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accuracy active constraints algorithm applied Bibliography for Section bound Cholesky factorization columns computed condition error condition number conjugate-gradient method consider constrained problem corresponding defined deleted denote descent direction diagonal discussed in Section eigenvalues elements equality constraints equations evaluations exact example feasible point Figure finite-difference interval formula forward-difference function value given gradient hence Hessian matrix ill-conditioned inequality constraints iteration Lagrange multiplier estimates least-squares problem linear combination linear constraints linear programming linear search linearly independent minimize minimum Newton method non-singular non-zero nonlinear constraints nonlinearly constrained objective function optimization problem orthogonal penalty function penalty parameter perturbation positive definite positive-definite problem functions procedure projected Hessian properties quadratic function quadratic program quasi-Newton method range-space rate of convergence rows satisfy scalar scaling search direction second-order sequence solution solve step length strategy subspace sufficiently techniques termination criteria transformation unconstrained univariate update vector zero