Scaling relationships between the copositive cone and the cone of sums of squares polynomials

International audienceSeveral NP-complete problems can be turned into convex problems by formulating them as optimization problems over the copositive cone. Unfortunately checking membership in the copositive cone is a co-NP-complete problem in itself. To deal with this problem, several approximation schemes have been developed. One of them is a hierarchy of cones introduced by P. Parrilo. Membership of these cones can be checked by deciding whether a certain polynomial can be written as a sum of squares, which can be done via semidefinite programming. It is known that for matrices of order n4. In particular a surprising result is found for the case n=5, establishing a direct link between the copositive cone and the semidefinite cone of that order

Scaling relationship between the copositive cone and Parrilo's first level approximation

<p>We investigate the relation between the cone of n x n copositive matrices and the approximating cone introduced by Parrilo. While these cones are known to be equal for n a parts per thousand currency sign 4, we show that for n a parts per thousand yen 5 they are not equal. This result is based on the fact that is not invariant under diagonal scaling. We show that for any copositive matrix which is not the sum of a nonnegative and a positive semidefinite matrix we can find a scaling which is not in . In fact, we show that if all scaled versions of a matrix are contained in for some fixed r, then the matrix must be in . For the 5 x 5 case, we show the more surprising result that we can scale any copositive matrix X into and in fact that any scaling D such that for all i yields . From this we are able to use the cone to check if any order 5 matrix is copositive. Another consequence of this is a complete characterisation of in terms of . We end the paper by formulating several conjectures.</p>

Irreducible elements of the copositive cone

<p>We call an element A of the n x n copositive cone C-n irreducible with respect to the nonnegative cone N-n if it cannot be written as a nontrivial sum A = C + N of a copositive matrix C and an elementwise nonnegative matrix N (note that our concept of irreducibility differs from the standard one normally studied in matrix theory). This property was studied by Baumert [2] who gave a characterisation of irreducible matrices. We demonstrate here that Baumert's characterisation is incorrect and give a correct version of his theorem which establishes a necessary and sufficient condition for a copositive matrix to be irreducible. For the case of 5 x 5 copositive matrices we give a complete characterisation of all irreducible matrices. We show that those irreducible matrices in C-5 which are not positive semidefinite can be parameterized in a semi-trigonometric way. Finally, we prove that every 5 x 5 copositive-matrix which is not the sum of a nonnegative and a semidefinite matrix can be expressed as the sum of a nonnegative matrix with zero diagonal and a single irreducible matrix. (C) 2013 Published by Elsevier Inc.</p>

Quantitative DWI implemented after DCE-MRI yields increased specificity for BI-RADS 3 and 4 breast lesions

PurposeTo assess if specificity can be increased when semiautomated breast lesion analysis of quantitative diffusion-weighted imaging (DWI) is implemented after dynamic contrast-enhanced (DCE-) magnetic resonance imaging (MRI) in the workup of BI-RADS 3 and 4 breast lesions larger than 1cm. Materials and MethodsIn all, 120 consecutive patients (mean-age, 48 years; age range, 23-75 years) with 139 breast lesions (1cm) were examined (2010-2014) with 1.5T DCE-MRI and DWI (b=0, 50, 200, 500, 800, 1000 s/mm(2)) and the BI-RADS classification and histopathology were obtained. For each lesion malignancy was excluded using voxelwise semiautomated breast lesion analysis based on previously defined thresholds for the apparent diffusion coefficient (ADC) and the three intravoxel incoherent motion (IVIM) parameters: molecular diffusion (D-slow), microperfusion (D-fast), and the fraction of D-fast (f(fast)). The sensitivity (Se), specificity (Sp), and negative predictive value (NPV) based on only IVIM parameters combined in parallel (D-slow, D-fast, and f(fast)), or the ADC or the BI-RADS classification by DCE-MRI were compared. Subsequently, the Se, Sp, and NPV of the combination of the BI-RADS classification by DCE-MRI followed by the IVIM parameters in parallel (or the ADC) were compared. ResultsIn all, 23 of 139 breast lesions were benign. Se and Sp of DCE-MRI was 100% and 30.4% (NPV=100%). Se and Sp of IVIM parameters in parallel were 92.2% and 52.2% (NPV=57.1%) and for the ADC 95.7% and 17.4%, respectively (NPV=44.4%). In all, 26 of 139 lesions were classified as BI-RADS 3 (n=7) or BI-RADS 4 (n=19). DCE-MRI combined with ADC (Se=99.1%, Sp=34.8%) or IVIM (Se=99.1%, Sp=56.5%) did significantly improve (P=0.016) Sp of DCE-MRI alone for workup of BI-RADS 3 and 4 lesions (NPV=92.9%). ConclusionQuantitative DWI has a lower NPV compared to DCE-MRI for evaluation of breast lesions and may therefore not be able to replace DCE-MRI; when implemented after DCE-MRI as problem solver for BI-RADS 3 and 4 lesions, the combined specificity improves significantly. J. Magn. Reson. Imaging 2016;44:1642-1649

Effectiveness of a Low-Calorie Weight Loss Program in Moderately and Severely Obese Patients

Aims: To compare effectiveness of a 1-year weight loss program in moderately and severely obese patients. Methods: The study sample included 311 obese patients participating in a weight loss program, which comprised a 12-week weight reduction phase (low-calorie formula diet) and a 40-week weight maintenance phase. Body weight and glucose and lipid values were determined at the beginning of the program as well as after the weight reduction and the weight maintenance phase. Participants were analyzed according to their BMI class at baseline (30-34.9 kg/m2; 35-39.9 kg/m2; 40-44.9 kg/m2; 45-49.9 kg/m2; ≥50 kg/m2). Furthermore, moderately obese patients (BMI 2) were compared to severely obese participants (BMI ≥ 40 kg/m2). Results: Out of 311 participants, 217 individuals completed the program. Their mean baseline BMI was 41.8 ± 0.5 kg/m2. Average weight loss was 17.9 ± 0.6%, resulting in a BMI of 34.3 ± 0.4 kg/m2 after 1 year (p Conclusion: 1-year weight loss intervention improves body weight as well as lipid and glucose metabolism not only in moderately, but also in severely obese individuals

High Glutathione and Glutathione Peroxidase-2 Levels Mediate Cell-Type-Specific DNA Damage Protection in Human Induced Pluripotent Stem Cells

Pluripotent stem cells must strictly maintain genomic integrity to prevent transmission of mutations. In human induced pluripotent stem cells (iPSCs), we found that genome surveillance is achieved via two ways, namely, a hypersensitivity to apoptosis and a very low accumulation of DNA lesions. The low apoptosis threshold was mediated by constitutive p53 expression and a marked upregulation of proapoptotic p53 target genes of the BCL-2 family, ensuring the efficient iPSC removal upon genotoxic insults. Intriguingly, despite the elevated apoptosis sensitivity, both mitochondrial and nuclear DNA lesions induced by genotoxins were less frequent in iPSCs compared to fibroblasts. Gene profiling identified that mRNA expression of several antioxidant proteins was considerably upregulated in iPSCs. Knockdown of glutathione peroxidase-2 and depletion of glutathione impaired protection against DNA lesions. Thus, iPSCs ensure genomic integrity through enhanced apoptosis induction and increased antioxidant defense, contributing to protection against DNA damage