{"id":{"repo_id":"rice","oai_identifier":"oai:repository.rice.edu:1911/89129"},"canonical_url":"https://search.dev.ndltd.org/etd/rice/oai:repository.rice.edu:1911/89129","repository":{"repo_id":"rice","name":"Rice University","base_url":"https://repository.rice.edu/server/oai/request"},"display":{"title":"A first-order method for the extremization of constrained and unconstrained functions","abstract":"The 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.","abstract_html":"The 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.","abstract_has_math":false,"creators":["Naqvi, Sarwar"],"institution":"Rice University","degree_name":"Master of Science","degree_level":"Masters","degree_discipline":"Engineering","degree_department":null,"school":null,"contributors":[],"advisors":["Huang, H. Y."],"committee_chairs":[],"committee_members":[],"year":1971,"date_issued":"1971","date_published":"1971","updated_at":"2026-07-24T04:10:41Z","subjects":[],"languages":["eng"],"rights":["Copyright 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."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1911/89129","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Huang, H. Y."]},{"key":"dc:creator","label":"Author","values":["Naqvi, Sarwar"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2016-04-21T12:01:33Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2016-04-21T12:01:33Z"]},{"key":"dc:date.issued","label":"Date","values":["1971"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Rice University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1911/89129"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The 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."]},{"key":"dc:title","label":"Title","values":["A first-order method for the extremization of constrained and unconstrained functions"]}]}],"canonical_facts":{"dc:contributor.advisor":["Huang, H. Y."],"dc:creator":["Naqvi, Sarwar"],"dc:date.accessioned":["2016-04-21T12:01:33Z"],"dc:date.available":["2016-04-21T12:01:33Z"],"dc:date.issued":["1971"],"dc:description.abstract":["The 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."],"dc:identifier.uri":["https://hdl.handle.net/1911/89129"],"dc:language.iso":["eng"],"dc:rights":["Copyright 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."],"dc:title":["A first-order method for the extremization of constrained and unconstrained functions"],"dc:type":["Thesis"],"thesis:degree_discipline":["Engineering"],"thesis:degree_level":["Masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Rice University"]},"updated_at":"2026-07-24T04:10:41Z"}