Effect of the subdivision strategy on convergence and efficiency of some global optimization algorithms

We investigate subdivision strategies that can improve the convergence and efficiency of some branch and bound algorithms of global optimization. In particular, a general class of so called weakly exhaustive simplicial subdivision processes is introduced that subsumes all previously known radial exhaustive processes. This result provides the basis for constructing flexible subdivision strategies that can be adapted to take advantage of various problem conditions.     

Convex programs with an additional reverse convex constraint

A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject toxD,g(x)0, whereD is a closed convex subset ofR ⁿ andf,g are convex finite functionsR ⁿ . Under suitable stability hypotheses, it is shown that a feasible point is optimal if and only if 0=max{g(x):xD,f(x)f( )}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ _k ,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS _k . The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.     

Hexacyanometalate molecular chemistry: di-, tri-, tetra-, hexa- and heptanuclear heterobimetallic complexes; control of nuclearity and structural anisotropy

Following a bottom-up approach to nanomaterials, we present a rational synthetic route to high-spin and anisotropic molecules based on hexacyanometalate [M(CN)(6)](3-) cores. Part 1 of this series was devoted to isotropic heptanuclear clusters; herein, we discuss the nuclearity and the structural anisotropy of nickel(II) derivatives. By changing either the stoichiometry, the nature of the terminal ligand, or the counterion, it is possible to tune the nuclearity of the polynuclear compounds and therefore to control the structural anisotropy. We present the synthesis and the characterisation by mass spectrometry, X-ray crystallography and magnetic susceptibility of bi-, tri-, tetra-, hexa- and heptanuclear species [M(CN)(n)(CN-M'L)(6-n)](m+) (with n=0-5; M=Cr(III), Co(III), M'=Ni(II); L=pentadentate ligand). Thus, with M=Cr(III), d(3), S=3/2, a dinuclear complex [Cr(III)(CN)(5)(CN-NiL(n))](9+), (L(n)=polydentate ligand) was built and characterised, showing a spin ground state, S(G)=5/2, with a ferromagnetic interaction J(Cr,Cu)=+18.5 cm(-1). With M=Co(III) (d(6), S=0) were built di-, tri-, tetra-, hexa and hepanuclear CoNi species: CoNi, CoNi(2), CoNi(3), CoNi(5) and CoNi(6). By a first approximation, they behave as one, two, three, five and six isolated nickel(II) complexes, respectively, but more accurate studies allow us to evaluate the weak antiferromagnetic coupling constant between two next-nearest neighbours M'-Co-M'.     

Expanded Pyridiniums: Bis‐cyclization of Branched Pyridiniums into Their Fused Polycyclic and Positively Charged Derivatives—Assessing the Impact of Pericondensation on Structural,Electrochemical, Electronic,and Photophysical Features

This study evaluates the impact of the extension of the π‐conjugated system of pyridiniums on their various properties. The molecular scaffold of aryl‐substituted expanded pyridiniums (referred to as branched species) can be photochemically bis‐cyclized into the corresponding fused polycyclic derivatives (referred to as pericondensed species). The representative 1,2,4,6‐tetraphenylpyridinium ( 1^H ) and 1,2,3,5,6‐pentaphenyl‐4‐(p‐tolyl)pyridinium ( 2^Me ) tetra‐ and hexa‐branched pyridiniums are herein compared with their corresponding pericondensed derivatives, the fully fused 9‐phenylbenzo[1,2]quinolizino[3,4,5,6‐def]phenanthridinium ( 1^H f ) and the hitherto unknown hemifused 9‐methyl‐1,2,3‐triphenylbenzo[h]phenanthro[9,10,1‐def]isoquinolinium ( 2^Me f ). Combined solid‐state X‐ray crystallography and solution NMR experiments showed that stacking interactions are barely efficient when the pericondensed pyridiniums are not appropriately substituted. The electrochemical study revealed that the first reduction process of all the expanded pyridiniums occurs at around ?1 V vs. SCE, which indicates that the lowest unoccupied molecular orbital (LUMO) remains essentially localized on the pyridinium core regardless of pericondensation. In contrast, the electronic and photophysical properties are significantly affected on going from branched to pericondensed pyridiniums. Typically, the number of absorption bands increases with extended activity towards the visible region (down to ca. 450 nm in MeCN), whereas emission quantum yields are increased by three orders of magnitude (at ca. 0.25 on average). A relationship is established between the observed differential impact of the pericondensation and the importance of the localized LUMO on the properties considered: predominant for the first reduction process compared with secondary for the optical and photophysical properties.     

On Global Optimality Conditions and Cutting Plane Algorithms

We discuss global optimality conditions and cutting plane algorithms for DC optimization. The discussion is motivated by certain incorrect results that have appeared recently in the literature on these topics. Incidentally, we investigate the relation of the Tikhonov reciprocity theorem to the optimality conditions for general nonconvex global optimization problems and show how the outer-approximation scheme developed earlier for DC programming can be used to solve a wider class of problems.     

On duality bound methods for nonconvex global optimization

A counter-example is given to several recently published results on duality bound methods for nonconvex global optimization.     

On the global minimization of a convex function under general nonconvex constraints

The problem of globally minimizing a convex function subject to general continuous inequality constraints is investigated. A convergent outer approximation method is proposed which systematically exploits the convexity of the objective function in order to transcend local optimality. Also the question of finding a good starting point by using a local approach is discussed.     

Linear programs with an additional rank two reverse convex constraint

An efficient algorithm is developed for solving linear programs with an additional reverse convex constraint having a special structure. Computational results are presented and discussed.     

Monotonic optimization for sensor cover energy problem

We study the sensor cover energy problem (SCEP) in wireless communication—a difficult nonconvex problem with nonconvex constraints. A local approach based on DC programming called DCA was proposed by Astorino and Miglionico (Optim Lett 10(2):355–368, 2016) for solving this problem. In the present paper, we propose a global approach to (SCEP) based on the theory of monotonic optimization. By using an appropriate reformulation of (SCEP) we propose an algorithm for finding quickly a local optimal solution along with an efficient algorithm for computing a global optimal solution. Computational experiments are reported which demonstrate the practicability of the approach.