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We present a unified study of first and second order necessary and sufficientoptimality conditions for minimax and Chebyshev optimisation problems with coneconstraints. First order optimality conditions for such problems can beformulated in several different forms: in terms of a linearised problem, interms of Lagrange multipliers (KKT-points), in terms of subdifferentials andnormal cones, in terms of a nonsmooth penalty function, in terms of cadres withpositive cadre multipliers, and in an alternance form. We describeinterconnections between all these forms of necessary and sufficient optimalityconditions and prove that seemingly different conditions are in factequivalent. We also demonstrate how first order optimality conditions can bereformulated in a more convenient form for particular classes of coneconstrained optimisation problems and extend classical second order optimalitycondition for smooth cone constrained problems to the case of minimax andChebyshev problems with cone constraints. The optimality conditions obtained inthis article open a way for a development of new efficient structure-exploitingmethods for solving cone constrained minimax and Chebyshev problems.

A Unified Study of Necessary and Sufficient Optimality Conditions for Minimax and Chebyshev Problems with Cone Constraints