Semidefinite programming method

We solve the nonlinear formulation of the semidefinite program by the augmented Lagrange multiplier method for constrained nonlinear optimization [28, 29]. Consider the augmented Lagrangian function... 

THE LOWER BOUND METHOD FOR DENSITY MATRICES AND SEMIDEFINITE PROGRAMMING... 

The central problem in electronic structure theory is to determine the ground state of a system of electrons, which is typically done variationally by minimizing the energy. The lower bound method can be invoked to achieve a feth-order approximation by replacing the variation minpgq5 (p,/ )g by the semidefinite program... 

SEMIDEFINITE PROGRAMMING FORMULATIONS AND PRIMAL-DUAL INTERIOR-POINT METHODS... 

It is fair to say that neither of these two approaches works especially well N-representability conditions in the spatial representation are virtually unknown and the orbital-resolved computational methods are promising, but untested. It is interesting to note that one of the most common computational algorithms (cf. Eq. (96)) can be viewed as a density-matrix optimization, although most authors consider only a weak A -representability constraint on the occupation numbers of the g-matrix [1, 4, 69]. Additional A-representability constraints could, of course, be added, but it seems unlikely that the resulting g-density functional theory approach would be more efficient than direct methods based on semidefinite programming [33, 35-37]. 

Variational minimization of the energy as a functional of the 2-RDM is expressible as a special convex optimization problem known as a semidefinite program [33, 37, 41,42, 53, 58]. The convexity of the A-representable set of 2-RDMs ensures a rigorous lower bound to the ground-state energy. Because the variational 2-RDM method... 

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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