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An algorithm for constrained optimization with applications to the design of mechanical structures

Publication . Barbarosie, Cristian; Lopes, Sérgio; Toader, Anca-Maria

We propose an algorithm for minimizing a functional under constraints. It uses _rst order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (which minimizes the objective functional) and a correction step related to the Newton method (which aims to solve the equality constraints). The linear combination between these two steps envolves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satis_ed in the limit (after convergence). Although the algorithm can be used as a general-purpose optimization tool, it is designed speci_cally for problems where _rst order derivatives of both objective and constraint functionals are available but not second order derivatives (as is often the case in structural optimization).

2018-09-14Conference object

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Propagation of regularity and positive definiteness: a constructive approach

Publication . Buescu, Jorge; Paixão, António; Oliveira, Claudemir

We show that, for positive de finite kernels, ifspecific forms of regularity (continuity, S-n-differentiability or holomorphy) hold locally on the diagonal, then they must hold globally on the whole domain of positive-definiteness. This local-toglobal propagation of regularity is constructively shown to be a consequence of the algebraic structure induced by the non-negativity of the associated bilinear forms up to order 5. Consequences of these results for topological groups and for positive definite and exponentially convex functions are explored.

2018Journal article

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Complex positive definite functions on strips

Publication . Buescu, Jorge; Coelho, Jose; Symeonides, A.

We characterize a holomorphic positive definite function f defined on a horizontal strip of the complex plane as the Fourier–Laplace transform of a unique exponentially finite measure on R. With this characterization, the classical theorems of Bochner on positive definite functions and of Widder on exponentially convex functions become respectively the real and imaginary sections of the corresponding complex integral representation. We provide minimal holomorphy assumptions for this characterization and derive conclusions for meromorphic functions under minimal positive definiteness conditions. Further characterizations are derived from conditions on the derivatives of f arising from the study of the usual concepts of moment, moment-generating function and characteristic function in this context.

2017-03Journal article

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