A Barrier Function Approach for Bilevel Optimization with Coupled Lower-Level Constraints: Formulation, Approximation and Algorithms

TL;DR

This work tackles BLOs with coupled lower-level constraints by introducing a log-barrier reformulation that renders the lower-level unconstrained and enables computable hypergradients. It proves convergence of hyperfunction values and hypergradients under mild assumptions, and provides a non-asymptotic convergence guarantee for the barrier formulation via an adaptive algorithm; it also establishes asymptotic convergence to a stationary point of the original BLO when SCSC holds. The approach covers both strongly convex and linear lower-level settings, including the LP case, and is supported by numerical experiments showing feasibility guarantees and strong performance, notably in price-setting applications. Overall, the paper delivers a unified, minimally-assumptive framework with rigorous convergence guarantees for barrier-based BLOs with coupled lower-level constraints, expanding applicability to LP-lower-level problems.

Abstract

In this paper, we consider bilevel optimization problem where the lower-level has coupled constraints, i.e. the constraints depend both on the upper- and lower-level variables. In particular, we consider two settings for the lower-level problem. The first is when the objective is strongly convex and the constraints are convex with respect to the lower-level variable; The second is when the lower-level is a linear program. We propose to utilize a barrier function reformulation to translate the problem into an unconstrained problem. By developing a series of new techniques, we proved that both the hyperfunction value and hypergradient of the barrier reformulated problem (uniformly) converge to those of the original problem under minimal assumptions. Further, to overcome the non-Lipschitz smoothness of hyperfunction and lower-level problem for barrier reformulated problems, we design an adaptive algorithm that ensures a non-asymptotic convergence guarantee. We also design an algorithm that converges to the stationary point of the original problem asymptotically under certain assumptions. The proposed algorithms require minimal assumptions, and to our knowledge, they are the first with convergence guarantees when the lower-level problem is a linear program. Numerical experiments are conducted to show the effectiveness of the proposed method.

Figures (3)

• Figure 1: $d_{s-1}$ is an output of Algorithm \ref{['algo:fo_lower_2']}, which indicates that there exists one point $y_0$ such that $h_i(x_{s-1},y_0)\leq -d_{s-1}$ for any $i$. By Lipschitz continuity of $h_i(x,y)$, when $x$ is in the ball $B_{x_{s-1}}(d_{s-1}/(2L_h))$, we have $h_i(x,y_0)\leq -d_{s-1}/2$ for any $i$. In view of Theorem \ref{['lem:local_M']}, we have a negative upper bound of $h_{i}(x,y^\ast_t(x))$ for any $i\in\{1,..,k\}$ and $x$ in the ball $B_{x_{s-1}}(d_{s-1}/(2L_h))$.
• Figure 2: Utilizing the negative upper bound of $h_{i}(x,y^\ast_t(x))$ in the ball $B_{x_{s-1}}(d_{s-1}/(2L_h))$ obtained from left figure, we estimate the Lipschitz smoothness constant of $\widetilde{\phi}_t(x)$ in this ball, and design the stepsize $\eta_{s-1}$. The design of $\eta_{s-1}$ ensures that $x_s$ is still in $B_{x_{s-1}}(d_{s-1}/(2L_h))$. We update and get $x_{s}$ by Step 2 of Algorithm \ref{['algo:fo']}. We also get the ball $B_{x_s}(d_s/(2L_h))$ by repeating the argument in the left figure, then update to the next point $x_{s+1}$ by repeating the above process.
• Figure 3: Find a spherical neighborhood that satisfies the first three properties, as shown by the solid hemisphere in the figure, then find a tubular neighborhood within it, as shown by the red dashed tube in the figure.

Theorems & Definitions (51)

• Remark 3.1
• Proposition 3.1
• Theorem 3.1: Lemma 1 in tsaknakis2023implicit
• Example 3.1
• Lemma 3.1: Optimality gap
• Theorem 3.2: Hyperfunction value convergence under Linear setting
• Definition 3.1: SCSC point
• Remark 3.2
• Proposition 3.2
• Theorem 3.3: Jacobian convergence under Linear setting