A feasible SQP-GS algorithm for nonconvex, nonsmooth constrained optimization

Tang, Chunming; Liu, Shuai; Jian, Jinbao; Li, Jianling

doi:10.1007/s11075-012-9692-5

Cited by 23 publications

(21 citation statements)

References 25 publications

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“…Tang, Liu, Jian, and Li have also proposed in [TLJL14] a feasible variant of the SQP-GS method in which the iterates are forced to remain feasible for the constraints and the objective function is monotonically decreasing throughout the optimization process. This opens the door to employing a two-phase approach common for solving some optimization problems, where phase 1 is responsible for attaining a feasible point and phase 2 seeks optimality while maintaining feasibility.…”

Section: Sqp-gs For Constrained Optimizationmentioning

confidence: 99%

Gradient Sampling Methods for Nonsmooth Optimization

Burke

Curtis

Lewis

et al. 2020

Numerical Nonsmooth Optimization

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This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method for minimizing smooth objectives. We then provide overviews of various enhancements that have been proposed to improve practical performance, as well as of several extensions that have been made in the literature, such as to solve constrained problems. The paper also includes clarification of certain technical aspects of the analysis of gradient sampling algorithms, most notably related to the assumptions one needs to make about the set of points at which the objective is continuously differentiable. Finally, we discuss possible future research directions.

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Section: Sqp-gs For Constrained Optimizationmentioning

confidence: 99%

Gradient Sampling Methods for Nonsmooth Optimization

Burke

Curtis

Lewis

et al. 2020

Numerical Nonsmooth Optimization

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“…We also note that our method is very different from that in [16] due to the proximalprojection strategy and the new descent test criterion. Some other recent methods for general-purpose nonsmooth constrained minimization problems can be found in, e.g., [13,32,17,35,36].…”

Section: Chunming Tang Jinbao Jian and Guoyin LImentioning

confidence: 99%

A proximal-projection partial bundle method for convex constrained minimax problems

Tang¹,

Jian²,

Li³

2019

Journal of Industrial &Amp; Management Optimization

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In this paper, we propose a partial bundle method for a convex constrained minimax problem where the objective function is expressed as maximum of finitely many convex (not necessarily differentiable) functions. To avoid complete evaluation of all component functions of the objective, a partial cutting-planes model is adopted instead of the traditional one. Based on the proximal-projection idea, at each iteration, an unconstrained proximal subproblem is solved first to generate an aggregate linear model of the objective function, and then another subproblem based on this model is solved to obtain a trial point. Moreover, a new descent test criterion is proposed, which can not only simplify the presentation of the algorithm, but also improve the numerical performance significantly. An explicit upper bound for the number of bundle resets is also derived. Global convergence of the algorithm is established, and some preliminary numerical results show that our method is very encouraging.

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“…We would like to inspect the reduction of the model function m after we have obtained a new trial point via the linear programming problem (26). Denote a general optimal solution of problem (26) by (x * , z * ) and by (x kl , z kl ) when it is necessary to indicate it is in the l-th minor iteration in the k-th major iteration. In the following lemma we derive the explicit expression of the reduction of the model fromx to x * .…”

Section: The Model Problem and Model Reductionmentioning

confidence: 99%

“…We will use the following set which essentially defines the model m(x). (25), 'a' be such that E i ≥ 0, ∀ i ∈ I, x * be the first component of an optimal solution of problem (26), and F be defined in (42).…”

Section: The Model Problem and Model Reductionmentioning

confidence: 99%

“…We have finished showing (60) which together with (59) yields Then line 16 of Algorithm 3 must be executed infinite times, which implies that there exists an infinite subsequence K * ⊆ N such that ρ l < − 1 min{∆ l ,1} for all l ∈ K * and K ′ := K ∩ K * is infinite. By the definition of ρ l , the Lipschitz continuity of f , and the feasibility of x l to problem (26) we have…”

Section: Convergence Analysismentioning

confidence: 99%

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A simple version of bundle method with linear programming

Liu

2018

Comput Optim Appl

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Bundle methods have been intensively studied for solving both convex and nonconvex optimization problems. In most of the bundle methods developed thus far, at least one quadratic programming (QP) subproblem needs to be solved in each iteration. In this paper, we exploit the feasibility of developing a bundle algorithm that only solves linear subproblems. We start from minimization of a convex function and show that the sequence of major iterations converge to a minimizer. For nonconvex functions we consider functions that are locally Lipschitz continuous and prox-regular on a bounded level set, and minimize the cutting-plane model over a trust region with infinity norm. The para-convexity of such functions allows us to use the locally convexified model and its convexity properties. Under some conditions and assumptions, we study the convergence of the proposed algorithm through the outer semicontinuity of the proximal mapping. Encouraging results of preliminary numerical experiments on standard test sets are provided.

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A feasible SQP-GS algorithm for nonconvex, nonsmooth constrained optimization

Cited by 23 publications

References 25 publications

Gradient Sampling Methods for Nonsmooth Optimization

Gradient Sampling Methods for Nonsmooth Optimization

A proximal-projection partial bundle method for convex constrained minimax problems

A simple version of bundle method with linear programming

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