Convex Cones

Bazaraa, M. S.; Shetty, C. M.

doi:10.1007/978-3-642-48294-6_3

Cited by 2 publications

(3 citation statements)

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“…According to our construction, this implies ν k i → ∞ and thus, by (4), we have (6). This shows δI = 0.…”

Section: The Global Regularizationmentioning

confidence: 71%

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A Study of One-Parameter Regularization Methods for Mathematical Programs with Vanishing Constraints

Hoheisel¹,

Pablos²,

Pooladian³

et al. 2020

Preprint

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Mathematical programs with vanishing constraints (MPVCs) are a class of nonlinear optimization problems with applications to various engineering problems such as truss topology design and robot motion planning. MPVCs are difficult problems from both a theoretical and numerical perspective: the combinatorial nature of the vanishing constraints often prevents standard constraint qualifications and optimality conditions from being attained; moreover, the feasible set is inherently nonconvex, and often has no interior around points of interest. In this paper, we therefore study and compare four regularization methods for the numerical solution of MPVCS. Each method depends on a single regularization parameter, which is used to embed the original MPVC into a sequence of standard nonlinear programs. Convergence results for these methods based on both exact and approximate stationary of the subproblems are established under weak assumptions. The improved regularity of the subproblems is studied by providing sufficient conditions for the existence of KKT multipliers. Numerical experiments, based on applications in truss topology design and an optimal control problem from aerothermodynamics, complement the theoretical analysis and comparison of the regularization methods. The computational results highlight the benefit of using regularization over applying a standard solver directly, and they allow us to identify two promising regularization schemes.

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“…According to our construction, this implies ν k i → ∞ and thus, by (4), we have (6). This shows δI = 0.…”

Section: The Global Regularizationmentioning

confidence: 71%

“…in contradiction to (6). Consequently our assumption was wrong and η G, * i η H, * i ≤ 0 for all i ∈ I 00 .…”

Section: The Global Regularizationmentioning

confidence: 83%

A Study of One-Parameter Regularization Methods for Mathematical Programs with Vanishing Constraints

Hoheisel¹,

Pablos²,

Pooladian³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…However, the numerical algorithms rarely terminate in the KKT point. Note that the stopping criteria of nonlinear program algorithms are usually approximate conditions in software, like [3,4,5,6,24]. In a recent work [9], Kanzow only need to compute inexact KKT points of variational inequalities to investigate some classes of quasi-variational inequalities.…”

mentioning

confidence: 99%

The inexact log-exponential regularization method for mathematical programs with vertical complementarity constraints

Pang¹,

Xu²,

Lv³

2019

Journal of Industrial &Amp; Management Optimization

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We study the convergence of the log-exponential regularization method for mathematical programs with vertical complementarity constraints (MPVCC). The previous paper assume that the sequence of Lagrange multipliers are bounded and it can be found by software. However, the KKT points can not be computed via Matlab subroutines exactly in many cases. We note that it is realistic to compute inexact KKT points from a numerical point of view. We prove that, under the MPVCC-MFCQ assumption, the accumulation point of the inexact KKT points is Clarke (C-) stationary point. The idea of inexact KKT conditions can be used to define stopping criteria for many practical algorithms. Furthermore, we introduce a feasible strategy that guarantees inexact KKT conditions and provide some numerical examples to certify the reliability of the approach. It means that we can apply the inexact regularization method to solve MPVCC and show the advantages of the improvement.

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Convex Cones

Cited by 2 publications

References 0 publications

A Study of One-Parameter Regularization Methods for Mathematical Programs with Vanishing Constraints

A Study of One-Parameter Regularization Methods for Mathematical Programs with Vanishing Constraints

The inexact log-exponential regularization method for mathematical programs with vertical complementarity constraints

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