  Task and Motion Planning (TAMP) integrates high-level task planning and
low-level motion planning to equip robots with the autonomy to effectively
reason over long-horizon, dynamic tasks. Optimization-based TAMP focuses on
hybrid optimization approaches that define goal conditions via objective
functions and are capable of handling open-ended goals, robotic dynamics, and
physical interaction between the robot and the environment. Therefore,
optimization-based TAMP is particularly suited to solve highly complex,
contact-rich locomotion and manipulation problems. This survey provides a
comprehensive review on optimization-based TAMP, covering (i) planning domain
representations, including action description languages and temporal logic,
(ii) individual solution strategies for components of TAMP, including AI
planning and trajectory optimization (TO), and (iii) the dynamic interplay
between logic-based task planning and model-based TO. A particular focus of
this survey is to highlight the algorithm structures to efficiently solve TAMP,
especially hierarchical and distributed approaches. Additionally, the survey
emphasizes the synergy between the classical methods and contemporary
learning-based innovations such as large language models. Furthermore, the
future research directions for TAMP is discussed in this survey, highlighting
both algorithmic and application-specific challenges.
