Journal of Optimization Theory and Applications, ISSN 0022-3239, 3/2010, Volume 144, Issue 3, pp. 511 - 531

...) that uses the regularized gap function of a quasi-variational inequality (QVI). The regularized gap function for QVI is in general not differentiable, but only directionally differentiable...

Generalized Nash equilibrium problems | Regularized gap functions | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Mathematics | Theory of Computation | Engineering, general | Applications of Mathematics | Quasivariational inequalities | Optimization | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | VARIATIONAL INEQUALITY | ALGORITHMS | Game theory | Algorithms | Studies | Equilibrium

Generalized Nash equilibrium problems | Regularized gap functions | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Mathematics | Theory of Computation | Engineering, general | Applications of Mathematics | Quasivariational inequalities | Optimization | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | VARIATIONAL INEQUALITY | ALGORITHMS | Game theory | Algorithms | Studies | Equilibrium

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 8/2012, Volume 53, Issue 4, pp. 737 - 748

The paper aims to obtain new local/global error bounds for quasi variational inequality problems in terms of the regularized gap function and the D-gap function...

Operations Research/Decision Theory | Error bounds | D-gap function | Computer Science, general | Optimization | Quasi variational inequality problem | Regularized gap function | Economics / Management Science | Real Functions | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Studies

Operations Research/Decision Theory | Error bounds | D-gap function | Computer Science, general | Optimization | Quasi variational inequality problem | Regularized gap function | Economics / Management Science | Real Functions | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Studies

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2016, Volume 2016, Issue 1, pp. 1 - 17

This paper is devoted to the study of gap functions of random generalized variational inequality problems in a fuzzy environment...

random fuzzy mapping | gap function | 90C30 | 49J52 | Analysis | 49J40 | strongly h -monotone | Mathematics, general | Mathematics | global error bound | Applications of Mathematics | regularized gap function | strongly h-monotone | MATHEMATICS | MATHEMATICS, APPLIED | MERIT FUNCTIONS | Fuzzy logic | Errors | Mathematical analysis | Inequalities | Mathematical models | Mapping | Vectors (mathematics) | Fuzzy

random fuzzy mapping | gap function | 90C30 | 49J52 | Analysis | 49J40 | strongly h -monotone | Mathematics, general | Mathematics | global error bound | Applications of Mathematics | regularized gap function | strongly h-monotone | MATHEMATICS | MATHEMATICS, APPLIED | MERIT FUNCTIONS | Fuzzy logic | Errors | Mathematical analysis | Inequalities | Mathematical models | Mapping | Vectors (mathematics) | Fuzzy

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2009, Volume 354, Issue 2, pp. 575 - 583

By using the regularized gap function for variational inequalities, Li and Peng introduced a new penalty function P α ( x...

Bounded level set | Exact penalty function | Merit function | Invexity | Unconstrained reformulation | Regularized gap function | Constrained optimization | Strong coerciveness | MATHEMATICS | MATHEMATICS, APPLIED | VARIATIONAL INEQUALITY PROBLEMS | UNCONSTRAINED MINIMIZATION

Bounded level set | Exact penalty function | Merit function | Invexity | Unconstrained reformulation | Regularized gap function | Constrained optimization | Strong coerciveness | MATHEMATICS | MATHEMATICS, APPLIED | VARIATIONAL INEQUALITY PROBLEMS | UNCONSTRAINED MINIMIZATION

Journal Article

5.
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Error Bounds of Regularized Gap Functions for Nonsmooth Variational Inequality Problems

Mathematical Programming, ISSN 0025-5610, 7/2007, Volume 110, Issue 2, pp. 405 - 429

We study the Clarke–Rockafellar directional derivatives of the regularized gap functions...

Mathematical and Computational Physics | Merit function | Error bound | Mathematics | Nonsmooth variational inequality problem | Regularized gap function | Mathematical Methods in Physics | 90C30 | Mathematics of Computing | Calculus of Variations and Optimal Control; Optimization | 49J52 | Numerical Analysis | 49J40 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | LOCAL UNIQUENESS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | error bound | nonsmooth variational inequality problem | merit function | regularized gap function | Algorithms | Equality | Studies | Mathematical programming

Mathematical and Computational Physics | Merit function | Error bound | Mathematics | Nonsmooth variational inequality problem | Regularized gap function | Mathematical Methods in Physics | 90C30 | Mathematics of Computing | Calculus of Variations and Optimal Control; Optimization | 49J52 | Numerical Analysis | 49J40 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | LOCAL UNIQUENESS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | error bound | nonsmooth variational inequality problem | merit function | regularized gap function | Algorithms | Equality | Studies | Mathematical programming

Journal Article

Fuzzy sets and systems, ISSN 0165-0114, 2019

Journal Article

Journal of computational and applied mathematics, ISSN 0377-0427, 01/2021, Volume 381, p. 113055

...) in fuzzy environment. Then, using the method of the nonlinear scalarization function, the regularized gap functions for GQVIP is established...

