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  • 1
    ISSN: 1573-2878
    Keywords: Multicriteria optimization ; vector-valued optimization ; maximum principle ; necessary conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal control problem with vector-valued cost is considered. The satisfaction of necessary conditions for this problem is related to the satisfaction of such conditions for the problems with individual (component) scalar-valued costs.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1573-2878
    Keywords: Calculus of variations ; stochastic optimization ; necessary conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A stochastic version of the modified Young's generalized necessary conditions in the calculus of variations is given in this paper. It is based on an extension of Minkowski's theorem on the existence of a flat support for a convex figure, and it generalizes the necessary conditions of Weierstrass and Euler in the classical theory of the calculus of variations to a class of admissible curves which are expressible in terms of a finite number of random parameters. The integrals which we consider here are in the general Denjoy sense, except those with respect to the random parameters, which exist in the Lebesgue sense defined on a probability space. The importance of our stochastic analysis lies in the completion that a minimum not attained in the classical sense may be, and frequently is, attained in the stochastic case.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1573-2878
    Keywords: Linear controlled systems ; optimal rendezvous ; necessary conditions ; sufficient conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A rendezvous problem of two linear controlled systems is investigated. The two systems are required to rendezvous with each other during a given nonzero time interval. A necessary condition and a sufficient condition are obtained.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Nonlinear systems ; calculus of variations ; necessary conditions ; maximum principle ; optimization theorems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is the purpose of this paper to develop and present new approaches to optimal control problems for which the state evolution equation is nonlinear. For bilinear systems in which the evolution equation is right invariant, it is possible to use ideas from differential geometry and Lie theory to obtain explicit closed-form solutions.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1573-2878
    Keywords: Discrete models ; partial difference equations ; Dieudonné-Rashevsky form ; noncausal systems ; necessary conditions ; derived cones ; separation of convex sets ; implicit function theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A general first-order dynamic representation for discrete systems with several independent variables is proposed, based on the Dieudonné-Rashevsky form for partial differential equations. This representation does not restrict consideration to causal systems. A minimum principle for such systems is proved, thus extending results known for discrete-time systems to the case of several independent variables. The proof requires only the classical implicit function theorem.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Optimization ; necessary conditions ; Kuhn-Tucker theorem ; multiplier rules ; separation of convex sets ; mathematical programming ; generalized derivatives
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the following abstract mathematical programming problem: in a setD, find an element that optimizes a real function φ0, subject to inequality constraints φ1⩽0, ..., φ p ⩽0 and equality constraints φ p+1=0, ..., φ p+q =0. Necessary conditions for this problem, like the Karush-Kuhn-Tucker theorem, can be seen as a consequence of separating with a hyperplane two convex sets inR p+q+1, the image space of the map Φ=(φ0, φ1, ..., φ p+q ). This paper reviews this approach and organizes it into a coherent way of looking at necessary conditions in optimization theory.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Calculus of variations ; simplest problem ; necessary conditions ; sufficient conditions ; strong conjugate point condition ; minimum principle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the theory of the calculus of variations for the simplest problem is reviewed. Simpler proofs are given for the classical conditions and a new necessary condition is presented which allows strong, necessary, and sufficient conditions to be stated for the first time.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Range of vector measure ; convexity ; Lyapunov's theorem ; optimal control theory ; necessary conditions ; bang-bang control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A constructive and simple proof is given of a generalization of a result of Lyapunov on the convexity of the range of vector measure.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Nonsmooth optimization ; necessary conditions ; semidifferentiable functions ; Lagrange multipliers ; image space
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract After the pioneering approach of Hestenes, which has not been sufficiently considered, the concept of image of a constrained extremum problem has been recognized recently to be a powerful tool in optimization. Hence in this paper, we consider a necessary optimality condition for a nondifferentiable extremum problem established by Hestenes in the image space and, by again exploiting the concept of image of the problem, a necessary condition embodying that of Hestenes is given. Moreover, the study is completed by expressing the previous result in the space where the unknowns of the given problem run. In this way, we obtain a Lagrangian-type necessary condition holding for a particular class of semidifferentiable problems.
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  • 10
    ISSN: 1573-2878
    Keywords: Bilevel programming ; value functions ; existence theorems ; necessary conditions ; nonsmooth analysis ; Lagrange multipliers
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We formulate in this paper several versions of the necessary conditions for general bilevel programming problems. The technique used is related to standard methods of nonsmooth analysis. We treat separately the following cases: Lipschitz case, differentiable case, and convex case. Many typical examples are given to show the efficiency of theoretical results. In the last section, we formulate the general multilevel programming problem and give necessary conditions of optimality in the general case. We illustrate then the application of these conditions by an example.
    Type of Medium: Electronic Resource
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