Core Concepts
The author introduces an alternative method for solving convex optimization problems using homotopy, focusing on transforming feasible sets continuously. The approach numerically outperforms state-of-the-art methods in various cases.
Abstract
Homotopy methods are explored as a novel approach to solving convex optimization problems. The paper introduces the concept of transforming feasible sets continuously through a homotopy technique. By tracking optimal solutions along this path, the method proves to be effective in various scenarios, outperforming existing methods. The study delves into semidefinite programs, hyperbolic programs, and convex optimization with single constraints, showcasing the superiority of the proposed approach numerically. Additionally, comparisons with traditional interior point methods highlight the efficiency and effectiveness of homotopy methods in certain classes of problems.
Stats
Interior point methods are state-of-the-art for solving convex optimization problems.
Homotopically changing a problem's feasible set is a key aspect of the proposed method.
The path of optimal solutions is determined by solving a system of differential equations.
Hyperbolic programming involves computing ground states using adiabatic quantum computing.
Real zero polynomials play a crucial role in defining rigidly convex sets.
Quotes
"Convex optimization encompasses efficiently solvable subclasses."
"Homotopies offer an alternative method for tackling convex optimization problems."
"The unique solution path is determined by solving differential equations."