Minmax robustness for multi-objective optimization problems

In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances.     

Phosphido‐diphosphine pincer group 3 complexes as efficient initiators for lactide polymerization

New Group 3 metal complexes of the type [LM(CH₂SiMe₃)₂(THF)_n] supported by tridentate phosphido‐diphosphine ligands [(o‐C₆H₄PR₂) ₂ PH; L1‐H : R = iPr; L2‐H : R = Ph] have been synthesized by reaction of L1‐H and L2‐H with [M(CH₂SiMe₃)₃(THF)₂)] (M = Y and Sc). All the new complexes [(o‐C₆H₄PR₂) ₂ PM(CH₂SiMe₃)₂(THF)_n] [M = Y, R = iPr (1), R = Ph (2); M = Sc, R = iPr (3), R = Ph (4)] were studied as initiators for the ring opening polymerization of lactide. The yttrium complexes ( 1 and 2 ) exhibited high activity and good polymerization control, shown by the linear fits in the plot of number‐averaged molecular weight (M_n) versus the percentage conversion and versus the monomer/initiator ratio and by the low polydispersity index values. Interestingly, very good molar‐mass control was observed even when L ‐Lactide was polymerized in the absence of solvent at 130 °C. A good molar‐mass control but lower activities were observed in the polymerization reaction of lactide promoted by the analogous scandium complexes 3 and 4 . © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 1374–1382, 2010     

Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations π such that the elements in the closed lower Bruhat interval [id,π] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id,π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner.Our characterisation connects the Poincaré polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups An and Bn. The expressions involve q-Stirling numbers of the second kind, and for the group An putting q=1 yields the poly-Bernoulli numbers defined by Kaneko.     

Bounds and Numerical Results for Homogenized Degenerated p-Poisson Equations

In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.