First-order penalty methods for bilevel optimization
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In this paper we study a class of unconstrained and constrained bilevel optimization problems in which the lower-level part is a convex optimization problem, while the upper-level part is possibly a nonconvex optimization problem. In particular, we propose penalty methods for solving them, whose subproblems turn out to be a structured minimax problem and are suitably solved by a first-order method developed in this paper. Under some suitable assumptions, an operation complexity of O(ε^-4logε^-1) and O(ε^-7logε^-1), measured by their fundamental operations, is established for the proposed penalty methods for finding an ε-KKT solution of the unconstrained and constrained bilevel optimization problems, respectively. To the best of our knowledge, the methodology and results in this paper are new.
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