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Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem
resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore
these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature. 相似文献
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René Meziat Diego Patiño Pablo Pedregal 《Computational Optimization and Applications》2007,38(1):147-171
We propose an alternative method for computing effectively the solution of non-linear, fixed-terminal-time, optimal control
problems when they are given in Lagrange, Bolza or Mayer forms. This method works well when the nonlinearities in the control
variable can be expressed as polynomials. The essential of this proposal is the transformation of a non-linear, non-convex
optimal control problem into an equivalent optimal control problem with linear and convex structure. The method is based on
global optimization of polynomials by the method of moments. With this method we can determine either the existence or lacking
of minimizers. In addition, we can calculate generalized solutions when the original problem lacks of minimizers. We also
present the numerical schemes to solve several examples arising in science and technology. 相似文献
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Here, we solve non-convex, variational problems given in the form
where u ∈ (W
1,∞(0, 1))
k
and is a non-convex, coercive polynomial. To solve (1) we analyse the convex hull of the integrand at the point a, so that we can find vectors and positive values λ1, . . . , λ
N
satisfying the non-linear equation
Thus, we can calculate minimizers of (1) by following a proposal of Dacorogna in (Direct Methods in the Calculus of Variations.
Springer, Heidelberg, 1989). Indeed, we can solve (2) by using a semidefinite program based on multidimensional moments.
We dedicate this work to our colleague Jesús Bermejo. 相似文献
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In this work we propose an exact semidefinite relaxation for non-linear, non-convex dynamical programs under discrete constraints
in the state variables and the control variables. We outline some theoretical features of the method and workout the solutions
of a benchmark problem in cybernetics and the classical inventory problem under discrete constraints. 相似文献
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R. Meziat 《Journal of Mathematical Sciences》2003,116(3):3303-3324
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