Boundary methods, semidefinite

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

The constraint matrices A,- in Eq. (19) contain the mappings among D, Q, and G, the contractions to D and g, and the fixed-trace condition. Semidefinite programs for quantum chemical Hamiltonians have been solved with a variety of algorithms [27, 33, 37, 54, 60]. In this work, we utilize the boundary-point method [26, 37], a type of quadratic regularization method. The floating-point... 

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