Primal SDP problem

Let us denote by S the space of block-diagonal real symmetric matrices (i.e., multiple symmetric matrices arranged diagonally in a unique large matrix) with prescribed dimensions, and by U " the m-dimensional real space. Given the constants C,Ai,A2,. .., A e S, and b e IR , an SDP problem is usually defined either as the primal SDP problem. 

Then we can dehne the primal SDP problem with free variables. 

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. 

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... 

As an instructive example, we consider the primal SDP formulation in detail. First, we show that the variational minimization of a two-particle system can be trivially formulated as a primal SDP problem. Next, we show how we constrain the eigenvalues of the 1-RDM between zero and one, and finally, how we set the SDP constraints to satisfy P and Q conditions simultaneously. 

The difficulty here is how to simultaneously constrain y and / y to be positive semidefinite. To formulate it as a primal SDP problem (Eq. (1)), we should express these two conditions as a positive semidefinite constraint over a single matrix let y be a block-diagonal matrix in which two symmetric matrices yj and y2 are arranged diagonally, and let us express the interrelation between these two matrices via linear constraints defined by the matrices Ap and the constants as in Eq. (1). That is. 

Summing up, the main points when formulating this variational calculation as a primal SDP problem are ... 

The inclusion of other known A-representability conditions like G, Tl, and T2 [14] in the variational calculation can be embedded into the primal SDP problem in a similar way. 

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). 

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. 

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. 

The success of Primal-Dual interior-point methods is due to its feature of computing reliable and highly precise solutions in a guaranteed time framework, although its computational cost can become prohibitively expensive for large-scale SDP problems. 

For the SDP problems arising from the variational calculation, in which we are interested, the theoretical number of floating-point operations required by parallel Primal-Dual interior-point method-based software scales as... 

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