[Paper Review] Optimality, identifiability, and sensitivity

Dmitriy Drusvyatskiy, Adrian S. Lewis|arXiv (Cornell University)|Jul 27, 2012

Optimization and Variational Analysis23 references4 citations

TL;DR

This paper introduces the concept of minimal identifiable sets in optimization, showing they are central to optimality conditions, sensitivity analysis, and active set methods. It establishes that identifiable manifolds are equivalent to partial smoothness, unifying key variational analysis concepts and providing a geometric foundation for convergence and stability in nonsmooth optimization problems.

ABSTRACT

Around a solution of an optimization problem, an "identifiable" subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are "minimal". This new notion lays a broad and intuitive variational-analytic foundation for optimality conditions, sensitivity, and active set methods.

Motivation & Objective

• To develop a variational-analytic framework for optimality and sensitivity in nonsmooth optimization using the concept of identifiability.
• To identify minimal identifiable sets as the most informative subsets for characterizing critical points under perturbations.
• To unify active set methods, critical cones, and sensitivity analysis through the notion of identifiable manifolds.
• To establish a precise connection between identifiable manifolds and the classical notion of partial smoothness.
• To provide a geometric and analytical foundation for convergence and stability in optimization algorithms via identifiable sets.

Proposed method

• Defines identifiability for set-valued mappings, particularly subdifferential mappings, as a property ensuring that sequences approaching a critical point with vanishing subgradients eventually lie within the identifiable set.
• Introduces locally minimal identifiable sets as the smallest such sets containing all nearby critical points under small perturbations.
• Uses critical cones and tangent cones to characterize the geometry of identifiable sets, showing that the closed convex hull of the tangent cone to an identifiable set equals the critical cone under regularity conditions.
• Applies prox-regularity and Clarke regularity to ensure stability and smooth approximations of the subdifferential mapping near the critical point.
• Establishes equivalence between identifiable manifolds and partial smoothness via four conditions: prox-regularity, smoothness on the manifold, sharpness of the normal cone, and inner semicontinuity of the subdifferential.
• Uses the proximal mapping as a functional analog of metric projection to extend results to function settings, though the full extension is omitted for brevity.

Experimental results

Research questions

• RQ1 What conditions ensure the existence and uniqueness of a locally minimal identifiable set at a critical point?
• RQ2 How are identifiable sets related to critical cones and active set structures in optimization?
• RQ3 In what sense is an identifiable manifold equivalent to partial smoothness in variational analysis?
• RQ4 How does identifiability simplify sensitivity analysis and optimality conditions?
• RQ5 What role do tangent cones and normal cones play in characterizing identifiable sets and their geometric structure?

Key findings

• A locally minimal identifiable set exists and is unique locally if it exists, and it consists precisely of all critical points of small linear perturbations of the objective function.
• Quadratic growth of a function around a critical point is equivalent to quadratic growth on its minimal identifiable set, simplifying optimality verification.
• The critical cone at a point equals the closed convex hull of the tangent cone to the minimal identifiable set under Clarke regularity and prox-regularity conditions.
• Identifiable manifolds are equivalent to partial smoothness: a C² manifold M is identifiable at x for v if and only if f is partly smooth with respect to M at x for v and v lies in the relative interior of the subdifferential.
• The subdifferential mapping of f is locally determined by its restriction to the minimal identifiable set, meaning the graph of ∂f near (x,f(x),v) coincides with that of ∂(f+δ_M) near (x,f(x),v).
• When f is Clarke regular and f+δ_M is prox-regular and smoothly derivable, the critical cone K_f(x,v) is exactly the closed convex hull of the tangent cone to M at x.