- Author:
- Farid Alizadeh-Dehkharghani
We consider the problem of optimizing a linear function of a symmetric matrix X, subject to linear constraints on the matrix and the additional condition that X be positive semi-definite; we call this problem the semi-definite programming problem (SDP).
First we develop a duality theory for SDP which is structurally similar to the duality theory in linear programming and is a straightforward generalization of the latter. We also show that various eigenvalue optimization problems may be formulated as semi-definite programs.
Second, we develop two primal-dual interior-point methods for solving (SDP) which again is direct generalization of interior methods for linear programming. Furthermore, we argue that just about any linear programming interior point method may be modified in a mechanical way to yield a method for the SDP problems. The proofs of convergence or polynomial-time solvability may also be extended to the SDP problem.
Third, we study applications of SDP in combinatorial optimization problems. In particular, we apply our interior point methods to obtain good bounds for the maximum clique problem in perfect graphs, graph partitioning (and in particular graph bisection) problems, maximum induced multipartite subgraph problem, and the general 0-1 integer programming problems.
Index Terms
- Combinatorial optimization with interior point methods and semi-definite matrices
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