SpaceX recently utilised a convex optimisation method to figure out the optimal landing route for the rocket, which was aided by real-time computer vision data. In a rocket, the landing trajectory should be replanned in real time to accomplish precision landing under substantial initial state deviation. When new data is acquired while in flight, rapid progression, a way of planning that can meet the real-time need, will be required in the future.
The landing trajectory planning issue is highly nonconvex due to nonlinear dynamics and many nonconvex state and control restrictions, making it challenging to solve in real time using existing nonconvex optimization algorithms.
In solving fast trajectory optimization issues, a study found that convex optimization approaches have a significant performance benefit. While theoretically possible, convex optimization can be used to obtain the optimal solution in polynomial time using classic convex methods.
What is Convex Optimization?
In the field of convex optimisation, the goal is to reduce a convex function’s area under a given constraint to the smallest possible value. Many classes of convex programmes have benefited from efficient algorithms based on convexity and its various implications. As a result, convex optimisation has had a significant impact on many scientific and engineering fields. Surprisingly, convex optimisation algorithms have been utilised to create counting issues for discrete objects like matroids. Convex optimisation techniques, have proven essential in a wide range of modern machine learning applications. The demand for convex optimisation methods has pushed the state of the art of convex optimisation itself, owing to larger and increasingly complicated input cases. Many convex optimisation problems can be solved in polynomial time, although mathematical optimisation is generally NP-hard.
Convex optimisation problems are more generic than linear programming problems, although they share some of the same desired characteristics: they can be solved rapidly and consistently. Using gradient descent or Newton’s method combined with line search to find an appropriate step size makes it easy to solve unconstrained convex optimisation problems. However, the equality constraints can also be eliminated with linear algebra or solved by solving the dual problem. Finally, by using an unconstrained convex optimisation technique on the objective function plus logarithmic barrier terms, convex optimisation with both linear equality and convex inequality constraints can be solved.
The following current methods can also be used to tackle convex optimisation problems:
- Bundle methods
- Subgradient projection methods
- Cutting-plane methods
- Ellipsoid method
- Subgradient method
- Dual subgradients and the drift-plus-penalty method
Here, the subgradient algorithms are extensively utilised because they are simple to implement. Subgradient methods used to solve a dual problem are known as dual subgradient methods. The drift-plus-penalty approach is similar to the dual subgradient method, except it takes the primal variables’ time average. Biconvex, pseudo-convex and quasiconvex functions are among the extensions of convex optimisation. In the topic of generalised convexity, often known as abstract convex analysis, extensions of the theory of convex analysis and iterative methods for approximately solving non-convex minimisation problems are found.
The problem of rocket landing trajectory planning is highly nonconvex due to nonlinear dynamics and numerous nonconvex state and control restrictions. As a result, using current nonconvex optimization approaches, it is difficult to solve in real time. According to a study, convex optimization methods provide a significant performance advantage when it comes to solving fast trajectory optimization problems. Recently, SpaceX employed a convex optimisation algorithm to figure out the ideal landing route for the rocket, with real-time computer vision data assisting with route identification. Let us wait and see what SpaceX does beyond the convex optimisation algorithm’s complexities. Certainly, additional AI advancements in space exploration are on the way in the near future. The contributions of Indian researchers will play a significant role in these breakthroughs.
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Nivash has a doctorate in Information Technology. He has worked as a Research Associate at a University and as a Development Engineer in the IT Industry. He is passionate about data science and machine learning.