Numerical Optimization for Dynamic Optimization Problems

Many complex decision making processes can be formulated as control problems. This can be incredibly valuable since control theory can also describe the best decision to make given a desired outcome (optimal control). Such control systems are leveraged every day in high stakes applications such as managing the flow of electricity in power grids, navigation in autonomous vehicles, algorithmic trading in financial markets and recommender systems for social media.

When describing such systems mathematically, we arrive at a dynamic optimisation problem which can be solved using various algorithms from the field of numerical optimization.

However, dynamic optimization problems can be challenging to solve and require significant computational resources to arrive at a decision. More often than not, we are required to solve the dynamic optimization problem multiple times per second in order to react quickly to changes in the operating environment. To enable the widespread adaptation of dynamic optimization for decision making, more efficient algorithms need to be developed to reduce the barrier to entry and the level of expertise required for implementation.

The existing solutions in the numerical optimization literature generally rely on complex hyperparameter tuning and are prone to numerical issues in specific edge cases. This can lead to unreliable control systems that return dangerous decisions and require the need for high-precision computing hardware.

The recent work by Prof. Kerrigan's group has been focusing on the use of residual-based optimisation algorithms which solve a sequence of feasibility problems to eventually arrive at the optimal decision. This novel approach avoids numerical instability issues, allows the use of low-precision hardware and guarantees that even in the event of the solver crashing, it will still return a decision that at least satisfies the decision's constraints.

The mathematical properties of the residual-based approach to numerical optimization is still in its infancy and although it is designed purely for general optimization problems, we hypothesise it is particularly suited to dynamic optimisation problems. The mathematical properties of residual-based optimization for dynamic optimisation problems will be analysed and exploited in the development of novel numerical optimisation algorithms.

If successful, software packages will be built in the Julia and Python programming languages for the ease in adoption of our methods.

Our research goals align with many of the EPSRC's research areas such as operational research, numerical analysis, machine learning, control engineering, signal processing, energy networks and robotics. Once the main theory and algorithms have been explored, we plan on working on dynamic optimization problems prevalent in many of these areas.