Numerical Methods for Constrained OptimizationPhilip E. Gill, P. E. Gill, William Allan Murray, Institute of Mathematics and Its Applications, National Physical Laboratory (Great Britain) |
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Page 43
... ( Solution of the Dual Program ) The solution of QP is defined by two vectors , the solution p , and & the vector of Lagrange multipliers associated with the constraints ÂT which satisfy λώ = g + Gp . We refer to ( p , ỏ ) as a solution ...
... ( Solution of the Dual Program ) The solution of QP is defined by two vectors , the solution p , and & the vector of Lagrange multipliers associated with the constraints ÂT which satisfy λώ = g + Gp . We refer to ( p , ỏ ) as a solution ...
Page 96
... Solution of Large Sparse Sets of Linear Equations 4.2 . Classification of Methods Methods for the solution of the linear equations Ax = b ( 4.2.1 ) can be divided into two classes : direct methods , in which A is transformed into a ...
... Solution of Large Sparse Sets of Linear Equations 4.2 . Classification of Methods Methods for the solution of the linear equations Ax = b ( 4.2.1 ) can be divided into two classes : direct methods , in which A is transformed into a ...
Page 150
... solution , and then taking the solution of this quadratic program as the new estimate x ( +1 ) of the solution to the nonlinear problem . An immediate question is whether such a sequence of estimates will in fact converge to a solution ...
... solution , and then taking the solution of this quadratic program as the new estimate x ( +1 ) of the solution to the nonlinear problem . An immediate question is whether such a sequence of estimates will in fact converge to a solution ...
Common terms and phrases
active constraints active set algorithm barrier function basis Chapter Cholesky factors compute condition number d₁ defined deleted descent direction diagonal direct-search direction of search efficient equality constraints equations estimates evaluations feasible point feasible region fill-in Fletcher formula Gill and Murray given Goldfarb gradient Hessian matrix implies inactive constraints inverse iteration jth column Lagrange multipliers Lagrangian function linear constraints linear programming linearly constrained problem linearly independent LQ factorization matrix G minimize F(x modified Newton method non-singular non-zero elements nonlinear constraints nonlinear programming objective function obtained orthogonal parameter penalty function positive definite procedure projection quadratic approximation quadratic function quadratic programming quasi-Newton methods rate of convergence reduced rows scalar search direction second derivatives second-order Section sequence simplex solution solving sparse sparse matrix stationary point step steplength storage strategy strong local minimum techniques Theorem unconstrained minimization updating variables vector vertex zero