 
Summary: INTERIOR POINT METHODS IN SEMIDEFINITE PROGRAMMING
WITH APPLICATIONS TO COMBINATORIAL OPTIMIZATION \Lambda
FARID ALIZADEH y
Abstract. We study the semidefinite programming problem (SDP), i.e the optimization problem
of a linear function of a symmetric matrix subject to linear equality constraints and the additional
condition that the matrix be positive semidefinite. First we review the classical cone duality as
is specialized to SDP. Next we present an interior point algorithm which converges to the optimal
solution in polynomial time. The approach is a direct extension of Ye's projective method for linear
programming. We also argue that many known interior point methods for linear programs can be
transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial
time complexity also carrying over in a similar fashion. Finally we study the significance of these
results in a variety of combinatorial optimizationproblems including the general 01 integer programs,
the maximum clique and maximum stable set problems in perfect graphs, the maximum kpartite
subgraph problem in graphs, and various graph partitioning and cut problems. As a result, we
present barrier oracles for certain combinatorial optimization problems (in particular, clique and
stable set problem for perfect graphs) whose linear programming formulation requires exponentially
many inequalities. Existence of such barrier oracles refutes the commonly believed notion that in
order to solve a combinatorial optimization problem with interior point methods, one needs its linear
programming formulation explicitly.
Key words. semidefinite programming, interior point methods, eigenvalue optimization, com
