Primal-dual interior point method

Wright, S. J. Primal-Dual Interior-Point Methods. SIAM, Philadelphia, PA (1999). 

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

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). 

Primal-Dual interior-point methods always compute the desired solution within a guaranteed time complexity framework. Moreover, we can always... 

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... 

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". 

From the table, we can see that the first-order method usually requires fewer floating-point operations and memory storage if compared with the Primal-Dual interior-point method. The unique drawback of the former method is that we cannot guarantee a convergence of the method in a certain time frame. 

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. 

Wright S (1997) Primal-dual interior-point methods. SIAM, Philadelphia... 

A number of optimization techniques can be directly applied to QP, such as Newton method, conjugate gradient, and primal dual interior-point method. But in fact, those methods are very hard to use, so they are not widely used in SVM. 

Mehrotra, S. 1992. On the implementation of a primal-dual interior point method. SIAM Journal on Optimization 4(2) 575-601. 

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