Algorithm 813: SPG—Software for Convex-Constrained Optimization
ACM Transactions on Mathematical Software (TOMS)
Ellipsoidal Approach to Box-Constrained Quadratic Problems
Journal of Global Optimization
Quartic Formulation of Standard Quadratic Optimization Problems
Journal of Global Optimization
Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization
ACM Transactions on Mathematical Software (TOMS)
Journal of Global Optimization
Behavior of DCA sequences for solving the trust-region subproblem
Journal of Global Optimization
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In this paper we consider the problem of minimizing a (possibly nonconvex) quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the first part of the paper, we show that (i) given a KKT point that is not a global minimizer, it is easy to find a "better" feasible point; (ii) strict complementarity holds at the local-nonglobal minimizer. In the second part of this paper, we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function. Using the unconstrained formulation we give, in the nonconvex case, a new second order necessary condition for global minimizers. In the third part of this paper, algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported.