This paper presents a solution to the so called equivalence problem which was introduced in [2]. The equivalence problem consists of a semismooth system of equations which has the following form: (Formula Presented) T...
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This paper presents a solution to the so called equivalence problem which was introduced in [2]. The equivalence problem consists of a semismooth system of equations which has the following form: (Formula Presented) This system can be formulated in a more simplified representation by ũ = M̃(x, y, z, w)(x, y, z, w) T = M̃(x̃)x̃ T The vector ũ can be interpreted as possible control parameters of a time-discrete system, whereas the vector x̃ can be seen as bargaining solution of a cooperative game. In [2] the bargaining solution is identical to the τ-value, which was introduced in [3]. For the special case of three actors the τ-value lies always in the core, if we assume 1-convexity. Under these assumptions, all properties are expressed by the general formulation of (2) where the functions f, g and h are of the form ξ(u) = min{u 1, . . . , u n} = - max{-u 1, . . . , -u n}. By exploiting the combinatorial structure of max-type functions we can show that a solution of (2) exists and may be found via Newton type methods which are treated in [1].
Clarke’s generalized gradient of the smooth composition of max-type functions isconsidered. Clarke’s generalized gradient of this class of functions is a convex hull of afinite number of points. Several elements of ...
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Clarke’s generalized gradient of the smooth composition of max-type functions isconsidered. Clarke’s generalized gradient of this class of functions is a convex hull of afinite number of points. Several elements of Clarke’s generalized gradient is *** elements of Clarke’s generalized gradient for this class of functions, at apoint, is transformed into computing gradients of a smooth function, at a point, bydetermining the compatibility of systems of linear inequalities.
The paper is devoted to two systems of nonsmooth equations. One is the system of equations of max-type functions and the other is the system of equations of smooth compositions of max-type functions. The Newton and ap...
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The paper is devoted to two systems of nonsmooth equations. One is the system of equations of max-type functions and the other is the system of equations of smooth compositions of max-type functions. The Newton and approximate Newton methods for these two systems are proposed. The Q-superlinear convergence of the Newton methods and the Q-linear convergence of the approximate Newton methods are established. The present methods can be more easily implemented than the previous ones, since they do not require an element of Clarke generalized Jacobian, of B-differential, or of b-differential, at each iteration point.
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