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**1 - 3**of**3**### SOME COMPOSITE-STEP CONSTRAINED OPTIMIZATION METHODS INTERPRETED VIA THE PERTURBED SEQUENTIAL QUADRATIC PROGRAMMING FRAMEWORK

, 2013

"... We consider the inexact restoration and the composite-step sequential quadratic programming (SQP) methods, and relate them to the so-called perturbed SQP framework. In particular, iterations of the methods in question are interpreted as certain structured perturbations of the basic SQP iterations. T ..."

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We consider the inexact restoration and the composite-step sequential quadratic programming (SQP) methods, and relate them to the so-called perturbed SQP framework. In particular, iterations of the methods in question are interpreted as certain structured perturbations of the basic SQP iterations. This gives a different insight into local behaviour of those algorithms, as well as improved or different local convergence and rate of convergence results. Key words: sequential quadratic programming; inexact restoration; perturbed SQP; composite-step SQP; superlinear convergence.

### Invited “Discussion Paper ” for TOP CRITICAL LAGRANGE MULTIPLIERS: WHAT WE CURRENTLY KNOW ABOUT THEM, HOW THEY SPOIL OUR LIFE, AND WHAT WE CAN DO ABOUT IT∗

, 2014

"... We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear ..."

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We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipli-ers. This is quite striking because, typically, the set of critical multipliers is “thin ” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly-constrained augmented Lagrangian method). In spite of clear computational

### STABILIZED SEQUENTIAL QUADRATIC PROGRAMMING: A SURVEY

, 2013

"... We review the motivation for, the current state-of-the-art in convergence results, and some open questions concerning the stabilized version of the sequential quadratic programming algorithm for constrained optimization. We also discuss the tools required for its local convergence analysis, globaliz ..."

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We review the motivation for, the current state-of-the-art in convergence results, and some open questions concerning the stabilized version of the sequential quadratic programming algorithm for constrained optimization. We also discuss the tools required for its local convergence analysis, globalization challenges, and extentions of the method to the more general variational problems.