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active constraints active set barrier function basis Chapter column compute condition number constrained optimization curvature defined deleted descent direction diagonal direction of search efficient equality constraints equations estimate evaluations feasible point feasible region first-order Fletcher formula Gill and Murray given Goldfarb gradient Hessian matrix implies inactive constraints inequality constraints inverse iteration Lagrange multipliers Lagrangian function linear constraints linearly constrained problem linearly independent minimize F(x modified Newton method Newton-type non-singular nonlinear constraints number of constraints objective function obtained orthogonal parameter pattern search penalty function positive definite possible problem minimize procedure projection quadratic approximation quadratic function quadratic programming quasi-Newton methods rate of convergence reduced rows satisfied as equalities scalar search direction second derivatives second-order Section sequence simplex solution solving sparse sparse matrix stationary point step strategy strong local minimum techniques Theorem transformation methods unconstrained minimization updating variables vector violated zero