Open Access Dissertation
Date of Award
Boris S. Mordukhovich
This dissertation conducts a second-order variational analysis for an important class on nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone. These second-order cone programs (SOCPs) are mathematically challenging due to the nonpolyhedrality of the underlying second-order cone while being important for various applications. The two main devices in our study are second epi-derivative and graphical derivative of the normal cone mapping which are proved to accumulate vital second-order information of functions/constraint systems under investigation. Our main contribution is threefold:
- proving the twice epi-differentiability of the indicator function of the second-order cone and of the augmented Lagrangian associated with SOCPs, and deriving explicit formulae for the calculation of the second epi-derivatives of both functions;
- establishing a precise formula-entirely via the initial data-for calculating the graphical derivative of the normal cone mapping generated by the constraint set of SOCPs without imposing any nondegeneracy condition;
- conducting a complete convergence analysis of the Augmented Lagrangian Method (ALM) for SOCPs with solvability, stability and local convergence analysis of both exact and inexact versions of the ALM under fairly mild assumptions.
These results have strong potentials for applications to SOCPs and related problems. Among those presented in this dissertation we mention characterizations of the uniqueness of Lagrange multipliers together with an error bound estimate for second-order cone constraints; of the isolated calmness property for solutions maps of perturbed variational systems associated with SOCPs; and also of (uniform) second-order growth condition for the augmented Lagrangian associated with SOCPs.
Nguyen, Hang Thi Van, "Variational Analysis In Second-Order Cone Programming And Applications" (2020). Wayne State University Dissertations. 2499.