Semidefinite programming

A semidefinite program may be written in two complementary formulations, which are known as the primal and dual programs. Eor convenience we define the map M that transforms any vector x) of length n into an h x h matrix M x) by creating each column of the matrix sequentially with the elements of the vector. The primal formulation of the semidefinite program may be expressed in general notation as... 

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... 

M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm. J. Chem. Phys. 114, 8282 (2001). 

D. A. Mazziotti, Realization of quanmm chemistry without wavefunctions through first-order semidefinite programming. Phys. Rev. Lett. 93, 213001 (2004). 

D. A. Mazziotti, First-order semidefinite programming for the direct determination of two-electron reduced density matrices with apphcation to many-electron atoms and molecules. J. Chem. Phys. 121, 10957 (2004). 

S. Burer and R. D. C. Monteiro, A nonhnear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Programming Ser. B 95, 329 (2003). 

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

E. Additional Properties of Matrix Representations E. Energy Lower Bounds 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... 

The gap between the energy and bottom eigenvalue is nonnegative. If the fe-matrix P moves to further decrease the energy, and if the Pauli matrix S moves to further increase the bottom eigenvalue, the gap narrows and possibly shrinks to zero. It is important to note that there are semidefinite programs where this gap cannot shrink to zero we discuss such an example later. However, In our special case where we vary fc-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. 

The semidefinite program we are considering is formulated in terms of the symmetric matrix H and the linear subspace of symmetric matrices S. For the... 

The preceeding lemma does most of the work in establishing the following fundamental theorem of semidefinite programming. 

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]. 

We first review the two papers [3, 4] where, using semidefinite programming, the 3-matrix of a one-dimensional superconductor was calculated for a... 

M. Fukuda, B. J. Braams, M. Nakata, M. L. Overton, J. K. Percus, M. Yamashita, and Z. Zhao, Large-scale semidefinite programs in electronic structure calculation. Math. Programming Ser. B... 

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

They employed a general-purpose semidefinite programming (SDP) software [2] for these calculations. 

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