and contraction methods for semidefinite programming

Recently, He (1992, 1994a,b, 1996) presented some iterative projection and contraction (PC) methods for linear variational inequalities. Its simplicity, robustness and ability to handle large problems has attracted a lot of attention. This paper extends the PC methods from linear programming to semidefinite programming (SDP). The SDP problem is transformed into an equivalent projection equation, which is solved by PC methods. Some promising numerical results obtained from a preliminary implementation are included.

[1]  Bingsheng He,et al.  A new method for a class of linear variational inequalities , 1994, Math. Program..

[2]  E. Polak.  Introduction to linear and nonlinear programming , 1973 .

[3]  Bin-Xin He.  Solving a class of linear projection equations , 1994 .

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

[5]  Bingsheng He,et al.  A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming , 1992 .

[6]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[7]  M. Overton.  On minimizing the maximum eigenvalue of a symmetric matrix , 1988 .

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

[9]  W. Glunt.  An alternating projections method for certain linear problems in a Hilbert space , 1995 .

[10]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[11]  R. Fletcher.  Semi-Definite Matrix Constraints in Optimization , 1985 .

[12]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[13]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..