Central path ($\mu >0$)

For a strictly feasible primal-dual conic optimization problem and a logarithmically homogeneous barrier, the central path consists of solutionsThe primal-dual gap is $⟨s,y⟩=\nu \mu$. Existence requires appropriate feasibility and boundedness hypotheses, rather than merely a full-rank constraint matrix. Linearizing these equations gives a central-path Newton system.

At a target barrier parameter $\mu$, let $r_p=Ax-b-s$, $r_d=A^{\top }y-c$, $r_c=y+\mu \nabla F(s)$. The Newton direction solvesFull column rank of $A$ and a positive-definite barrier Hessian give a positive-definite reduced matrix. Backtracking must keep both cone variables interior; solving the linear equations alone does not guarantee that a full step stays inside the cones.