## Minimization with One Linear Equality Constraint and Bounds on the Variables |

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active set algebraic Algorithm for SLCP Appendix approximate projected Hessian bound choice column computer storage condition number constraint release convergence corresponding denote density estimation descent direction diagonal matrix direction of negative equality constraint problem exact minimization factors of H feasible direction feasible point feasible region function evaluations Gill and Murray go to Step gradient H O H inequality constraints iterate Lagrange multipliers Lemma line search linear constraints linear equality constraint machine epsilon matrix H minimize f(x n-t n-t nonsingular null space numerical stability objective function optimization problem orthogonal orthogonal matrix permutation matrix positive definite procedure Proof quadratic programming regularized Newton method result satisfies 9 search direction secant algorithm secant method secant update single linear equality solution special structure stationary point subproblem Suppose Theorem 3.2 triangular factorization unit lower triangular vector Vf(x Z V f(x)Z ZTHZ ZTVf(x