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Vietnam Journal of Mathematics 40:2&3(2012) 201-229

On Second-Order Conditions for Nonsmooth

Problems with Constraints

Anulekha Dhara¹, Dinh The Luc² and Phan Nhat Tinh³

¹Department of Mathematics, Indian Institute of Technology Gandhinagar,

VGEC Campus, Chandkheda, Ahmedabad, 382424 Gujarat, India

²Department of Mathematics, University of Avignon, 33 Rue Louis Pasteur,

84000 Avignon, France, and CIID, Vietnam Academy of Science and

Technology, Vietnam

³Faculty of Mathematics, Hue University of Sciences,77 Nguyen Hue,

Hue, Vietnam

Dedicated to Professor Phan Quoc Khanh on the occasion of his 65th birthday

Received January 2, 2012

Revised May 20, 2012

Abstract. To study the sufficiency of an optimization problem, one either imposes some convexity assumptions or consider second order optimality conditions. In this paper we establish second order optimality conditions for nonsmooth optimization problems by considering second order approximations of the functions involved and by introducing the concept of second order tangentiability.

2000 Mathematics Subject Classification. 49K27, 90C30, 90C34.

Keywords. Second order tangent sets, second order approximations, second order optimality conditions, tangentiability, minimax problem.

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