Error bounds | Scalarization | Fuzzy mappings | Regularized gap functions | Strong mixed vector generalized quasi-variational inequality problems

Error bounds | Scalarization | Fuzzy mappings | Regularized gap functions | Strong mixed vector generalized quasi-variational inequality problems

Journal Article

8.
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Gap function and global error bounds for generalized mixed quasi variational inequalities

Applied Mathematics and Computation, ISSN 0096-3003, 06/2015, Volume 260, pp. 71 - 81

In this paper, we obtain some gap functions for generalized mixed quasi variational inequality problems in terms of regularized gap function and D-gap function...

Residual vector | Error bounds | Strong monotone mapping | D-gap function | Regularized gap function | AUXILIARY PRINCIPLE | MATHEMATICS, APPLIED | MERIT FUNCTIONS | EQUILIBRIUM PROBLEMS

Residual vector | Error bounds | Strong monotone mapping | D-gap function | Regularized gap function | AUXILIARY PRINCIPLE | MATHEMATICS, APPLIED | MERIT FUNCTIONS | EQUILIBRIUM PROBLEMS

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 11/2013, Volume 407, Issue 2, pp. 270 - 280

.... Our aim is to obtain local/global error bounds for inverse quasi-variational inequality problems in terms of different gap functions/merit functions i.e...

Inverse quasi-variational inequality problem | Error bounds | [formula omitted]-gap function | Regularized gap function | Residual gap function | D-gap function | MATHEMATICS | MATHEMATICS, APPLIED | PROJECTION METHOD | UNCONSTRAINED MINIMIZATION | Equality

Inverse quasi-variational inequality problem | Error bounds | [formula omitted]-gap function | Regularized gap function | Residual gap function | D-gap function | MATHEMATICS | MATHEMATICS, APPLIED | PROJECTION METHOD | UNCONSTRAINED MINIMIZATION | Equality

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 11/2014, Volume 163, Issue 2, pp. 413 - 438

A well-known technique for the solution of quasi-variational inequalities (QVIs) consists in the reformulation of this problem as a constrained or unconstrained optimization problem by means of so-called gap functions...

mathrm{PC}^1$$ PC 1 function | DC optimization | Nonconvex duality | Mathematics | Theory of Computation | Optimization | Regularized gap function | Calculus of Variations and Optimal Control; Optimization | Set-valued mapping | Operations Research/Decision Theory | 90C33 | Dual gap function | 49M29 | Quasi-variational inequality | Conjugate function | 65K10 | Applications of Mathematics | Engineering, general | (Formula presented.) function | PC1 function | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | OPTIMIZATION | Studies | Optimization algorithms | Mapping | Mathematical functions | Inequality | Operators | Constraints | Mathematical analysis | Inequalities | Mathematical models | Standards

mathrm{PC}^1$$ PC 1 function | DC optimization | Nonconvex duality | Mathematics | Theory of Computation | Optimization | Regularized gap function | Calculus of Variations and Optimal Control; Optimization | Set-valued mapping | Operations Research/Decision Theory | 90C33 | Dual gap function | 49M29 | Quasi-variational inequality | Conjugate function | 65K10 | Applications of Mathematics | Engineering, general | (Formula presented.) function | PC1 function | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | OPTIMIZATION | Studies | Optimization algorithms | Mapping | Mathematical functions | Inequality | Operators | Constraints | Mathematical analysis | Inequalities | Mathematical models | Standards

Journal Article

JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION, ISSN 1547-5816, 05/2020, Volume 16, Issue 3, pp. 1261 - 1272

In this paper, the Clarke generalized Jacobian of the generalized regularized gap function for a nonmonotone Ky Fan inequality is studied...

AUXILIARY PRINCIPLE | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | MONOTONE EQUILIBRIUM PROBLEMS | Error bound | Ky Fan inequality | NONSMOOTH | DESCENT METHODS | COMBINED RELAXATION METHOD | descent method | regularized gap function

AUXILIARY PRINCIPLE | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | MONOTONE EQUILIBRIUM PROBLEMS | Error bound | Ky Fan inequality | NONSMOOTH | DESCENT METHODS | COMBINED RELAXATION METHOD | descent method | regularized gap function

Journal Article

Computational and Applied Mathematics, ISSN 0101-8205, 11/2018, Volume 37, Issue 5, pp. 5935 - 5950

.... Using the nonlinear scalarization method, we first propose gap functions for such problems...

