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  • 1
    Electronic Resource
    Electronic Resource
    Existence of solutions of certain systems of non-linear inequalities (1968)
    Springer
    Numerische Mathematik 12 (1968), S. 327-334 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In recent years there has been increasing interest in the problem of the existence of solutions to systems of linear inequalities, however little has been done in the general case of non-linear inequalities. In this paper a certain class of non-linear inequalities is considered. Several existence theorems are established which generalize certain recent results in linear inequalities.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02162513
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  • 2
    Electronic Resource
    Electronic Resource
    The complementarity problem (1972)
    Springer
    Mathematical programming 2 (1972), S. 107-129 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract For a given mapF from then-dimensional Euclidean spaceE n into itself, we consider the problem of finding a nonnegative vectorx inE n whose imageF(x) is also nonnegative and such that the two vectors are orthogonal. This problem is refered to in the literature as thecomplemcntarity problcm. The importance of the complementarity problem lies in the fact that it is the unifying mathematical form for a wide range of problems arising in different fields such as mathematical programming, game theory, economics, mechanics, etc ⋯ This paper is concerned mainly with the question of the existence of a solution. Several existence theorems are given under various conditions on the mapF. These theorems cover the cases whenF is nonlinear nondifferentiable, nonlinear but differentiable, and affine.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584538
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  • 3
    Electronic Resource
    Electronic Resource
    An existence theorem for the complementarity problem (1976)
    Springer
    Journal of optimization theory and applications 19 (1976), S. 227-232 
    Details
    ISSN: 1573-2878
    Keywords: Complementarity problem ; convex cones ; existence theorems ; mathematical programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE n ,C* its polar cone,M:C→E n a map, andq a vector inE n . The complementarity problem (q|M) overC is that of finding a solution to the system $$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$ It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q 0 ∈ intC*, then it has a solution for allq ∈E n .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934094
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  • 4
    Electronic Resource
    Electronic Resource
    Complementarity problems over cones with monotone and pseudomonotone maps (1976)
    Springer
    Journal of optimization theory and applications 18 (1976), S. 445-454 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear complementarity problems over cones ; pseudomonotone maps ; mathematical programming ; variational inequalities ; duality theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The notion of a monotone map is generalized to that of a pseudomonotone map. It is shown that a differentiable, pseudoconvex function is characterized by the pseudomonotonicity of its gradient. Several existence theorems are established for a given complementarity problem over a certain cone where the underlying map is either monotone or pseudomonotone under the assumption that the complementarity problem has a feasible or strictly feasible point.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932654
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  • 5
    Electronic Resource
    Electronic Resource
    A note on Khatchian's algorithm (1980)
    Springer
    Journal of optimization theory and applications 32 (1980), S. 595-597 
    Details
    ISSN: 1573-2878
    Keywords: Khatchian's algorithm ; duality theorem ; linear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A short proof of some properties of Khatchian's algorithm is presented using the duality theorem of linear programming.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934042
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  • 6
    Electronic Resource
    Electronic Resource
    The nonlinear complementarity problem with applications, part 1 (1969)
    Springer
    Journal of optimization theory and applications 4 (1969), S. 87-98 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The main result in this paper is an existence and uniqueness theorem for the following nonlinear complementarity problem: Given a mapping from then-dimensional Euclidean spaceE n into itself, find a nonnegative vector inE n whose image, under the given mapping, is also nonnegative, the two vectors being orthogonal to each other. It is shown that the above problem has a unique solution if the given mapping is continuous and strongly monotone on the nonnegative orthantE + n ofE n . It is also shown that a sufficient condition for a differentiable mapping to be strongly monotone on an open set is that all the eigenvalues of the symmetric part of its Jacobian be bounded below by a positive constant on the given set.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00927414
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Seven kinds of monotone maps (1990)
    Springer
    Journal of optimization theory and applications 66 (1990), S. 37-46 
    Details
    ISSN: 1573-2878
    Keywords: Monotone maps ; generalized monotone maps ; convex functions ; generalized convex functions ; first-order conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Known as well as new types of monotone and generalized monotone maps are considered. For gradient maps, these generalized monotonicity properties can be related to generalized convexity properties of the underlying function. In this way, pure first-order characterizations of various types of generalized convex functions are obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940531
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    The nonlinear complementarity problem with applications, part 2 (1969)
    Springer
    Journal of optimization theory and applications 4 (1969), S. 167-181 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The main existence and uniqueness theorem of Part 1 is applied to three specific problems, namely, (a) the symmetric, dual, nonlinear programs of Dantzig, Eisenberg, and Cottle, (b) the saddle point problem of a differentiable scalar function over an unbounded product set, and (c) the equilibrium point problem of ann-person game.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00930577
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  • 9
    Electronic Resource
    Electronic Resource
    Characterizations of generalized monotone maps (1993)
    Springer
    Journal of optimization theory and applications 76 (1993), S. 399-413 
    Details
    ISSN: 1573-2878
    Keywords: Generalized monotone maps ; generalized convex functions ; one-dimensional maps ; first-order characterizations ; affine maps
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalized convexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939374
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  • 10
    Electronic Resource
    Electronic Resource
    Generalized complementarity problem (1971)
    Springer
    Journal of optimization theory and applications 8 (1971), S. 161-168 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A general complementarity problem with respect to a convex cone and its polar in a locally convex, vector-topological space is defined. It is observed that, in this general setting, the problem is equivalent to a variational inequality over a convex cone. An existence theorem is established for this general case, from which several of the known results for the finite-dimensional cases follow under weaker assumptions than have been required previously. In particular, it is shown that, if the given map under consideration is strongly copositive with respect to the underlying convex cone, then the complementarity problem has a solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932464
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