Newton-Type Methods for Optimization and Variational Problems

Izmailov, A. F.; Solodov, M. V.

doi:10.1007/978-3-319-04247-3

Cited by 161 publications

(180 citation statements)

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“…the set of all such indices. Taking into account the convergence of {(x k , λ k , µ k )} to (x,λ,μ), using the result on local convergence of sSQP [12] (with a certain quantitative improvement using [14, Theorem 1 (b)]; see also [31,Chapter 7]) and employing the error bound (5), we conclude that for all k ∈ K large enough it holds thatλ k + η 0 ∈ [λ min ,λ max ], µ k + ζ 0 ∈ [0,μ max ] and…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Here, we only mention that sSQP has local superlinear convergence under the second-order sufficient optimality condition only, without any constraints qualification assumptions [12] (for equality-constrained problems, even the weaker noncriticality condition is enough [29]). This should be contrasted with the usual SQP method [6,20] (see also [31,Chapter 4]), which in addition requires relatively strong regularity condition on the constraints (while sSQP needs nothing at all). We note that very few globalizations of the local sSQP scheme have been proposed so far, all very recently.…”

Section: Introductionmentioning

confidence: 99%

“…Recall that, according to [23,Lemma 2] and [14, Theorem 2] (see also [31,Section 1.3.3]), the SOSC (4) implies the following error bound:…”

Section: Introductionmentioning

confidence: 99%

“…In fact, this error bound is equivalent to the assumption that the multiplier (λ,μ) ∈ M(x) is noncritical, as defined in [30]; see also [26] and [31,Section 1.3]. As in what follows we would only invoke this notion for the equality-constrained case, we give here the corresponding simpler definition.…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [12,14,22,30,37] for the origins of the method and developments in this area; see also [31,Chapter 7]. Here, we only mention that sSQP has local superlinear convergence under the second-order sufficient optimality condition only, without any constraints qualification assumptions [12] (for equality-constrained problems, even the weaker noncriticality condition is enough [29]).…”

Section: Introductionmentioning

confidence: 99%

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Combining stabilized SQP with the augmented Lagrangian algorithm

2015

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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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Section: ⊓ ⊔mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Recall that, according to [23,Lemma 2] and [14, Theorem 2] (see also [31,Section 1.3.3]), the SOSC (4) implies the following error bound:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Combining stabilized SQP with the augmented Lagrangian algorithm

2015

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Near‐optimal control of a class of output‐constrained systems using recurrent neural network: A control‐barrier function approach

Rad-Moghadam

Farrokhi

2023

Optim Control Appl Methods

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This paper proposes a near‐optimal controller design for the constrained nonlinear affine systems based on a Recurrent Neural Network (RNN) and Extended State Observers (ESOs). For this purpose, after defining a finite‐horizon integral‐type performance index, the prediction over the horizon is performed using the Taylor expansion that converts the primary problem into a finite‐dimensional optimization. In comparison with other controllers of the similar structure, the proposed method is capable of dealing with output constraints by employing the Control Barrier Function (CBF). The class of the output and input constraints are of the box‐type. Moreover, whereas several safe control approaches are proposed regardless of the performance of the closed‐loop system, this paper aims at achieving a near‐optimal performance as far as the constraints permit. As a result, a constrained optimization problem is achieved, where the online solution is obtained using a rapidly convergent RNN. Stability and the ease of implementation are some of the advantages of this network making the algorithm more reliable. Moreover, integrated stability analysis of the closed‐loop system that includes the dynamic RNN reveals that the closed‐loop system is stable in the sense of the Lyapunov stability theory. The effectiveness of the proposed control method in terms of the tracking performance and constraint satisfaction is illustrated through a simulating example of two‐inverted pendulums system. The results indicated advantages of the proposed method as compared with the recently published methods in well‐known literature.

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A Local Search Scheme for the Inequality-Constrained Optimal Control Problem

Strekalovsky

2021

Mathematical Optimization Theory and Operations Research

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Newton-Type Methods for Optimization and Variational Problems

Cited by 161 publications

References 0 publications

Combining stabilized SQP with the augmented Lagrangian algorithm

Combining stabilized SQP with the augmented Lagrangian algorithm

Near‐optimal control of a class of output‐constrained systems using recurrent neural network: A control‐barrier function approach

A Local Search Scheme for the Inequality-Constrained Optimal Control Problem

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