58E35 | Computational Mathematics and Numerical Analysis | Strong vector quasiequilibrium | Strong monotonicity | Mathematics | Regularized gap function | 49J53 | Mathematical Applications in Computer Science | 90C25 | 90C33 | Error bounds | Applications of Mathematics | Mathematical Applications in the Physical Sciences | Scalarization methods | EXISTENCE | SYSTEM | MATHEMATICS, APPLIED | STABILITY | DESCENT METHODS | MERIT FUNCTIONS | SOLUTION SETS | LOWER SEMICONTINUITY

58E35 | Computational Mathematics and Numerical Analysis | Strong vector quasiequilibrium | Strong monotonicity | Mathematics | Regularized gap function | 49J53 | Mathematical Applications in Computer Science | 90C25 | 90C33 | Error bounds | Applications of Mathematics | Mathematical Applications in the Physical Sciences | Scalarization methods | EXISTENCE | SYSTEM | MATHEMATICS, APPLIED | STABILITY | DESCENT METHODS | MERIT FUNCTIONS | SOLUTION SETS | LOWER SEMICONTINUITY

Journal Article

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Smoothness properties of a regularized gap function for quasi-variational inequalities

Optimization Methods and Software, ISSN 1055-6788, 07/2014, Volume 29, Issue 4, pp. 720 - 750

...) problem using a regularized gap function approach. For a special class of QVIs, this gap function is continuously differentiable everywhere, in general, however, it has nondifferentiability points...

convex inequalities | Hadamard directional differentiability | Gâteaux differentiability | Fréchet differentiability | finite-dimensional quasi-variational inequalities | generalized Nash equilibrium problem | generalized moving set | regularized gap function | MATHEMATICS, APPLIED | MULTIPLIERS | Gateaux differentiability | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | IMPULSE CONTROL | OPTIMIZATION | Frechet differentiability | PROGRAMMING ALGORITHM | Mathematical analysis | Inequalities | Software | Mathematical models | Continuity | Smoothness | Optimization | Computer programs

convex inequalities | Hadamard directional differentiability | Gâteaux differentiability | Fréchet differentiability | finite-dimensional quasi-variational inequalities | generalized Nash equilibrium problem | generalized moving set | regularized gap function | MATHEMATICS, APPLIED | MULTIPLIERS | Gateaux differentiability | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | IMPULSE CONTROL | OPTIMIZATION | Frechet differentiability | PROGRAMMING ALGORITHM | Mathematical analysis | Inequalities | Software | Mathematical models | Continuity | Smoothness | Optimization | Computer programs

Journal Article

Journal of inequalities and applications, ISSN 1029-242X, 06/2020, Volume 2020, Issue 1, pp. 1 - 16

...) and to obtain error bounds for this kind of MSVIQVIP in terms of the residual gap function, the regularized gap function, and the D-gap function...

EXISTENCE | MATHEMATICS, APPLIED | L-i-Lipschitz continuous mapping | Generalized f-projection operator | Regularized gap function | Residual gap function | MATHEMATICS | h-v-Lipschitz continuous mapping | MERIT FUNCTIONS | Error bounds | D-gap function | Mixed set-valued vector inverse quasi-variational inequality | GAP FUNCTIONS | H $\mathfrak{H}$ -ϑ-Lipschitz continuous mapping

EXISTENCE | MATHEMATICS, APPLIED | L-i-Lipschitz continuous mapping | Generalized f-projection operator | Regularized gap function | Residual gap function | MATHEMATICS | h-v-Lipschitz continuous mapping | MERIT FUNCTIONS | Error bounds | D-gap function | Mixed set-valued vector inverse quasi-variational inequality | GAP FUNCTIONS | H $\mathfrak{H}$ -ϑ-Lipschitz continuous mapping

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2015, Volume 2015, Issue 1, pp. 1 - 19

In this paper, we define some new notions of gap functions for set-valued mixed quasi-variational inequalities under suitable conditions...

error bounds | residue vector | Mathematics | 90C30 | 49J53 | Analysis | 49C25 | 90C33 | Mathematics, general | sub-differential | Applications of Mathematics | strong monotone mapping | regularized gap function | Errors | Inequalities

error bounds | residue vector | Mathematics | 90C30 | 49J53 | Analysis | 49C25 | 90C33 | Mathematics, general | sub-differential | Applications of Mathematics | strong monotone mapping | regularized gap function | Errors | Inequalities

Journal Article

16.
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Error bounds of regularized gap functions for weak vector variational inequality problems

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2014, Volume 2014, Issue 1, pp. 1 - 7

In this paper, by the nonlinear scalarization method, a global error bound of a weak vector variational inequality is established via a regularized gap function...

