A Mathematical Theory of Robust Optimisation Under Deep Uncertainty for Development Planning in Northern Nigeria

This dissertation develops an original mathematical theory of robust optimisation under deep uncertainty, specifically tailored to long-horizon development planning problems in Northern Nigeria where severe data scarcity, structural model uncertainty, and irreducible ambiguity about future states preclude the specification of reliable probability distributions over planning outcomes. Classical stochastic optimisation requires well-defined probability distributions over uncertain parameters, an assumption that is frequently untenable in the development planning contexts of Northern Nigeria, where data infrastructure is weak, structural change is rapid, and the range of plausible futures is genuinely ambiguous. The dissertation extends and generalises the info-gap decision theory framework of Ben-Haim to a class of nonlinear programming problems with multi-level nested uncertainty structures, proving novel robustness theorems that characterise the relationship between decision robustness and expected performance under fractional sets of uncertainty. A duality theory for robust counterpart problems with non-convex uncertainty sets is developed, establishing tractable reformulations for polynomial and signomial programming problems under ellipsoidal and interval uncertainty. The mathematical framework is applied to a multi-year infrastructure investment allocation problem for Northern Nigerian states under ambiguity about population growth, commodity price trajectories, and climate-related infrastructure damage rates. Pareto frontier characterisations of the robustness-performance tradeoff are derived analytically and computed numerically for the planning application. Keywords: robust optimisation, deep uncertainty, info-gap theory, development planning, Northern Nigeria