Analysis and Algorithms for Nonsmooth, Nonlinear Optimisation Problems in Chemical Engineering

Abstract

This thesis develops optimisation methodologies for challenging hierarchical and combinatorial problems arising in chemical engineering, with particular emphasis on bilevel optimisation and quantum annealing. The first part of the thesis addresses optimistic bilevel optimisation problems with a convex lower-level problem. The main analytical difficulty arises from the implicit dependence of the upper-level problem on the solution of a parametric lower-level problem. Under standard regularity assumptions, the local differentiability of the lower-level solution mapping is established. An adjoint-based representation of the gradient of the reduced upper-level problem is derived, avoiding explicit construction of the sensitivity Jacobian and reducing gradient evaluation to the solution of a linear adjoint system. Building on this analysis, an augmented Lagrangian framework is developed for the reduced upper-level problem, integrating adjoint-based gradients within a projected quasi-Newton scheme. Convergence to Karush–Kuhn–Tucker (KKT) points of the reduced problem is established under standard assumptions. Furthermore, equivalence between these KKT points and S-stationary solutions of the associated mathematical programme with complementarity constraints (MPCC) is proven under MPEC-LICQ. Numerical experiments on benchmark bilevel problems demonstrate robust recovery of known optimal solutions. The second part of the thesis evaluates the potential of quantum annealing for combinatorial optimisation in chemical engineering. A representative pooling-blending benchmark problem is reformulated as a quadratic unconstrained binary optimisation (QUBO) model compatible with quantum annealers. Embedding strategies, hardware constraints, and performance characteristics are analysed using simulated and hardware-inspired architectures. Comparisons with classical solvers assess solution quality and scalability, providing a systematic evaluation of the feasibility and limitations of quantum annealing for structured engineering problems. Overall, the thesis contributes both a theoretically grounded and computationally validated framework for hierarchical optimisation and a critical assessment of emerging quantum technologies for combinatorial optimisation problems in process systems engineering.