Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 21 Sep 2020 00:51:43 GMT2020-09-21T00:51:43Z5041A Merit Function Approach for Direct Searchhttp://hdl.handle.net/10316/45560Title: A Merit Function Approach for Direct Search
Authors: Gratton, Serge; Vicente, Luís Nunes
Abstract: In this paper it is proposed to equip direct-search methods with a general procedure to minimize an objective function, possibly nonsmooth, without using derivatives and subject to constraints on the variables. One aims at considering constraints, most likely nonlinear or nonsmooth, for which the derivatives of the corresponding functions are also unavailable. The novelty of this contribution relies mostly on how relaxable constraints are handled. Such constraints, which can be relaxed during the course of the optimization, are taken care of by a merit function and, if necessary, by a restoration procedure. Constraints that are unrelaxable, when present, are treated by an extreme barrier approach. One is able to show that the resulting merit function direct-search algorithm exhibits global convergence properties for first-order stationary constraints. As in the progressive barrier method [C. Audet and J. E. Dennis Jr., SIAM J. Optim., 20 (2009), pp. 445--472], we provide a mechanism to indicate the transfer of constraints from the relaxable set to the unrelaxable one.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/455602014-01-01T00:00:00ZAn indicator for the switch from derivative-free to derivative-based optimizationhttp://hdl.handle.net/10316/44581Title: An indicator for the switch from derivative-free to derivative-based optimization
Authors: Gratton, Serge; Soualmi, Nacer; Vicente, Luís Nunes
Abstract: In some optimization problems found in applications, the derivatives of the objective function can be computed or approximated but at an expensive cost, and it is desirable to know when to use derivative-free methods (such as direct search, for instance) or derivative-based methods (such as gradient or quasi-Newton methods). Derivative-free methods may achieve a steady initial progress for some problems, but after some advance they may also become slower or even stagnate due to the lack of derivatives. It is thus of interest to provide a way to appropriately switch from a derivative-free method to a derivative-based one. In this paper, we develop a family of indicators for such a switch based on the decrease properties of both classes of methods (typically used when deriving worst case complexity bounds).
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10316/445812017-01-01T00:00:00ZA contribution to the conditioning of the total least squares problemhttp://hdl.handle.net/10316/13634Title: A contribution to the conditioning of the total least squares problem
Authors: Baboulin, Marc; Gratton, Serge
Abstract: We derive closed formulas for the condition number of a linear function
of the total least squares solution. Given an over determined linear systems Ax = b,
we show that this condition number can be computed using the singular values and
the right singular vectors of [A; b] and A. We also provide an upper bound that
requires the computation of the largest and the smallest singular value of [A; b] and
the smallest singular value of A. In a numerical example, we compare these values
with the condition estimate given in [17].
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/136342009-01-01T00:00:00ZUsing dual techniques to derive componentwise and mixed condition numbers for a linear functional of a linear least squares solutionhttp://hdl.handle.net/10316/11225Title: Using dual techniques to derive componentwise and mixed condition numbers for a linear functional of a linear least squares solution
Authors: Baboulin, Marc; Gratton, Serge
Abstract: We prove duality results for adjoint operators and product norms in
the framework of Euclidean spaces. We show how these results can be used to derive
condition numbers especially when perturbations on data are measured componentwise
relatively to the original data. We apply this technique to obtain formulas for
componentwise and mixed condition numbers for a linear functional of a linear least
squares solution. These expressions are closed when perturbations of the solution
are measured using a componentwise norm or the in nity norm and we get an upper
bound for the Euclidean norm.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/112252008-01-01T00:00:00Z