Interior Point Algorithm

Two Primal-dual interior point algorithms are presented for theproblem of maximizing the smallest eigenvalue of a symmetric matrixover diagonal perturbations. These algorithms prove to be simple,robust, and efficient. Both algorithms are based on transforming theTechnische Universitat Graz, Institut fur Mathematik, Kopernikusgasse 24, A-8010Graz, Austria. Research support by Christian Doppler Laboratorium fur DiskreteOptimierung.yProgram in Statistics & Operations Research,...

The quantum many-body problem can be rephrased as a variational determination of the two-body reduced density matrix, subject to a set of N-representability constraints. The mathematical problem has the form of a semidefinite program. We adapt a standard primal-dual interior point algorithm in order to exploit the specific structure of the physical problem. In particular the matrix-vector product can be calculated very efficiently. We have applied the proposed algorithm to a pairing-type Hamiltonian and studied the computational aspects of the method. The standard N-representability conditions perform very well for this problem.

The setE of extreme points which are also efficient are of basic importance in defining the efficiency frontier, from which the observations for all other DMUs are evaluated in DEA. A significant question which we address is “What variations in the data can be tolerated before the membership inE is changed?” This topic is explored using (1) a simple illustrative example, and (2) production data for 30 independent oil companies during the period 1983–1985. Data were allowed to vary simultaneously for all observations and in different subsets determined by random drawings of data for points both inE and not inE. The results were found to be robust in this study, thereby lending further support to earlier studies which also found these classifications into efficient and inefficient performers to be robust in DEA. Technical developments for these new methods of sensitivity analysis are supplied. These developments feature an application of analytic center (interior point) algorithms which ensure that the Strong Complementary Slackness Condition (SCSC) is fulfilled. The solutions satisfy a mathematical condition called “centrality”. Generally, the solutions are at interior points calledanalytic centers. At these interior points, continuity of the input-output ratios ensures that DMUs inE remain inE for at least small relative variations in the data, while empirically these properties have been found to extend to much larger variations in the data sets.

Abstract. The present report establishes the computational issues that will be considered for the implementation of hybrid optimization approaches ori-ented to automated parameter estimation problems. The proposed hybrid optimization approaches are based on the coupling of the ...

We discuss fast implementations of primal-dual interior-point methods for semidefinite programs derived from the Kalman-Yakubovich-Popov lemma, a class of problems that are widely encountered in control and signal processing applications. By exploiting problem structure we achieve a reduction of the complexity by several orders of magnitude compared to general-purpose semidefinite programming solvers.