Global optimization of general constrained grey-box models: new method and its application to constrained PDEs for pressure swing adsorption

This paper introduces a novel methodology for the global optimization of general constrained grey-box problems. A grey-box problem may contain a combination of black-box constraints and constraints with a known functional form. The novel features of this work include (i) the selection of initial samples through a subset selection optimization problem from a large number of faster low-fidelity model samples (when a low-fidelity model is available), (ii) the exploration of a diverse set of interpolating and non-interpolating functional forms for representing the objective function and each of the constraints, (iii) the global optimization of the parameter estimation of surrogate functions and the global optimization of the constrained grey-box formulation, and (iv) the updating of variable bounds based on a clustering technique. The performance of the algorithm is presented for a set of case studies representing an expensive non-linear algebraic partial differential equation simulation of a pressure swing adsorption system for \(\hbox {CO}_{2}\). We address three significant sources of variability and their effects on the consistency and reliability of the algorithm: (i) the initial sampling variability, (ii) the type of surrogate function, and (iii) global versus local optimization of the surrogate function parameter estimation and overall surrogate constrained grey-box problem. It is shown that globally optimizing the parameters in the parameter estimation model, and globally optimizing the constrained grey-box formulation has a significant impact on the performance. The effect of sampling variability is mitigated by a two-stage sampling approach which exploits information from reduced-order models. Finally, the proposed global optimization approach is compared to existing constrained derivative-free optimization algorithms.     

Convex envelopes for edge-concave functions

Deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions. In this paper we describe the structure of the polyhedral convex envelopes of edge-concave functions over polyhedral domains using geometric arguments. An algorithm for computing the facets of the convex envelope over hyperrectangles in ³ is described. Sufficient conditions are described under which the convex envelope of a sum of edge-concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions.Author to whom all correspondence should be addressed.     

A vibrational and DFT study of M(diimine)(dithiolate) complexes and their complexation route

A complete vibrational spectra analysis of the Pd(phen)(bdt), the free ligands, where phen=1,10-phenanthroline and bdt=1,2-benzenedithiolate and the starting material of its synthesis, Pd(phen)Cl(2), is performed in this paper. The molecular geometry, binding and spectroscopic properties for the aforementioned compounds are studied in detail by FT-IR, Raman and DFT methods using B3LYP functional together with basis sets of valence triple-zeta quality. Further, changes in FT-IR and Raman spectra during complexation are monitored revealing the electron delocalization over ligands. They are also consistent with pi-back donation theory.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Convex relaxation for solving posynomial programs

Convex underestimation techniques for nonlinear functions are an essential part of global optimization. These techniques usually involve the addition of new variables and constraints. In the case of posynomial functions \({x_1^{\alpha _1 } x_2^{\alpha _2 }\ldots x_n^{\alpha _n } ,}\) logarithmic transformations (Maranas and Floudas, Comput. Chem. Eng. 21:351–370, 1997) are typically used. This study develops an effective method for finding a tight relaxation of a posynomial function by introducing variables y _j and positive parameters β _j , for all α _j > 0, such that \({y_j =x_j^{-\beta _j }}\) . By specifying β _j carefully, we can find a tighter underestimation than the current methods.     

Conservation laws and hierarchies of potential symmetries for certain diffusion equations

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989; G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806-811]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations, and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.     

On the invariants of two dimensional linear parabolic equations

We consider the most general two dimensional linear parabolic equations. Motivated by the recent work of Ibragimov et al. [1], [2], [3] we construct differential invariants, semi-invariants and invariant equations. These results are achieved with the employment of the equivalence group admitted by this class of parabolic equations. We derive those variable coefficient equations of this class of linear parabolic equations that can be mapped into constant coefficient equations. Further applications are presented.     

Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP Problems

In Akrotirianakis and Floudas (2004) we presented the theoretical foundations of a new class of convex underestimators for C ² nonconvex functions. In this paper, we present computational experience with those underestimators incorporated within a Branch-and-Bound algorithm for box-conatrained problems. The algorithm can be used to solve global optimization problems that involve C ² functions. We discuss several ways of incorporating the convex underestimators within a Branch-and-Bound framework. The resulting Branch-and-Bound algorithm is then used to solve a number of difficult box-constrained global optimization problems. A hybrid algorithm is also introduced, which incorporates a stochastic algorithm, the Random-Linkage method, for the solution of the nonconvex underestimating subproblems, arising within a Branch-and-Bound framework. The resulting algorithm also solves efficiently the same set of test problems.     

On the scattering of spherical electromagnetic waves by a layered sphere

A spherical electromagnetic wave is scattered by a layered sphere.The exact expressions of the scattered and interior fields areobtained by solving the corresponding boundary-value problem,by means of a combination of Sommerfeld's and T-matrix methods.A recursive algorithm with respect to the number of layers isextracted for the computation of the fields in every layer.The far-field pattern and the scattering cross-sections aredetermined in terms of the physical and geometrical characteristicsof the scatterer. As the point-source tends to infinity, theknown results for plane wave incidence are recovered. Numericalresults are presented for several cases and various parametersof the layered spherical scatterer.     

Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes

Convex underestimators of nonconvex functions, frequently used in deterministic global optimization algorithms, strongly influence their rate of convergence and computational efficiency. A good convex underestimator is as tight as possible and introduces a minimal number of new variables and constraints. Multilinear monomials over a coordinate aligned hyper-rectangular domain are known to have polyhedral convex envelopes which may be represented by a finite number of facet inducing inequalities. This paper describes explicit expressions defining the facets of the convex and concave envelopes of trilinear monomials over a box domain with bounds of opposite signs for at least one variable. It is shown that the previously used approximations based on the recursive use of the bilinear construction rarely yield the convex envelope itself.