Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination

Abstract

Abstract A new active set based algorithm is proposed that uses the conjugate gradient method to explorethe face of the feasible region defined by the current iterate and the reduced gradient projection with the fixedsteplength to expand the active set. The precision of approximate solutions of the auxiliary unconstrained problemsis controlled by the norm of violation of the Karush-Kuhn-Tucker conditions at active constraints and the scalarproduct of the reduced gradient with the reduced gradient projection. The modifications were exploited to find therate of convergence in terms of the spectral condition number of the Hessian matrix, to prove its finite terminationproperty even for problems whose solution does not satisfy the strict complementarity condition, and to avoidany backtracking at the cost of evaluation of an upper bound for the spectral radius of the Hessian matrix. Theperformance of the algorithm is illustrated on solution of the inner obstacle problems. The result is an importantingredient in development of scalable algorithms for numerical solution of elliptic variational inequalities.