  Convex optimization encompasses a wide range of optimization problems,
containing many efficiently solvable subclasses. Interior point methods are
currently the state-of-the-art approach for solving such problems, particularly
effective for classes like semidefinite programming, quadratic programming, and
geometric programming. However, their success hinges on the construction of
self-concordant barrier functions for the feasible sets. In this work, we
introduce an alternative method for tackling convex optimization problems,
employing a homotopy. With this technique, the feasible set of a trivial
optimization problem is continuously transformed into the target one, while
tracking the solutions. We conduct an analysis of this approach, focusing on
its application to semidefinite programs, hyperbolic programs, and convex
optimization problems with a single convexity constraint. Moreover, we
demonstrate that our approach numerically outperforms state-of-the-art methods
in several interesting cases.
