Abstract Newtonian Frameworks and Their Applications

Izmailov, A. F.; Kurennoy, Alexey

doi:10.1137/120899194

Cited by 9 publications

(8 citation statements)

References 28 publications

(73 reference statements)

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“…Under the conditions of semistability and hemistability (these two conditions are satisfied if Robinson's strong regularity holds at the solution), the author proved the superlinear convergence of the Newton's method (quadratic convergence if f is C 1,1 ). We note that these results were generalized by Izmailov and Solodov [25] and Izmailov and Kurennoy [26] to the case f (x)+F (x) 0 with a smooth single-valued map f and a set-valued mapping F by using an inexact Josephy-Newton method. We note that in the papers [10,25], the authors considered a single-valued approximation, and that in the current paper we authorized the approximation to be set-valued.…”

Section: Commentarymentioning

confidence: 68%

“…The case of single-valued approximations A k : X × X → Y of the function f was studied in [10,25,26] and [12,Section 6C]. It is well known that specific choices of (A k ) k∈N0 lead to various methods for solving (1.3).…”

Section: Instead Of Considering a General Inclusion Findmentioning

confidence: 99%

“…The key assumption is the calmness of the solution mapping (which is equivalent to metric subregularity of the inverse mapping) of the perturbed generalized equation f (z) + F (z) + p 0. In [26], the authors extended and improved the local convergence analysis of the Newtonian iterative methods developed earlier in [10,18] by using both single-valued and set-valued approximations.…”

Section: Instead Of Considering a General Inclusion Findmentioning

confidence: 99%

See 2 more Smart Citations

Newton's Method for Solving Inclusions Using Set-Valued Approximations

Adly¹,

Cibulka²,

Ngai³

2015

SIAM J. Optim.

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International audienceResults on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton- type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton’s method, the nonsmooth Newton’s method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton’s method is discussed

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Section: Commentarymentioning

confidence: 68%

Section: Instead Of Considering a General Inclusion Findmentioning

confidence: 99%

Section: Instead Of Considering a General Inclusion Findmentioning

confidence: 99%

See 1 more Smart Citation

Newton's Method for Solving Inclusions Using Set-Valued Approximations

Adly¹,

Cibulka²,

Ngai³

2015

SIAM J. Optim.

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“…A problem with direct optimization techniques is the number of constraints, which can be large for applied problems. For example, in Sequential Linear Programming [1], which is a widely used direct optimization method [2][3][4][5], the problem is formulated to be solved using simplex algorithm. This requires working with not only the main constraints, but also additional constraints on the upper and lower limits of the variables, which can increase the total number of constraints, signi cantly, for problems with a large number of variables [6], or in sequential quadratic programming [7][8][9], the number of constraints is at least equal to the sum of the number of design variables and the Lagrange multipliers, which can make the procedure of optimization very time-taking.…”

Section: Introductionmentioning

confidence: 99%

Structural Optimization by Spherical Interpolation of Objective Function and Constraints

Meshki

Joghataie

2016

Scientia Iranica

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A new method for structural optimization is presented for successive approximation of the objective function and constraints in conjunction with Lagrange multipliers approach. The focus is on presenting the methodology with simple examples. The basis of the iterative algorithm is that after each iteration, it brings the approximate location of the estimated minimum closer to the exact location, gradually. In other words, instead of the linear or parabolic term used in Taylor expansion, which works based on a short step length, an arch is used that has a constant curvature but a longer step length. Using this approximation, the equations of optimization involve the Lagrange multipliers as the only unknown variables. The equations which depend on the design variables are decoupled linearly as these variables are directly obtained. One mathematical example is solved to explain in details how the method works. Next, the method is applied to the optimization of a simple truss structure to explain how the method can be used in structural optimization. The same problems have been solved by penalty method and compared. The results from both methods have been the same. However, because of the long step length and reduction in the number of variables, the speed of convergence has been higher in the presented method.

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“…We refer to [3] and [13,25] for state-of-the-art on global and local convergence properties of Aug-L methods, respectively. For many other issues, see [5].…”

Section: Introductionmentioning

confidence: 99%

Combining stabilized SQP with the augmented Lagrangian algorithm

2015

Self Cite

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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 advantage 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 stepsize) 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 direction 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 local convergence rate properties of sSQP under the same weak assumptions. Numerical results on degenerate problems and comparisons with some alternatives are reported.

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Abstract Newtonian Frameworks and Their Applications

Cited by 9 publications

References 28 publications

Newton's Method for Solving Inclusions Using Set-Valued Approximations

Newton's Method for Solving Inclusions Using Set-Valued Approximations

Structural Optimization by Spherical Interpolation of Objective Function and Constraints

Combining stabilized SQP with the augmented Lagrangian algorithm

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