On the symmetric travelling salesman problem I: Inequalities

We investigate several classes of inequalities for the symmetric travelling salesman problem with respect to their facet-defining properties for the associated polytope. A new class of inequalities called comb inequalities is derived and their number shown to grow much faster with the number of cities than the exponentially growing number of subtour-elimination constraints. The dimension of the travelling salesman polytope is calculated and several inequalities are shown to define facets of the polytope. In part II (On the travelling salesman problem II: Lifting theorems and facets) we prove that all subtour-elimination and all comb inequalities define facets of the symmetric travelling salesman polytope.     

Convexifactors,generalized convexity and vector optimization

In this article we study a recently introduced notion of non-smooth analysis, namely convexifactors. We study some properties of the convexifactors and introduce two new chain rules. A new notion of non-smooth pseudoconvex function is introduced and its properties are studied in terms of convexifactors. We also present some optimality conditions for vector minimization in terms of convexifactors.     

Reconstruction of n-dimensional convex bodies from surface tensors

In this paper, we derive uniqueness and stability results for surface tensors. Further, we develop two algorithms that reconstruct shape of n-dimensional convex bodies. One algorithm requires knowledge of a finite number of surface tensors, whereas the other algorithm is based on noisy measurements of a finite number of harmonic intrinsic volumes. The derived stability results ensure consistency of the two algorithms. Examples that illustrate the feasibility of the algorithms are presented.     

L~p空间中的Fejér和Hermite-Hadamard型不等式

给出了二阶导数属于Lp空间时Fejér和Hermite-Hadamard型不等式的推广,得到两个新结果.     

Effect of substitution positions of alkyl side chains in phenanthrodithiophene–isoindigo copolymers: The enhancement of crystallinity and control of molecular orders

The synthesis, characterization, and solar cell application of newly developed two semiconducting polymers containing phenanthro[1,2‐b:8,7‐b′]dithiophene (PDT) and an isoindigo (IID) unit are described. In addition, a relationship between substitution position of side chains and molecular weights of the polymers and their solar cell performance are also discussed. Because of the installation of alkyl side chains onto sterically less hindered positions, PDT‐IID copolymers 12OD‐2 and 8OD‐2 have stronger intermolecular interaction than that of the previously reported copolymer 12OD . In low‐M_n polymers 12OD‐2 and 8OD‐2 formed high‐crystalline thin film with higher face‐on ratio than that of 12OD , but their unsuitable large‐scale phase separation suppressed their efficient photocurrent generation, leading to poor PCE of 2–3%. However, the surface morphology of 12OD‐2 and 8OD‐2 blended films are drastically improved by increasing M_n, which leads to the enhancement of J_sc and higher PCE of up to 4.3%. However, high‐M_n polymers 12OD‐2 and 8OD‐2 formed high‐crystalline film with about 10–15% lower face‐on ratio than that of high‐M_n polymer 12OD , leading to poor hole transporting ability, and thus lower J_sc and PCE. From this result, too much strong intermolecular interaction promotes the formation of unsuitable edge‐on orientation in blended films. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2018 , 56, 1757–1767     

On safe tractable approximations of chance constraints

A natural way to handle optimization problem with data affected by stochastic uncertainty is to pass to a chance constrained version of the problem, where candidate solutions should satisfy the randomly perturbed constraints with probability at least 1 − ?. While being attractive from modeling viewpoint, chance constrained problems “as they are” are, in general, computationally intractable. In this survey paper, we overview several simulation-based and simulation-free computationally tractable approximations of chance constrained convex programs, primarily, those of chance constrained linear, conic quadratic and semidefinite programming.     

Highly oriented and ordered microstructures in block copolymer films

Block copolymer (BCP) films with long-range lateral ordering and orientation are crucial for many applications. Here, we report a simple, versatile strategy based on a solution casting procedure, to produce millimeter thick film of BCPs with highly oriented nanostructures. Transmission electron microscope (TEM), small angle X-ray scattering (SAXS), and Hansen solubility parameters were used to study the morphology and interactions of the system. A variety of BCP-solvent pairs were investigated. Factors including set-up geometry, BCP characteristics, solvent evaporation, surface tension, and interactions, such as solvent-BCP, solvent-substrate, and BCP-substrate were examined. A mechanism is proposed to describe the observed long-range lateral ordering and orientation in films up to 1 mm in thickness. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 1369–1375     

Hjelmslev quadrilateral central configurations

A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices. Using mutual distances as coordinates, we show that any four-body central configuration forming a Hjelmslev quadrilateral must be a right kite configuration.     

Invertibility of a linear combination of two matrices and partial orderings

In this paper, we present some results relating different matrix partial orderings and the invertibility of a linear combination of EP matrices.     

Pathwise superhedging for time-dependent barrier options on càdlàg paths—Finite or infinite tradeable European,One-Touch,lookback or forward starting options

We establish pathwise duality using simple predictable trading strategies for the robust hedging problem associated with a barrier option whose payoff depends on the terminal level and the infimum of a càdlàg strictly positive stock price process, given tradeable European options at all strikes at a single maturity. The result allows for a significant dimension reduction in the computation of the superhedging cost, via an alternate lower-dimensional formulation of the primal problem as a convex optimization problem, which is qualitatively similar to the duality which was formally sketched using linear programming arguments in Duembgen and Rogers [10] for the case where we only consider continuous sample paths. The proof exploits a simplification of a classical result by Rogers (1993) which characterizes the attainable joint laws for the supremum and the drawdown of a uniformly integrable martingale (not necessarily continuous), combined with classical convex duality results from Rockefellar (1974) using paired spaces with compatible locally convex topologies and the Hahn–Banach theorem. We later adapt this result to include additional tradeable One-Touch options using the Kertz and Rösler (1990) condition. We also compute the superhedging cost when in the more realistic situation where there is only finite tradeable European options; for this case we obtain the full duality in the sense of quantile hedging as in Soner (2015), where the superhedge works with probability $1?\epsilon$ where $\epsilon$ can be arbitrarily small), and we obtain an upper bound for the true pathwise superhedging cost. In Section 5, we extend our analysis to include time-dependent barrier options using martingale coupling arguments, where we now have tradeable European options at both maturities at all strikes and tradeable forward starting options at all strikes. This set up is designed to approximate the more realistic situation where we have a finite number of tradeable Europeans at both maturities plus a finite number of tradeable forward starting options.¹