A first-order method for the extremization of constrained and unconstrained functions

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dc.contributor.advisorHuang, H. Y.en_US
dc.creatorNaqvi, Sarwaren_US
dc.date.accessioned2016-04-21T12:01:33Zen_US
dc.date.available2016-04-21T12:01:33Zen_US
dc.date.issued1971en_US
dc.description.abstractThe problem of extremizing a function f(xl subject to the constraint cc(x) = 0 is considered. Here, f is a scalar, x an n-vector, and cp a q-vector, where 0 < q < n. This problem is transformed into that of minimizing the unconstrained function R(x, X), where x and X are regarded as independent variables. The q-vector X is the Lagrange multiplier associated with the constraint and the function R(x, X) is the performance index measuring the cumulative error in the optimum condition and the constraint. The minimum R(x, X) = 0 of the performance index is sought by applying quadratically convergent algorithms for unconstrained function minimization: the (n+qj-vector Y = [x,^]T is the independent variable associated with the performance index R(y). Since the performance index R(y) involves the first derivatives f and cp^, the gradient G(y) = R^(y), which is employed in quadratically convergent algorithms, involves the second derivatives f and cp . To avoid the explicit use of these second derivatives, a two-point determination of the gradient G(y) is developed: the (n+q)-vector G(y) is computed numerically through only two evaluations of the function R(y). Concerning the one-dimensional determination of the stepsize a, a two point quasilinearization search is developed. This method requires only two evaluations of the function R(y), but preserves the eventual quadratic convergence of the quasilinearization method. Two terminating conditions are investigated: exact search and one-cycle search. Thus, the method presented here is a first-order method. For the ideal case of a quadratic function subject to a linear constraint, it converges to the solution in n+q iterations, at most. The total computational effort involved is equivalent to, at most, 3(n+q) + 1 evaluations of the function R(y). Three numerical examples are given using both the exact search and the one-cycle search. The results are presented in terms of number of iterations and number of function evaluations for convergence.en_US
dc.format.digitalOriginreformatted digitalen_US
dc.format.extent32 ppen_US
dc.identifier.callnoThesis M.E. 1971 NAQVIen_US
dc.identifier.citationNaqvi, Sarwar. "A first-order method for the extremization of constrained and unconstrained functions." (1971) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/89129">https://hdl.handle.net/1911/89129</a>.en_US
dc.identifier.digitalRICE0166en_US
dc.identifier.urihttps://hdl.handle.net/1911/89129en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.titleA first-order method for the extremization of constrained and unconstrained functionsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMechanical Engineeringen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelMastersen_US
thesis.degree.nameMaster of Scienceen_US
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