arXiv subjects

Nguyen T. V. Hang

Publications and source records attributed to Nguyen T. V. Hang.

7 recordsLinked to original sources

arXiv2026

Stabilized SQP Method for Convex Piecewise Linear-Quadratic Composite Optimization Problems

This paper addresses the local convergence analysis of the stabilized sequential quadratic programming method for a class of composite optimization problems with convex piecewise linear- quadratic (CPLQ) modeling functions. We demonstrate that an appropriate second-order sufficient condition (SOSC) -formulated for composite problems- alone is sufficient to guarantee local superlin- ear convergence of the primal-dual sequence generated by this method. Furthermore, we examine the relaxation of the SOSC and show that it can be replaced with a weaker noncriticality of the associated Lagrange multiplier, provided, the modeling function is polyhedral and a relative interior condition is satisfied. We also present counterexamples showing that the relative interior condition is indispensable and that this relaxation fails for general CPLQ modeling functions.

math.OC↗

arXiv2026

A fresh look into variational analysis of $\mathcal C^2$-partly smooth functions

$\mathcal C^2$-partial smoothness of functions has been an important subject of research in optimization, on both theoretical and algorithmic aspects, since it was first introduced by Lewis in 2002. Our work aims at providing a fresh variational analysis viewpoint on the class of $\mathcal C^2$-partly smooth functions. Namely, we explore the relationship between $\mathcal C^2$-partial smoothness and strict twice epi-differentiability and demonstrate that functions from the latter class are always strictly twice epi-differentiable. On the other hand, we provide two examples to show that the opposite conclusion does not hold in general. As a consequence of our analysis, we calculate the second subderivative of $\mathcal C^2$-partly smooth functions. Applications to stability analysis of related generalized equations involving a general perturbation and to asymptotic analysis of the well-known sample average approximation method for stochastic programs with $\mathcal C^2$-partly smooth regularizers are also given.

math.OC↗

arXiv2024

Smoothness of Subgradient Mappings and Its Applications in Parametric Optimization

We demonstrate that the concept of strict proto-differentiability of subgradient mappings can play a similar role as smoothness of the gradient mapping of a function in the study of subgradient mappings of prox-regular functions. We then show that metric regularity and strong metric regularity are equivalent for a class of generalized equations when this condition is satisfied. For a class of composite functions, called C2-decomposable, we argue that strict proto-differentiability can be characterized via a simple relative interior condition. Leveraging this observation, we present a characterization of the continuous differentiability of the proximal mapping for this class of function via a certain relative interior condition. Applications to the study of strong metric regularity of the KKT system of a class of composite optimization problems are also provided.

math.OC↗

arXiv2023

Convergence of Augmented Lagrangian Methods for Composite Optimization Problems

Local convergence analysis of the augmented Lagrangian method (ALM) is established for a large class of composite optimization problems with nonunique Lagrange multipliers under a second-order sufficient condition. We present a new second-order variational property, called the semi-stability of second subderivatives, and demonstrate that it is widely satisfied for numerous classes of functions, important for applications in constrained and composite optimization problems. Using the latter condition and a certain second-order sufficient condition, we are able to establish Q-linear convergence of the primal-dual sequence for an inexact version of the ALM for composite programs.

math.OC↗

arXiv2021

Augmented Lagrangian Method for Second-Order Cone Programs under Second-Order Sufficiency

This paper addresses problems of second-order cone programming important in optimization theory and applications. The main attention is paid to the augmented Lagrangian method (ALM) for such problems considered in both exact and inexact forms. Using generalized differential tools of second-order variational analysis, we formulate the corresponding version of second-order sufficiency and use it to establish, among other results, the uniform second-order growth condition for the augmented Lagrangian. The latter allows us to justify the solvability of subproblems in the ALM and to prove the linear primal-dual convergence of this method.

math.OC↗

arXiv2021

Local Convergence Analysis of Augmented Lagrangian Methods for Piecewise Linear-Quadratic Composite Optimization Problems

Second-order sufficient conditions for local optimality have been playing an important role in local convergence analysis of optimization algorithms. In this paper, we demonstrate that this condition alone suffices to justify the linear convergence of the primal-dual sequence, generated by the augmented Lagrangian method for piecewise linear-quadratic composite optimization problems, even when the Lagrange multiplier in this class of problems is not unique. Furthermore, we establish the equivalence between the second-order sufficient condition and the quadratic growth condition of the augmented Lagrangian problem for this class of composite optimization problems.

math.OC↗

arXiv2017

Second-oder analysis in second-oder cone programming

The paper conducts a second-order variational analysis for an important class of nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone $Q$. From one hand, we prove that the indicator function of $Q$ is always twice epi-differentiable and apply this result to characterizing the uniqueness of Lagrange multipliers at stationary points together with an error bound estimate in the general second-order cone setting involving ${\cal C}^2$-smooth data. On the other hand, we precisely calculate the graphical derivative of the normal cone mapping to $Q$ under the weakest metric subregularity constraint qualification and then give an application of the latter result to a complete characterization of isolated calmness for perturbed variational systems associated with second-order cone programs. The obtained results seem to be the first in the literature in these directions for nonpolyhedral problems without imposing any nondegeneracy assumptions.

math.OC↗