Optimality conditions and newton-type methods for mathematical programs with vanishing constraints

Izmailov, A. F.; Pogosyan, A. L.

doi:10.1134/s0965542509070069

Cited by 24 publications

(10 citation statements)

References 12 publications

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“…This fact might be used to establish some nonsmooth versions of Newton Method for MPCC (cf. [54,58]). This is an issue of current research.…”

Section: Characterization Of Strong Stability For C-stationary Pointsmentioning

confidence: 99%

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Topological Aspects of Nonsmooth Optimization

Shikhman

2012

Springer Optimization and Its Applications

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The goal of this paper is to compare alternative stationarity notions in structured nonsmooth optimization (SNO). Here, nonsmoothness is caused by complementarity, vanishing, orthogonality type, switching, or disjunctive constraints. On one side, we consider geometrically motivated notions of N -, N -, and N -stationarity in terms of Fréchet, Mordukhovich, and Clarke normal cones to the feasible set, respectively. On the other side, we advocate the notion of topologically relevant T-stationarity, which adequately captures the global structure of SNO. Our main findings say that (a) N -stationary points include all local minimizers; (b) Nstationary points, which are not N -stationary, correspond to the singular saddle points of first order; (c) T-stationary points, which are not N -stationary, correspond to the regular saddle points of first order; (d) N -stationary points, which are not T-stationary, are irrelevant for optimization purposes. Overall, a hierarchy of stationarity notions for SNO is established.

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“…This fact might be used to establish some nonsmooth versions of Newton Method for MPCC (cf. [54,58]). This is an issue of current research.…”

Section: Characterization Of Strong Stability For C-stationary Pointsmentioning

confidence: 99%

“…It is motivated by the fact that the constraint G m does not play any role whenever H m is active. We refer to [43,44,45,46,57,56] for more details on optimality conditions, constraint qualifications, sensitivity and numerical methods for MPVC. Note that additional constraints G m (x) ≥ 0, m = 1, .…”

Section: Chaptermentioning

confidence: 99%

Topological Aspects of Nonsmooth Optimization

Shikhman

2012

Springer Optimization and Its Applications

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“…Further, Hoheisel and Kanzow [31] investigated necessary and sufficient optimality conditions through Abadie and Guignard type constraint qualifications for mathematical programs with vanishing constraints. For more details on the MPVC, we refer to [16,32,33] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Semidefinite Multiobjective Mathematical Programming Problems with Vanishing Constraints Using Convexificators

et al. 2021

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In this paper, we establish Fritz John stationary conditions for nonsmooth, nonlinear, semidefinite, multiobjective programs with vanishing constraints in terms of convexificator and introduce generalized Cottle type and generalized Guignard type constraints qualification to achieve strong S—stationary conditions from Fritz John stationary conditions. Further, we establish strong S—stationary necessary and sufficient conditions, independently from Fritz John conditions. The optimality results for multiobjective semidefinite optimization problem in this paper is related to two recent articles by Treanta in 2021. Treanta in 2021 discussed duality theorems for special class of quasiinvex multiobjective optimization problems for interval-valued components. The study in our article can also be seen and extended for the interval-valued optimization motivated by Treanta (2021). Some examples are provided to validate our established results.

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“…The theoretical foundations, i.e. optimality, stationarity, criticality and constraint qualifications have been established in the literature by Hoheisel et al [13][14][15][16][17] and other authors [10,22]. Numerical schemes based on smoothing and relaxation were studied by Hoheisel et al [1,2,18,19].…”

Section: Introductionmentioning

confidence: 99%

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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Optimality conditions and newton-type methods for mathematical programs with vanishing constraints

Cited by 24 publications

References 12 publications

Topological Aspects of Nonsmooth Optimization

Topological Aspects of Nonsmooth Optimization

Semidefinite Multiobjective Mathematical Programming Problems with Vanishing Constraints Using Convexificators

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

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