error bound | Analysis | Mathematics, general | Mathematics | Applications of Mathematics | weak vector variational inequality | regularized gap function | Weak vector variational inequality | Error bound | Regularized gap function | MATHEMATICS | MATHEMATICS, APPLIED | OPTIMIZATION PROBLEMS | NONSMOOTH | Errors | Nonlinearity | Vectors (mathematics) | Mathematical analysis | Inequalities

error bound | Analysis | Mathematics, general | Mathematics | Applications of Mathematics | weak vector variational inequality | regularized gap function | Weak vector variational inequality | Error bound | Regularized gap function | MATHEMATICS | MATHEMATICS, APPLIED | OPTIMIZATION PROBLEMS | NONSMOOTH | Errors | Nonlinearity | Vectors (mathematics) | Mathematical analysis | Inequalities

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2007, Volume 334, Issue 2, pp. 1022 - 1038

Under the condition that the involved function F is locally Lipschitz, but not necessarily differentiable, we investigate the regularized gap function defined by a generalized distance function...

Error bound | Nonsmooth variational inequality problems | Nonlinear complementarity problems | Regularized gap function | Merit function | MATHEMATICS | MATHEMATICS, APPLIED | nonlinear complementarity problems | LOCAL UNIQUENESS | error bound | ERROR-BOUNDS | nonsmooth variational inequality problems | merit function | regularized gap function | Algorithms | Equality

Error bound | Nonsmooth variational inequality problems | Nonlinear complementarity problems | Regularized gap function | Merit function | MATHEMATICS | MATHEMATICS, APPLIED | nonlinear complementarity problems | LOCAL UNIQUENESS | error bound | ERROR-BOUNDS | nonsmooth variational inequality problems | merit function | regularized gap function | Algorithms | Equality

Journal Article

Mathematical Control and Related Fields, ISSN 2156-8472, 2014, Volume 4, Issue 3, pp. 365 - 379

.... Employing this result, we obtain the exact formulations of the Clarke directional derivatives of the regularized gap functions for nonsmooth quasi-variational...

Nonsmooth Quasi-variational inequality | Clarke directional derivative | Regularized gap function | MATHEMATICS | MARGINAL FUNCTIONS | MATHEMATICS, APPLIED | Nonsmooth quasi-variational inequality | CONVEX | OPTIMIZATION | ERROR-BOUNDS | regularized gap function

Nonsmooth Quasi-variational inequality | Clarke directional derivative | Regularized gap function | MATHEMATICS | MARGINAL FUNCTIONS | MATHEMATICS, APPLIED | Nonsmooth quasi-variational inequality | CONVEX | OPTIMIZATION | ERROR-BOUNDS | regularized gap function

Journal Article

Pacific Journal of Optimization, ISSN 1348-9151, 04/2012, Volume 8, Issue 2, pp. 197 - 215

...) is investigated by means of a variational inequality formulation. This problem is then solved by finding a stationary point of the gap function and the regularized gap function...

Variational inequality problem | Gap function | Complementarity problem | Eigenvalue problem | Regularized gap function | MATHEMATICS, APPLIED | gap function | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | QUADRATIC PROGRAMS | NEWTON METHOD | ALGORITHM | complementarity problem | SUBJECT | variational inequality problem | eigenvalue problem | regularized gap function

Variational inequality problem | Gap function | Complementarity problem | Eigenvalue problem | Regularized gap function | MATHEMATICS, APPLIED | gap function | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | QUADRATIC PROGRAMS | NEWTON METHOD | ALGORITHM | complementarity problem | SUBJECT | variational inequality problem | eigenvalue problem | regularized gap function

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 12/2016, Volume 171, Issue 3, pp. 887 - 905

The paper deals with the gap function approach for equilibrium problems with locally Lipschitz data...

Mathematics | Theory of Computation | Optimization | Gap functions | Descent methods | Calculus of Variations and Optimal Control; Optimization | 90C33 | Nonsmooth equilibrium problem | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | 49M37 | Monotonicity | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ALGORITHM | REGULARIZED GAP FUNCTIONS | Studies | Mathematical models | Equilibrium | Mathematical analysis | Paper | Derivatives | Joints | Descent | Continuity (mathematics)

Mathematics | Theory of Computation | Optimization | Gap functions | Descent methods | Calculus of Variations and Optimal Control; Optimization | 90C33 | Nonsmooth equilibrium problem | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | 49M37 | Monotonicity | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ALGORITHM | REGULARIZED GAP FUNCTIONS | Studies | Mathematical models | Equilibrium | Mathematical analysis | Paper | Derivatives | Joints | Descent | Continuity (mathematics)

Journal Article

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