Sharp Minima for Multiobjective Optimization in Banach Spaces

We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some of which are new even in the single objective setting.This research was supported by a Central Research Grant of The Hong Kong Polytechnic University (Grant No. G-T 507). Research of the first author was also supported by the National Natural Science Foundation of PR China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, China (Grant No. 2003A002M).     

Analytic Functions and Integrable Hierarchies–Characterization of Tau Functions

We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.     

Some Remarks on the Minty Vector Variational Inequality

In this paper, we establish some relations between a Minty vector variational inequality and a vector optimization problem under pseudoconvexity or pseudomonotonicity, respectively. Our results generalize those of Ref. 1.     

Persistence in discrete optimization under data uncertainty

An important question in discrete optimization under uncertainty is to understand the persistency of a decision variable, i.e., the probability that it is part of an optimal solution. For instance, in project management, when the task activity times are random, the challenge is to determine a set of critical activities that will potentially lie on the longest path. In the spanning tree and shortest path network problems, when the arc lengths are random, the challenge is to pre-process the network and determine a smaller set of arcs that will most probably be a part of the optimal solution under different realizations of the arc lengths. Building on a characterization of moment cones for single variate problems, and its associated semidefinite constraint representation, we develop a limited marginal moment model to compute the persistency of a decision variable. Under this model, we show that finding the persistency is tractable for zero-one optimization problems with a polynomial sized representation of the convex hull of the feasible region. Through extensive experiments, we show that the persistency computed under the limited marginal moment model is often close to the simulated persistency value under various distributions that satisfy the prescribed marginal moments and are generated independently.     

The 2-hamiltonian cubes of graphs

This paper deals with the problem of characterizing the pairs of vertices x,y in a connected graph G such that G³ - {x,y} is hamiltonian, where G³ is the cube of G. It is known that the cube G³ is 2-hamiltonian if G is 2-connected. In this paper, we first prove the stronger result that G³ - {x,y} is hamiltonian if either x or y is not a cut-vertex of G, and then proceed to characterize those cut-vertices x and y of G such that G³ -{x,y} is hamiltonian. As a simple consequence of these, we obtain Schaar's characterization of a connected graph G such that G³ is 2-hamiltonian.     

Application of the Π theorem of dimensional analysis to electron scattering in multi-component systems

Summary Monte Carlo simulation of electron scattering in solids is considered and an extremely simplified picture of the process is demonstrated by a semi-empirical application of the Π theorem of dimensional analysis. The case of electron beam lithography is presented as an example. A set of parameters fully characterising the process is derived, and their dependence is investigated on electron energy, substrate thickness and the material physical constants, for a great number of single- and multi-component substrates. A single generalised argument is shown to be able to effectively describe the properties of forward and backward scattering of any substrate, in the elastic regime. The physical laws governing the process are approximated through analytical functions of this argument. These functions are theoretically deduced from very simple models and empirically demonstrated by fitting the numerical data. The efficacy of the proposed method, applicable to all elemental and composite substrate materials, is shown in the 5 to 150 keV energy range, with substrate thickness varying from 0.002 to 8 μm. The leading role of diamond as substrate material, emerging from such an analysis, is finally emphasised.     

On a Conjecture on Directed Cycles in a Directed Bipartite Graph

Let D = (V ₁, V ₂; A) be a directed bipartite graph with |V ₁| = |V ₂| = n ≥ 2. Suppose that d _D (x) + d _D (y) ≥ 3n for all x ∈ V ₁ and y ∈ V ₂. Then, with one exception, D contains two vertex-disjoint directed cycles of lengths 2n ₁ and 2n ₂, respectively, for any positive integer partition n = n ₁ + n ₂. This proves a conjecture proposed in [9].     

Approximate Augmented Lagrangian Functions and Nonlinear Semidefinite Programs

In this paper, an approximate augmented Lagrangian function for nonlinear semidefinite programs is introduced. Some basic properties of the approximate augmented Lagrange function such as monotonicity and convexity are discussed. Necessary and sufficient conditions for approximate strong duality results are derived. Conditions for an approximate exact penalty representation in the framework of augmented Lagrangian are given. Under certain conditions, it is shown that any limit point of a sequence of stationary points of approximate augmented Lagrangian problems is a KKT point of the original semidefinite program and that a sequence of optimal solutions to augmented Lagrangian problems converges to a solution of the original semidefinite program.     

Cation and anion coordination chemistry of palladium(II) with polyazacycloalkanes. Thermodynamic and structural studies

The interaction of PdCl ₄ ^2– with the macrocyclic ligands of the series [3k]aneN _k has been studied both in solution and in the solid state. [18]aneN₆ and [21]aneN₇ form both mono- and binuclear Pd²⁺ complexes, whose stability constants have been determined in 0.5 mol dm^–3 NaCl at 298.15 K. [21]aneN₇ also forms, in solution, a trinuclear species in which an amino group deprotonates to bridge two Pd²⁺ ions, as observed in the solid state. The crystal structure of the complexes [Pd₂([18]aneN₆)Cl₂][ClO₄]₂ and [Pd₃([21]aneN₇)Cl₃][ClO₄]₂ · H₂O have been solved by single crystal X-ray analysis. C₁₂H₃₀N₆Cl₄O₈Pd₂: monoclinic, space group C2/m,a = 10.876(2),b = 18.117(2),c = 7.043(2) Å, = 113.78(2)°,V = 1270(12) ų,Z = 2,D _calc = 1.92 g cm^-3, = 16.94 cm^–1.R = 0.063,R _w = 0.059. C₁₄H₃₆N₇CI₅O₉Pd₃: orthorhombic, space groupPcab,a = 13.125(7),b = 13.213(3),c = 33.570(5) Å,V = 5822(3) ų,Z = 8,D _calc = 2.15 g cm^–3, = 21.20 cm^–1.R = 0.074,R _w = 0.061. In very acidic solutions the polyammonium cations (H _k [3k]aneN _k ) ^k+ interact with PdCl ₄ ^2– forming second sphere coordinated species. These reactions have been followed by a microcalorimetric technique in 2 mol dm^–3 HCl solutions. The slowness of the reactions of (H₁₀[30]aneN₁₀)¹⁰⁺ with PdCl ₄ ^2– has been interpreted in terms of inclusion of the anion into the receptor's cavity as shown by the crystal structure of [(PdCl₄)(H₁₀[30]aneN₁₀)][PdCl₄]₂Cl₄: triclinic, space group PT,a = 7.760(3),b = 11.448(4),c = 13.399(11) Å, = 96.31(8)°, = 104.50(6)°, = 92.30(3)°,Z = 1.R = 0.046 andR _w = 0.039.This paper is dedicated to the memory of the late Dr C. J. Pedersen.