We present new interstellar dust models which have been derived by simultaneously fitting the far-ultraviolet to near-infrared extinction, the diffuse infrared (IR) emission and, unlike previous models, the elemental abundance constraints on the dust for different interstellar medium abundances, including solar, F and G star, and B star abundances. The fitting problem is a typical ill-posed inversion problem, in which the grain size distribution is the unknown, which we solve by using the method of regularization. The dust model contains various components: PAHs, bare silicate, graphite, and amorphous carbon particles, as well as composite particles containing silicate, organic refractory material, water ice, and voids. The optical properties of these components were calculated using physical optical constants. As a special case, we reproduce the Li & Draine (2001) results, however their model requires an excessive amount of silicon, magnesium, and iron to be locked up in dust: about 50 ppm (atoms per million of H atoms), significantly more than the upper limit imposed by solar abundances of these elements, about 34, 35, and 28 ppm, respectively. A major conclusion of this paper is that there is no unique interstellar dust model that simultaneously fits the observed extinction, diffuse IR emission, and abundances constraints.

70 pages, 23 figures, accepted for publication in the Astrophysical Journal Supplement