On the composition of convex envelopes for quadrilinear terms

International audienceWithin the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting

Error bounds for monomial convexification in polynomial optimization

Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of $[0,1]^n$. This implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately. Our main error bounds depend primarily on the degree of the monomial, making them easy to compute. Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints. As part of this analysis, we also derive the convex hull of a multilinear monomial over $[-1,1]^n$.Comment: 33 pages, 2 figures, to appear in journa

Friedel-crafts alkylation over Al-incorporated mesoporous honeycomb

Catalytic activity of Friedel-Crafts alkylation of toluene with benzyl alcohol has been measured over the Al-incorporated mesoporous honeycomb. The honeycomb was fabricated using MCM-48 and pseudobohemite as inorganic binder and the incorporation of aluminum was performed either by direct implementation of AlCl3 or by slurry mixing before the extrusion. Hydrothermal stability and compressive strength can be improved with the increase of the aluminum content. High catalytic activity of Friedel-Crafts alkylation was observed for the honeycomb containing Al initially in the slurry mixture

Rgs19 regulates mouse palatal fusion by modulating cell proliferation and apoptosis in the MEE

Palatal development is one of the critical events in craniofacial morphogenesis. During fusion of the palatal shelves, removal of the midline epithelial seam (MES) is a fundamental process for achieving proper morphogenesis of the palate. The reported mechanisms for removing the MES are the processes of apoptosis, migration or general epithelial-to-mesenchymal transition (EMT) through modulations of various signaling molecules including Wnt signaling. RGS19, a regulator of the G protein signaling (RGS) family, interacts selectively with the specific α subunits of the G proteins (Gαi, Gαq) and enhances their GTPase activity. Rgs19 was reported to be a modulator of the Wnt signaling pathway. In mouse palatogenesis, the restricted epithelial expression pattern of Rgs19 was examined in the palatal shelves, where expression of Wnt11 was observed. Based on these specific expression patterns of Rgs19 in the palatal shelves, the present study examined the detailed developmental function of Rgs19 using AS-ODN treatments during in vitro palate organ cultivations as a loss-of-function study. After the knockdown of Rgs19, the morphological changes in the palatal shelves was examined carefully using a computer-aided three dimensional reconstruction method and the altered expression patterns of related signaling molecules were evaluated using genome wide screening methods. RT-qPCR and in situ hybridization methods were also used to confirm these array results. These morphological and molecular examinations suggested that Rgs19 plays important roles in palatal fusion through the degradation of MES via activation of the palatal fusion related and apoptotic related genes. Overall, inhibition of the proliferation related and Wnt responsive genes by Rgs19 are required for proper palatal fusion. © 2012 Elsevier Ireland Ltd. All rights reserved.link_to_subscribed_fulltex