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.

翻译：凸优化涵盖了一系列广泛的优化问题，其中包含许多可高效求解的子类。内点法是当前解决此类问题的最前沿方法，尤其适用于半定规划、二次规划和几何规划等类别。然而，其成功依赖于为可行集构建自和谐障碍函数。本文提出了一种替代方法，通过引入同伦来求解凸优化问题。该技术将简单优化问题的可行集连续变换为目标可行集，同时追踪解的变化轨迹。我们对这一方法进行了分析，重点关注其在半定规划、双曲规划以及具有单一凸性约束的凸优化问题中的应用。此外，我们证明，在多个有趣案例中，本方法在数值性能上优于现有最先进方法。