Convex optimization encompasses a wide range of optimization problems that contain 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 feasible sets. In this work, we investigate and develop a homotopy-based approach to solve convex optimization problems. While homotopy methods have been considered in optimization before, their potential for general convex programs remains underexplored. This approach gradually transforms the feasible set of a trivial optimization problem into the target one while tracking solutions by solving a differential equation, in contrast to traditional central path methods. We establish a criterion that ensures that the homotopy method correctly solves the optimization problem and prove the existence of such homotopies for several important classes, including semidefinite and hyperbolic programs. Furthermore, we demonstrate that our approach numerically outperforms state-of-the-art methods in hyperbolic programming, highlighting its practical advantages.

翻译：凸优化涵盖了一系列优化问题，其中包含许多可高效求解的子类。内点法是目前解决此类问题的最先进方法，尤其在半定规划、二次规划和几何规划等类别中表现出色。然而，其成功与否取决于可行集自协调障碍函数的构造。本文研究并发展了一种基于同伦的方法来解决凸优化问题。尽管同伦方法在优化领域已有先例，但其在一般凸规划中的潜力仍未得到充分探索。与传统中心路径方法不同，该方法通过求解微分方程追踪解路径，逐步将平凡优化问题的可行集变换为目标可行集。我们建立了一个确保同伦方法正确求解优化问题的准则，并证明了若干重要类别（包括半定规划和双曲规划）存在此类同伦变换。此外，我们通过数值实验证明，在双曲规划中本方法优于现有最先进方法，凸显了其实际优势。