Feasible generalized monotone line search SQP algorithm for nonlinear minimax problems with inequality constraints

Jian, Jinbao; Quan, Ran; Zhang, Xue-Lu

doi:10.1016/j.cam.2006.05.034

Cited by 21 publications

(12 citation statements)

References 14 publications

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“…So, many authors have applied the idea of SQP method to present effective algorithms for solving the minimax problems, such as in Refs. [4][5][6][7][8][9][10][11][12]. It is a key problem of various SQP methods to overcome the so-called Maratos effect [13] under suitable conditions, for example, to solve one or more additional quadratic programs or systems of linear equations, or compute explicit correction directions.…”

Section: Introductionmentioning

confidence: 99%

A sequential quadratically constrained quadratic programming method for unconstrained minimax problems

Jian

Chao

2010

Journal of Mathematical Analysis and Applications

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In this paper, a sequential quadratically constrained quadratic programming (SQCQP) method for unconstrained minimax problems is presented. At each iteration the SQCQP method solves a subproblem that involves convex quadratic inequality constraints and a convex quadratic objective function. The global convergence of the method is obtained under much weaker conditions without any constraint qualification. Under reasonable assumptions, we prove the strong convergence, superlinearly and quadratic convergence rate.

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Section: Introductionmentioning

confidence: 99%

A sequential quadratically constrained quadratic programming method for unconstrained minimax problems

Jian

Chao

2010

Journal of Mathematical Analysis and Applications

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“…Thus, 0 ∈ int D(x * ) and with the use of Theorems 2.3 and 2.6 one can conclude that x * is a local minimiser of problem (38) at which the first order growth condition holds true. However, let us check that a generalised complete alternance does not exist at x * .…”

Section: Constrained Minimax Problemsmentioning

confidence: 76%

“…Let us check optimality conditions at the point x * = 0. Firstly, note that x * is a not an isolated point of problem (38), since for any t ≥ 0 the point x(t) = (0, 0, t) T is feasible. One has I(x * ) = {1, 2}, ∇g 1 (x * ) = (0, 1, 0) T , and…”

Section: Constrained Minimax Problemsmentioning

confidence: 99%

“…Many methods for solving minimax problems are based on an application of nonlinear programming algorithms to this equivalent reformulation of a minimax problem (see such methods based on, e.g. sequential quadratic programming methods [34,38,54,61], sequential quadratically constrained quadratic programming methods [9,36,37], interior point methods [47,55], augmented Lagrangian methods [29][30][31], etc.). On the other hand, efficient, superlinearly or even quadratically convergent methods for solving minimax problems can be also based on a convenient characterisation of an optimal solution of a minimax problem, that is, on optimality conditons that are specific for minimax or Chebyshev problems (cf.…”

Section: Introductionmentioning

confidence: 99%

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A Unified Study of Necessary and Sufficient Optimality Conditions for Minimax and Chebyshev Problems with Cone Constraints

Dolgopolik

2020

Preprint

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This article is devoted to a detailed analysis of first and second order necessary and sufficient optimality conditions for minimax and Chebyshev optimisation problems with cone constraints. First order optimality conditions for such problems can be formulated in several different forms: in terms of a linearised problem, in terms of Lagrange multipliers (KKT-points), in terms of subdifferentials and normal cones, in terms of a nonsmooth penalty function, in terms of cadres with positive cadre multipliers, and in an alternance form. We describe interconnections between all these forms of necessary and sufficient optimality conditions and prove that seemingly different conditions are in fact equivalent. We also demonstrate how first order optimality conditions can be reformulated in a more convenient form for particular classes of cone constrained optimisation problems and extend classical second order optimality condition for smooth cone constrained problems to the case of minimax and Chebyshev problems with cone constraints. The optimality conditions obtained in this article open a way for a development of new efficient structureexploiting methods for solving cone constrained minimax and Chebyshev problems.

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“…Problems with inequality constraints can be reformulated in the above form by introducing slack variables. Moreover, nonlinear constrained programming is significant for solving engineering optimization problems in other forms, for example, the minimax problems (Jian, Quan, and Zhang 2007;Wang and Zhang 2008;Han, Jian, and Li 2011;Jian et al 2014) and dynamic optimization problems (Hu, Ong, and Teo 2002;Mohammed and Zhang 2013;Liu, Li, and Liu 2015;Zhang et al 2015). The filter method, as an alternative to merit functions for nonlinear constrained programming, was first proposed by Fletcher and Leyffer (2002).…”

Section: Introductionmentioning

confidence: 99%

An efficient interior-point algorithm with new non-monotone line search filter method for nonlinear constrained programming

Wang

Liu

Zhang

2016

Engineering Optimization

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An efficient primal-dual interior-point algorithm using a new nonmonotone line search filter method is presented for nonlinear constrained programming, which is widely applied in engineering optimization. The new non-monotone line search technique is introduced to lead to relaxed step acceptance conditions and improved convergence performance. It can also avoid the choice of the upper bound on the memory, which brings obvious disadvantages to traditional techniques. Under mild assumptions, the global convergence of the new non-monotone line search filter method is analysed, and fast local convergence is ensured by second order corrections. The proposed algorithm is applied to the classical alkylation process optimization problem and the results illustrate its effectiveness. Some comprehensive comparisons to existing methods are also presented. ARTICLE HISTORY

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Feasible generalized monotone line search SQP algorithm for nonlinear minimax problems with inequality constraints

Cited by 21 publications

References 14 publications

A sequential quadratically constrained quadratic programming method for unconstrained minimax problems

A sequential quadratically constrained quadratic programming method for unconstrained minimax problems

A Unified Study of Necessary and Sufficient Optimality Conditions for Minimax and Chebyshev Problems with Cone Constraints

An efficient interior-point algorithm with new non-monotone line search filter method for nonlinear constrained programming

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