Yes, there are several additional assumptions and problem structures that could potentially enable first-order methods to find absolute optimal solutions for simple bilevel problems:
Strong Convexity: If both the upper-level and lower-level functions are strongly convex, this could facilitate the use of first-order methods to achieve absolute optimal solutions. Strong convexity ensures that the functions have unique minimizers and that the optimization landscape is well-behaved, which can help in overcoming the intractability issues highlighted in the paper.
Smoothness Conditions: Imposing stronger smoothness conditions, such as Lipschitz continuity of the gradients for both functions, can also aid in the convergence of first-order methods. If the gradients are Lipschitz continuous, it may be possible to derive tighter bounds on the convergence rates, potentially allowing for absolute optimal solutions under certain conditions.
Special Structure of the Lower-Level Problem: If the lower-level problem exhibits a specific structure, such as being a linear program or having a closed-form solution, this could simplify the optimization process. In such cases, first-order methods could leverage the structure to compute exact solutions for the lower-level problem, thus enabling the upper-level optimization to be solved more effectively.
Error Bound Conditions: Introducing error bound conditions on the lower-level function can provide a framework within which first-order methods can operate more effectively. These conditions can help in establishing relationships between the upper-level and lower-level solutions, potentially leading to absolute optimal solutions.
Tighter Coupling Between Levels: If the upper-level and lower-level problems are more tightly coupled, meaning that the solution of one significantly influences the other, this could create a scenario where first-order methods can effectively navigate the optimization landscape to find absolute optimal solutions.
By exploring these additional assumptions and problem structures, researchers can potentially enhance the capabilities of first-order methods in solving simple bilevel problems, moving closer to achieving absolute optimal solutions.