Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Fukushima, M. Mathematical Programming, 53(1-3):99–110, January, 1992.

Paper doi abstract bibtex

Whether or not the general asymmetricvariational inequality problem can be formulated as a ditterentiable optimization problem has been an open question. This paper gives an affirmativeanswer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuonsly differentiable whenever the mapping involved in the latter problem is continuously ditIerentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem, We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.

@article{fukushima_equivalent_1992,
	title = {Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems},
	volume = {53},
	issn = {0025-5610, 1436-4646},
	url = {http://link.springer.com/10.1007/BF01585696},
	doi = {10.1007/BF01585696},
	abstract = {Whether or not the general asymmetricvariational inequality problem can be formulated as a ditterentiable optimization problem has been an open question. This paper gives an affirmativeanswer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuonsly differentiable whenever the mapping involved in the latter problem is continuously ditIerentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem, We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.},
	language = {en},
	number = {1-3},
	urldate = {2024-02-13},
	journal = {Mathematical Programming},
	author = {Fukushima, Masao},
	month = jan,
	year = {1992},
	pages = {99--110},
}

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