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  • 1
    Electronic Resource
    Electronic Resource
    s.l. : American Chemical Society
    Analytical chemistry 38 (1966), S. 715-720 
    ISSN: 1520-6882
    Source: ACS Legacy Archives
    Topics: Chemistry and Pharmacology
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A nonlinear programming problem with inequality constraints and with unknown vectorx is converted to an unconstrained minimization problem in unknownsx and λ, where λ is a vector of Lagrange multipliers. It is shown that, if the original problem possesses standard convexity properties, then local minima of the associated unconstrained problem are in fact global minima of that problem and, consequently, Kuhn-Tucker points for the original problem. A computational procedure based on the conjugate residual scheme is applied in thexλ-space to solve the associated unconstrained problem. The resulting algorithm requires only first-order derivative information on the functions involved and will solve a quadratic programming problem in a finite number of steps.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; penalty function methods ; gradient projection methods ; convergence analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new programming algorithm for nonlinear constrained optimization problems is proposed. The method is based on the penalty function approach and thereby circumyents the necessity to maintain feasibility at each iteration, but it also behaves much like the gradient projection method. Although only first-order information is used, the algorithm converges asymptotically at a rate which is independent of the magnitude of the penalty term; hence, unlike the simple gradient method, the asymptotic rate of the proposed method is not affected by the ill-conditioning associated with the introduction of the penalty term. It is shown that the asymptotic rate of convergence of the proposed method is identical with that of the gradient projection method.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Game theory ; stability ; contraction mappings
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with a class of noncooperative games ofn players that are defined byn reward functions which depend continuously on the action variables of the players. This framework provides a realistic model of many interactive situations, including many common models in economics, sociology, engineering, and political science. The concept of Nash equilibrium is a suitable companion to such models. A variety of different sufficient conditions for existence, uniqueness, and stability of a Nash equilibrium point have been previously proposed. By sharpening the noncooperative aspect of the framework (which is really only implicit in the original framework), this paper attempts to isolate one set of “natural” conditions that are sufficient for existence, uniqueness, and stability. It is argued thatl ∞ quasicontraction is such a natural condition. The concept of complete stability is introduced to reflect the full character of noncooperation. It is then shown that, in the linear case, the condition ofl ∞ quasicontraction is both necessary and sufficient for complete stability.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1573-2878
    Keywords: Pareto efficiency ; welfare theorems ; benefit functions ; duality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper develops several optimization principles relating the fundamental concepts of Pareto efficiency and competitive equilibria. The beginning point for this development is the introduction of a new function describing individual preferences, closely related to willingness-to-pay, termed the benefit function. An important property of the benefit function is that it can be summed across individuals to obtain a meaningful measure of total benefit relative to a given set of utility levels; and the optimization principles presented in the paper are based on maximization of this total benefit. Specifically, it is shown that, under appropriate technical assumptions, a Pareto-efficient allocationX maximizes the total benefit relative to the utility levels it yields. Conversely, if an allocationX yields zero benefit and maximizes the total benefit function, then that allocation is Pareto efficient. The Lagrange multipliersp of the benefit maximization problem serve as prices; and the (X,p) pair satisfies a generalized saddle-point property termed a Lagrange equilibrium. This in turn is equivalent, under appropriate assumptions, to a competitive equilibrium. There are natural duals to all of the results stated above. The dual optimization principle is based on a surplus function which is a function of prices. The surplus is the total income generated at pricesp, minus the total income required to obtain given utility levels. The dual optimization principle states that prices that are dual (or indirect) Pareto efficient minimize total surplus and render it zero. Conversely, a set of prices that minimizes total surplus and renders it zero is a dual Pareto efficient set of prices. The results of the paper can be viewed as augmenting the first and second theorems of welfare economics (and their duals) to provide a family of results that relate the important economic concepts of Pareto efficiency, equilibrium, dual (or indirect) Pareto efficiency, total benefit, Lagrange equilibrium, and total surplus.
    Type of Medium: Electronic Resource
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