Notes on bilevel optimization

based on “A review on bilevel optimization From classical to evolutionary approaches and applications" as well as other materials

Bilevel Optimization

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\[\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)\]

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Classical Approaches

Given the difficult nature of bilevel problems, it is not surprising that much of the classical literature considers bilevel problems that are mathematically well-behaved; i.e., contains functions that are linear, quadratic or convex. Strong assumptions like continuous differentiability and lower semi- continuity are quite common.

Single-level Reduction

In , the problem is reduced into a single level problem by replacing penalty term with its KKT conditions. Also, there are many ways to set penalty.

For instance, in they consider using penalty term to convert bilevel optimization problems to minimax problems.

Penalty Function Methods

Distill chose a numerical inline citation style to improve readability of citation dense articles and because many of the benefits of longer citations are obviated by displaying more information on hover.

Trust-region Methods

Not interested now.

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Moreau Envelope for bilevel optimization

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