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Through the 1 H and 13 C NMR measurements for the symmetrical b-diketones such as 2,4-pentanedione and 1,3-diphenyl-1,3-propanedione and unsymmetrical one such as 1-phenyl-1,3-butanedione at various concentrations and temperatures, we confirmed that 1-phenyl-1,3-butanedione in CDCl 3 exists as monomers in its relatively low concentration. In addition, the 1-phenyl-1,3-butanedione in CDCl 3 exists not as a keto-form but as two kinds of cis-enol forms. The proton transfer between the two kinds of cis-enols for 1-phenyl-1,3-butanedione was discussed thermodynamically; it is concluded that the OH proton of enol of 1-phenyl-1,3-butanedione is considerably located near the oxygen atom attached to the carbon atom linking to a phenyl group.

We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem (SDP) with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the box constraints and then onto a set defined by a linear matrix inequality. By exploiting structures of the two projections, we show the same convergence properties can be obtained for the proposed method as Birgin's method where the exact orthogonal projection onto the intersection of two convex sets is performed. Using the convergence properties, we prove that the proposed algorithm attains the optimal value or terminates in a finite number of iterations. The efficiency of the proposed method is illustrated with the numerical results on randomly generated synthetic/deterministic data and gene expression data, in comparison with other methods including the inexact primaldual path-following interior-point method, the adaptive spectral projected gradient method, and the adaptive Nesterov's smooth method. For the gene expression data, our results are compared with the quadratic approximation for sparse inverse covariance estimation method. We show that our method outperforms the other methods in obtaining a better optimal value fast.Key words. Dual spectral projected gradient methods, log-determinant semidefinite programs with linear constraints, dual problem, theoretical convergence results, computational efficiency. AMS Classification. 90C20, 90C22, 90C25, 90C26.

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