Finding largest small polygons with GloptiPoly

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many instances are already solved in the literature, namely for all odd $n$, and for $n=4, 6$ and 8. Thus, for even $n\geq 10$, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for $n=10$ and $n=12$. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic

Semidefinite approximations of projections and polynomial images of semialgebraic sets

Given a compact semialgebraic set S of R^n and a polynomial map f from R^n to R^m, we consider the problem of approximating the image set F = f(S) in R^m. This includes in particular the projection of S on R^m for n greater than m. Assuming that F is included in a set B which is simple (e.g. a box or a ball), we provide two methods to compute certified outer approximations of F. Method 1 exploits the fact that F can be defined with an existential quantifier, while Method 2 computes approximations of the support of image measures.The two methods output a sequence of superlevel sets defined with a single polynomial that yield explicit outer approximations of F. Finding the coefficients of this polynomial boils down to computing an optimal solution of a convex semidefinite program. We provide guarantees of strong convergence to F in L^1 norm on B, when the degree of the polynomial approximation tends to infinity. Several examples of applications are provided, together with numerical experiments

Effectiveness and tolerability of pegylated interferon alfa-2b in combination with ribavirin for treatment of chronic hepatitis C: the PegIntrust Study

Background and study aims : Large international clinical trials conducted in the past 5 years rapidly improved the treatment of chronic hepatitis C; however, it is unclear whether the advances seen in clinical trials are being paralleled by similar improvements in routine clinical practice. PegIntrust is a Belgian community-based trial evaluating the sustained virological response. Patients and Methods : Observational study of 219 patients receiving pegylated interferon alfa-2b (1.5 mu g/kg/wk) and weight. based ribavirin (800-1200 mg/day) for 48 weeks. Primary study end point was sustained virological response (SVR), defined as undetectable HCV RNA 6 months after the completion of treatment. Results : In total, 108 patients (49.3 %) had undetectable HCV RNA at the end of therapy, 91(41.6%) attaining SVR. Of the 111 patients without an end-of-treatment response, 28 were non-responders, and 21 had virological breakthrough. In total, 134 patients attained early virological response (EVR); 88 (65.7%) of those patients attained SVR. In contrast, 82 (96.5 %) of the 85 patients who did not attain EVR also did not attain SVR. Age, fibrosis score and baseline viral load were identified as important predictors of treatment outcome. The most frequently reported serious adverse events resulting in treatment discontinuation were anemia (n = 10), fatigue/asthenia/malaise (n = 6) and fever (n = 3). Conclusion : Our data indicate that treatment of chronic hepatitis C with PEG-IFN alfa-2b plus weight-based ribavirin results in favourable treatment outcomes in a Belgian cohort of patients treated in community-based clinical practice. (Ada gastroenterol. belg., 2010, 73, 5-11)

Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape

Exploiting symmetries in SDP-relaxations for polynomial optimization

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.Comment: (v3) Minor revision. To appear in Math. of Operations Researc

RISK and SAFE signaling pathway interactions in remote limb ischemic perconditioning in combination with local ischemic postconditioning

Local ischemic postconditioning (IPost) and remote ischemic perconditioning (RIPer) are promising methods to decrease ischemia–reperfusion (I/R) injury. We tested whether the use of the two procedures in combination led to an improvement in cardioprotection through a higher activation of survival signaling pathways. Rats exposed to myocardial I/R were allocated to one of the following four groups: Control, no intervention at myocardial reperfusion; IPost, three cycles of 10-s coronary artery occlusion followed by 10-s reperfusion applied at the onset of myocardial reperfusion; RIPer, 10-min limb ischemia followed by 10-min reperfusion initiated 20 min after coronary artery occlusion; IPost+RIPer, IPost and RIPer in combination. Infarct size was significantly reduced in both IPost and RIPer (34.25 ± 3.36 and 24.69 ± 6.02%, respectively) groups compared to Control (54.93 ± 6.46%, both p < 0.05). IPost+RIPer (infarct size = 18.04 ± 4.86%) was significantly more cardioprotective than IPost alone (p < 0.05). RISK pathway (Akt, ERK1/2, and GSK-3β) activation was enhanced in IPost, RIPer, and IPost+RIPer groups compared to Control. IPost+RIPer did not enhance RISK pathway activation as compared to IPost alone, but instead increased phospho-STAT-3 levels, highlighting the crucial role of the SAFE pathway. In IPost+RIPer, a SAFE inhibitor (AG490) abolished cardioprotection and blocked both Akt and GSK-3β phosphorylations, whereas RISK inhibitors (wortmannin or U0126) abolished cardioprotection and blocked STAT-3 phosphorylation. In our experimental model, the combination of IPost and RIPer improved cardioprotection through the recruitment of the SAFE pathway. Our findings also indicate that cross talk exists between the RISK and SAFE pathways

Cold-spray of aluminium with and without ceramic combined with plasma electrolytic oxidation

Cold spraying is developing rapidly and has a wide spectrum of applications: protection against corrosion, preparation of high conductivity coatings, repair of damaged metal components, metal additive manufacturing, thermal management. Plasma electrolytic oxidation (PEO) is widely used to improve the corrosion and wear resistance of lightweight metals such as aluminum alloys by the formation of a ceramic oxide layer. When applied on aluminium alloys, the PEO oxide layer is mainly composed with γ-Al2O3 (in major proportion) and corundum, α-Al2O3. Increasing the proportion of corundum is an objective in order to improve the corrosion and wear resistance properties. For this purpose, a duplex treatment combining cold spray and PEO is investigate [1]. Particularly it consists in firstly cold spraying an aluminium coating containing a certain amount of α-Al2O3 which is then PEO processed. Our results show that, while the projected alpha alumina particles are observable on cross sections of untreated samples and for short PEO treatment times (lower than 20 min), this is no longer the case for long PEO treatment times (over than 35 min) (see the figure). In fact, under these conditions, it is no longer possible to observe the alpha alumina particles in the PEO layer. More interesting still, the proportion of alpha alumina in the layers resulting from a duplex treatment is greater than that obtained in the layer resulting from a cold spray treatment by scanning electron microscope

Approximate volume and integration for basic semi-algebraic sets

Given a basic compact semi-algebraic set \K\subset\R^n, we introduce a methodology that generates a sequence converging to the volume of \K. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on \K can be approximated as closely as desired, and so permits to approximate the integral on \K of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed