Sensitivity analysis for parametric nonlinear programming: A tutorial
François Pacaud
TL;DR
This tutorial systematically organizes sensitivity analysis for parametric nonlinear programming, tracing developments from regularity-based results (IFT under LICQ/SSOSC and SCS) to handling degeneracy (non-unique multipliers) and extending to conic programs. It blends theory (Fiacco, Robinson, lexicographic derivatives) with practical numerical methods (HSD embedding, CasADi, sIpopt, path-following) to compute forward and adjoint sensitivities as well as value-function derivatives. Key contributions include explicit conditions for continuity and differentiability of primal-dual solutions and the value function under various constraint qualifications, plus robust treatments of degeneracy and approximate interior-point solutions. The discussion highlights the practical impact for differentiable programming, stochastic and bilevel problems, and efficient sensitivity computation in large-scale or structured optimization problems.
Abstract
This tutorial provides an overview of the current state-of-the-art in the sensitivity analysis for nonlinear programming. Building upon the fundamental work of Fiacco, it derives the sensitivity of primal-dual solutions for regular nonlinear programs and explores the extent to which Fiacco's framework can be extended to degenerate nonlinear programs with non-unique dual solutions. The survey ends with a discussion on how to adapt the sensitivity analysis for conic programs and approximate solutions obtained from interior-point algorithms.