Loss of Superlinear Convergence for an SQP-Type Method with Conic Constraints

Diehl, Moritz; Jarre, Florian; Vogelbusch, Christoph

doi:10.1137/050625977

Cited by 12 publications

(3 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These methods will be briefly reviewed in this section. Furthermore, other related methods were studied by Jarre [33], Kanzow, Nagel, Kato and Fukushima [36], Diehl, Jarre and Vogelbusch [16], Kanno and Takewaki [34], Li and Sun [46] and Zhu and Zhu [78], for example.…”

Section: Sequential Sdp Methodsmentioning

confidence: 99%

A Survey of Numerical Methods for Nonlinear Semidefinite Programming

Yamashita¹,

Yabe

2015

JORSJ

View full text Add to dashboard Cite

Nonlinear semidefinite programming (SDP) problems have received a lot of attentions because of large variety of applications. In this paper, we survey numerical methods for solving nonlinear SDP problems. Three kinds of typical numerical methods are described; augmented Lagrangian methods, sequential SDP methods and primal-dual interior point methods. We describe their typical algorithmic forms and discuss their global and local convergence properties which include rate of convergence.

show abstract

Section: Sequential Sdp Methodsmentioning

confidence: 99%

A Survey of Numerical Methods for Nonlinear Semidefinite Programming

Yamashita¹,

Yabe

2015

JORSJ

View full text Add to dashboard Cite

show abstract

“…All of the methods discussed here formulate and solve convex subproblems in each iteration, and in particular use only positive semidefinite Hessian approximations. While this is desirable for both theoretical and practical reasons, there is a price to pay for convexity of the subproblems: it generally limits the convergence speed to only linear, even if arbitrary positive semidefinite -but bounded -Hessian approximations are used, as shown in [10] for the simple example problem of minimizing the nonconvex objective f 0 (w…”

Section: Outlinementioning

confidence: 99%

Survey of sequential convex programming and generalized Gauss-Newton methods

2021

Self Cite

View full text Add to dashboard Cite

We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, generally nonconvex, but differentiable functions on the other hand. All methods from this class use only first order derivatives of the nonlinear functions and sequentially solve convex optimization problems. All of them are different generalizations of the classical Gauss-Newton (GN) method. We focus on the smooth constrained case and on three methods to address it: Sequential Convex Programming (SCP), Sequential Convex Quadratic Programming (SCQP), and Sequential Quadratically Constrained Quadratic Programming (SQCQP). While the first two methods were previously known, the last is newly proposed and investigated in this paper. We show under mild assumptions that SCP, SCQP and SQCQP have exactly the same local linear convergence – or divergence – rate. We then discuss the special case in which the solution is fully determined by the active constraints, and show that for this case the KKT conditions are sufficient for local optimality and that SCP, SCQP and SQCQP even converge quadratically. In the context of parameter estimation with symmetric convex loss functions, the possible divergence of the methods can in fact be an advantage that helps them to avoid some undesirable local minima: generalizing existing results, we show that the presented methods converge to a local minimum if and only if this local minimum is stable against a mirroring operation applied to the measurement data of the estimation problem. All results are illustrated by numerical experiments on a tutorial example.

show abstract

“…The purpose of this paper is to review some recent algorithmic developments in solving nonlinear semidefinite optimization problems (also called nonlinear semidefinite programs). As a natural extension of (linear) semidefinite programs (SDP), nonlinear semidefinite programs (NSDP) arise in various application fields such as system control and financial engineering [1,12,20,21,30] and has caught much attention on its theory and algorithms [3,5,7,8,9,10,13,23,24,25,31,32,33,34,35,36,37,46,48,49,51,54,55,57,61,62,63]. While the mathematical formats of NSDP may be different in various applications, we find it is convenient to start with the following general model…”

mentioning

confidence: 99%

On methods for solving nonlinear semidefinite optimization problems

Sun¹

2011

Numerical Algebra, Control &Amp; Optimization

View full text Add to dashboard Cite

The nonlinear semidefinite optimization problem arises from applications in system control, structural design, financial management, and other fields. However, much work is yet to be done to effectively solve this problem. We introduce some new theoretical and algorithmic development in this field.In particular, we discuss first and second-order algorithms that appear to be promising, which include the alternating direction method, the augmented Lagrangian method, and the smoothing Newton method. Convergence theorems are presented and preliminary numerical results are reported.

show abstract

Loss of Superlinear Convergence for an SQP-Type Method with Conic Constraints

Cited by 12 publications

References 5 publications

A Survey of Numerical Methods for Nonlinear Semidefinite Programming

A Survey of Numerical Methods for Nonlinear Semidefinite Programming

Survey of sequential convex programming and generalized Gauss-Newton methods

On methods for solving nonlinear semidefinite optimization problems

Contact Info

Product

Resources

About