Eric C. Kerrigan

Modern second order solvers for convex optimisation, such as interior point methods, rely on primal dual information and are difficult to warm start, limiting their applicability in real time control. We propose the PVM, a duality free framework that reformulates the constrained problem as the unconstrained minimisation of a value function. The resulting problem always has a solution, yields a certificate of infeasibility and is amenable to warm starting. We develop a second order algorithm for Quadratic Programming based on the PPA and semismooth Newton methods, and establish sufficient conditions for superlinear convergence to an arbitrarily small neighbourhood of the solution. Numerical experiments on a MPC problem demonstrate competitive performance with state of the art solvers from a cold start and up to 70\% reduction in Newton iterations when warm starting.

Many nonlinear optimal control and optimization problems involve constraints that combine continuous dynamics with discrete logic conditions. Standard approaches typically rely on mixed-integer programming, which introduces scalability challenges and requires specialized solvers. This paper presents an exact reformulation of broad classes of logical constraints as binary-variable-free expressions whose differentiability properties coincide with those of the underlying predicates, enabling their direct integration into nonlinear programming models. Our approach rewrites arbitrary logical propositions into conjunctive normal form, converts them into equivalent max--min constraints, and applies a smoothing procedure that preserves the exact feasible set. The method is evaluated on two benchmark problems, a quadrotor trajectory optimization with obstacle avoidance and a hybrid two-tank system with temporal logic constraints, and is shown to obtain optimal solutions more consistently and efficiently than existing binary variable elimination techniques.

We revisit the problem of physics-informed regression, and propose a method that directly computes the state at the prediction point, simultaneously with the derivative and curvature information of the existing samples. We frame each prediction as a constrained optimisation problem, leveraging multivariate Taylor series expansions and explicitly enforcing physical laws. Each individual query can be processed with low computational cost without any pre- or re-training, in contrast to global function approximator-based solutions such as neural networks. Our comparative benchmarks on a reaction-diffusion system show competitive predictive accuracy relative to a neural network-based solution, while completely eliminating the need for long training loops, and remaining robust to changes in the sampling layout.

We present a novel, input-output data-driven approach to uncertainty model identification. As the true bounds and distributions of system uncertainties ultimately remain unknown, we depart from the goal of identifying the uncertainty model and instead look for minimal concrete statements that can be made based on an uncertain system model and available input-output data. We refer to this as unfalsifying an uncertainty model. Two different unfalsification approaches are taken. The optimistic approach determines the smallest uncertainties that could explain the given data, while the pessimistic approach finds the largest possible uncertainties suggested by the data. The pessimistic problem is revealed to be a semi-infinite program, which is solved using the local reduction algorithm. It is also shown that the optimistic and pessimistic approaches to uncertainty model unfalsification are mathematical duals. Finally, both approaches are tested using an uncertain linear model with data from a simulated nonlinear system.

Direct collocation (DC) is a widely used method for solving dynamic optimization problems (DOPs), but its implementation simplicity and computational efficiency are limited for challenging problems. For DOPs involving singular arcs, DC solutions often exhibit significant fluctuations along the singular arc, accompanied by large residual errors between collocation points, where the dynamic constraints are enforced as equality constraints. In this paper, we introduce the direct transcription method of integrated residual regularized direct collocation (IRRDC). This approach enforces dynamic constraints using a combination of point-wise residual constraints (expressed as either equalities or inequalities) and a penalty term on the integrated residual error, which helps reduce errors between collocation points. IRRDC retains the implementation simplicity of DC while improving both solution accuracy and efficiency, particularly for challenging problem types. Through the examples, we demonstrate that for problems where traditional DC results in excessive fluctuations, IRRDC effectively suppresses fluctuations and yields solutions with greater accuracy — at least two orders of magnitude lower in various error measures in relation to the dynamic and path constraints.

Robust optimal or min-max model predictive control (MPC) approaches aim to guarantee constraint satisfaction over a known, bounded uncertainty set while minimizing a worst-case performance bound. Traditionally, these methods compute a trajectory that meets the desired properties over a fixed prediction horizon, apply a portion of the resulting input, and then re-solve the MPC problem using newly obtained measurements at the next time step. However, this approach fails to account for the fact that the control trajectory will be updated in the future, potentially leading to conservative designs. In this paper, we present a novel update-aware robust optimal MPC algorithm for decreasing horizon problems on nonlinear systems that explicitly accounts for future control trajectory updates. This additional insight allows our method to provably expand the feasible solution set and guarantee improved worst-case performance bounds compared to existing techniques. Our approach formulates the trajectory generation problem as a sequence of nested existence-constrained semi-infinite programs (SIPs), which can be efficiently solved using local reduction techniques. To demonstrate its effectiveness, we evaluate our approach on a planar quadrotor problem, where it clearly outperforms an equivalent method that does not account for future updates at the cost of increased computation time.

Generalized semi-infinite programs (generalized SIPs) are problems featuring a finite number of decision variables but an infinite number of constraints. They differ from standard SIPs in that their constraint set itself depends on the choice of the decision variable. Generalized SIPs can be used to model robust optimal control problems where the uncertainty itself is a function of the state or control input, allowing for a less conservative alternative to assuming a uniform uncertainty set over the entire decision space. In this work, we demonstrate how any generalized SIP can be converted to an existence-constrained SIP through a reformulation of the constraints and solved using a local reduction approach, which approximates the infinite constraint set by a finite number of scenarios. This transformation is then exploited to solve nonlinear robust optimal control problems with state-dependent uncertainties. We showcase our proposed approach on a planar quadrotor simulation where it recovers the true generalized SIP solution and outperforms a SIP-based approach with uniform uncertainty bounds.