Dual SDP formulation

Considering such recent relevance of SDP in quantum chemistry, this chapter discusses some practical aspects of this variational calculation of the 2-RDM formulated as an SDP problem. We first present the definition of an SDP problem, and then the primal and dual SDP formulations of the variational calculation of the 2-RDM as SDP problems (Section II), an efficient algorithm to solve the SDP problems the primal-dual interior-point method (Section III), a brief section about alternative and also efficient augmented Lagrangian methods (Section IV), and some computational aspects when solving the SDP problems (Section V). 

In this case, the variables for the primal SDP problem with free variables (Eq. (3)) and the dual SDP problem with equality constraints (Eq. (4)) are X,x) G S X IR and y G IR , respectively. Therefore the size of an SDP problem depends now on the size of each block-diagonal matrix of X, m, and s. We should also mention that the problem as represented by Eq. (4) is the preferred format for the dual SDP formulation of the variational calculation, which we present in the next section, too. 

Eq. (4)) [14—16], A key point here to understand the difference between these two formulations is that the dual SDP formulation (Eq. (4)) is not the dual of the primal SDP formulation (Eq. (1)). Both formulations produce two distinct pairs of primal and dual SDP problems, which mathematically describe the same fermionic system. Since their mathematical formulations differ, this implies differences in the computational effort to solve them. 

If we employ the dual SDP formulation and include the P, Q, G, Tl, and T2 conditions, the number of rows/columns of the largest block-diagonal matrices scale as 3r /16 again, while m scales as 3r" /64 and s as /A. 

The advantages of the dual SDP formulation are clear when comparing Tables I and II. First, notice that the sizes of the block-diagonal matrices are unchanged in both formulations. There is also an additional constraint = c in the dual SDP formulation, which is absent in the primal SDP formulation. Then, while the size m of equality constraint in the primal SDP formulation (see Eq. (1)) corresponds to the dimensions of the Q, G, Tl, and T2 matrices included in the formulation and scales as 25r /576, the dimension m of the variable vector y e R " in the dual SDP formulation (see Eq. (4)) corresponds to the dimension of the 2-RDM and scales merely as 3r" /64. The difference becomes more remarkable when more //-representabUity conditions are considered in these primal or dual SDP formulations. Computational implications when solving the SDP problems employing the primal and dual SDP formulations are discussed in Section V. 

Theoretical Number of Floating-Point Operations per Iteration (FLOPI), Maximum Number of Major Iterations, and Memory Usage for the Parallel Primal-Dual Interior-Point Method (pPDIPM) and for the First-Order Method (RRSDP) Applied to Primal and Dual SDP Formulations". 

We can also conclude that if we employ the Primal-Dual interior-point method, the dual SDP formulation provides a more reduced mathematical description of the variational calculation of the 2-RDM than employing the primal SDP formulation. The former formulation also allows us to reach a faster computational solution. On the other hand, the number of floating-point operations and the memory storage of RRSDP do not depend on the primal or dual SDP formulations. 

In this section, we focus on how to formulate the variational calculation of the 2-RDM as an SDP problem. In fact, it can always be formulated as a primal SDP problem (Eq. (1)) [1, 8-13] or as a dual SDP problem with equality constraints... 

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