Results

**11 - 16**of**16**### 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 ..."

Abstract
- Add to MetaCart

(Show Context)
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

### GLOBALIZING STABILIZED SQP BY SMOOTH PRIMAL-DUAL EXACT PENALTY FUNCTION∗

, 2014

"... An iteration of the stabilized sequential quadratic programming method (sSQP) consists in solving a certain quadratic program in the primal-dual space, regularized in the dual variables. The advantage with respect to the classical sequential quadratic programming (SQP) is that no constraint qualific ..."

Abstract
- Add to MetaCart

An iteration of the stabilized sequential quadratic programming method (sSQP) consists in solving a certain quadratic program in the primal-dual space, regularized in the dual variables. The advantage with respect to the classical sequential quadratic programming (SQP) is that no constraint qualifications are required for fast local convergence (i.e., the problem can be degenerate). In particular, for equality-constrained problems the superlinear rate of convergence is guaranteed under the only assumption that the primal-dual starting point is close enough to a stationary point and a noncritical Lagrange multiplier pair (the latter being weaker than the second-order sufficient optimality condition). However, unlike for SQP, designing natural globally convergent algorithms based on the sSQP idea proved quite a challenge and, currently, there are very few proposals in this direction. For equality-constrained problems, we suggest to use for the task linesearch for the smooth two-parameter exact penalty function, which is the sum of the Lagrangian with squared penalizations of the violation of the constraints and of the violation of the Lagrangian stationarity with respect to primal variables. Reasonable global convergence properties are established. Moreover,

### COMBINING STABILIZED SQP WITH THE AUGMENTED LAGRANGIAN ALGORITHM∗

, 2014

"... For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advan-tage of the well-known robust behav ..."

Abstract
- Add to MetaCart

(Show Context)
For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advan-tage of the well-known robust behavior of the augmented Lagrangian algorithm, including on problems with degenerate constraints, and at the same time try to reduce the overall algorithm locally to sSQP (which gives fast local convergence rate under weak assumptions). Specifically, the algorithm first verifies whether the primal-dual sSQP step (with unit step-size) makes good progress towards decreasing the violation of optimality conditions for the original problem, and if so, makes this step. Otherwise, the primal part of the sSQP direc-tion is used for linesearch that decreases the augmented Lagrangian, keeping the multiplier estimate fixed for the time being. The overall algorithm has reasonable global convergence guarantees, and inherits strong convergence rate properties of sSQP under the same weak assumptions. Numerical results on degenerate problems and comparisons with some alterna-tives are reported. Key words: stabilized sequential quadratic programming; augmented Lagrangian; superlinear con-vergence; global convergence.

### 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 ..."

Abstract
- Add to MetaCart

(Show Context)
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.

### SHARP PRIMAL SUPERLINEAR CONVERGENCE RESULTS FOR SOME NEWTONIAN METHODS FOR CONSTRAINED OPTIMIZATION

, 2010

"... As is well known, Q-superlinear or Q-quadratic convergence of the primal-dual sequence generated by an optimization algorithm does not, in general, imply Q-superlinear convergence of the primal part. Primal convergence, however, is often of particular interest. For the sequential quadratic programm ..."

Abstract
- Add to MetaCart

(Show Context)
As is well known, Q-superlinear or Q-quadratic convergence of the primal-dual sequence generated by an optimization algorithm does not, in general, imply Q-superlinear convergence of the primal part. Primal convergence, however, is often of particular interest. For the sequential quadratic programming (SQP) algorithm, local primal-dual quadratic convergence can be established under the assumptions of uniqueness of the Lagrange multiplier associated to the solution and the second-order sufficient condition. At the same time, previous primal Q-superlinear convergence results for SQP required strengthening of the first assumption to the linear independence constraint qualification. In this paper, we show that this strengthening of assumptions is actually not necessary. Specifically, we show that once primal-dual convergence is assumed or already established, for primal superlinear rate one needs only a certain error bound estimate. This error bound holds, for example, under the second-order sufficient condition, which is needed for primal-dual local analysis in any case. Moreover, in some situations even second-order sufficiency can be relaxed to the weaker assumption that the multiplier in question is noncritical. Our study is performed for a rather general perturbed SQP framework which covers, in addition to SQP and quasi-Newton SQP, some other algorithms as well. For example, as a byproduct, we obtain primal Q-superlinear convergence results for the linearly constrained (augmented) Lagrangian methods for which no primal Q-superlinear rate of convergence results were previously available. Another application of the general framework is sequential quadratically constrained quadratic programming methods. Finally, we discuss some difficulties with proving primal superlinear convergence for the stabilized version of SQP.