Optimization semidefinite programming

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 search for optimal solutions to both the energy problem and the spectral optimization problem typically starts with matrices P and S that have a positive gap. Iterations are designed to move P so that the energy is decreased, and to move S so that the bottom eigenvalue is increased, and such motions cause the gap to narrow. It is important that there are semidefinite programs where this gap cannot shrink to zero, and we discuss such an example later. However, in our special case where we vary fe-matrices and Pauli matrices, as we have defined them, the gap shrinks to zero. This is an important result for both theoretical and practical reasons a proof is supplied below. 

Here we briefly sketch two directions in research on algorithms for semidefinite programming. A more complete discussion can be found in M. Todd s Semidefinite Optimization [18], or in the Handbook of Semidefinite Programming edited by WoUcowicz et al. [19]. 

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... 

Kanno Y, Takewaki I (2006) Sequential semidefinite program for maximum robustness design of stmctures under load uncertainties. J Optim Theory Appl 130(2) 265-287... 

If H is positive semidefinite, then problem (QP) is a convex program. Thus, the KKT conditions are sufficient in this case, and any solution to this system will yield a global optimal solution to (QP). When H is indefinite, then local optimal solutions which are not global optimal solutions may... 

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