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Since more than three decades, interior-point methods proved very useful foroptimization, from linear over semidefinite to conic (and partly beyondnon-convex) programming; despite the fact that already in the semidefinite case(even when strong duality holds) "hard" problems are known. We shade a light ona rather surprising restriction in the non-commutative world (of semidefiniteprogramming), namely "commutative" paths and propose a new family of solversthat is able to use the full richness of "non-commutative" search directions:(primal) feasible-interior-point methods. Beside a detailed basic discussion,we illustrate some variants of "non-commutative" paths and provide a simpleimplementation for further (problem specific) investigations.

Primal-dual interior-point Methods for Semidefinite Programming from an algebraic point of view, or: Using Noncommutativity for Optimization