Minimization of functions

Armijo, L. Minimization of Functions Having Lipschitz Continuous First Partial Derivatives. Pac J Math 16 1-3 (1966). 

The rates of chemical reactions are in linear dependence on thermody namic rushes of the reaction groups of several reactants. When the pertinent kinetic schemes are reducible to a set of the intermediate monomolecular reactions (see Section 1.3.2), the minimization of functional ( Aor ) can be used to find the stationary state of these systems that are far from equilibrium. Let us demonstrate this. 

Abadie, J., and Carpentier, J. (1969), Generalization of the Wolfe Reduced Gradient Method to the Case of Nonlinear Constraints, in Optimization, R. Fletcher, Ed., Academic Press, New York. Armijo, L. (1966), Minimization of Functions having Lipschitz Continuous First-Partial Derivatives, Pacific J. Mathematics, Vol. 16, No. 1, pp. 1-3. 

Armijo, L. (1966) Minimization of functions having Lipschitz continuous first-partial derivatives. Padjic Journal of Mathematics, 16 (1), 1-3. 

The values of J. and were determined through all the experimental data or Jobs obtained from variation of the mixture volume-ratio of dioxane and 1 0 solvents (Xj = dioxane/(dioxane+DMSO)) after a minimization of function < ) as follows(8) ... 

We emphasize that the only fundamental physical restriction of the Onsager method is due to the second virial tq)proximation, i.e, the condition d> 1. The use of the variation procedure is simply a way of simplifying the calculation. The integral equation which arises in precise minimization of functional (1.1) can be solved numerically with a high degree of precision this was done in [11,12]. As a result, the following was obtained ... 

We try to estimate the function H(u), noted H, by minimization of the quadratic residual error... 

The general constrained optimization problem can be considered as minimizing a function of n variables F(x), subject to a series of m constraints of the fomi C.(x) = 0. In the penalty fiinction method, additional temis of the fomi. (x), a.> 0, are fomially added to the original fiinction, thus... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

C.D. Maranas, IP. Androulakis and C.A. Floudas, A deterministic global optimization approach for the protein folding problem, pp. 133-150 in Global minimization of nonconvex energy functions molecular conformation and protein folding (P. M. Pardalos et al., eds.), Amer. Math. Soc., Providence, RI, 1996. 

Ihmce for IM applied to Newtonian dynamics 7=1 and the if" term in X, is absent. Following minimization of the IM dynamics function to obtain X"+ yn+i jg obtained from the second equation of system (10). 

G. Ramachandran and T. Schlick. Beyond optimization Simulating the dynamics of supercoiled DNA by a macroscopic model. In P. M. Pardalos, D. Shal-loway, and G. Xue, editors. Global Minimization of Nonconvex Energy Functions Molecular Conformation and Protein Folding, volume 23 of DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, pages 215-231, Providence, Rhode Island, 1996. American Mathematical Society. 

J Chem. Phys., 52, 431 (1970)] is a relatively inexpensive one and can be used for calculations on quite large molecules. It is minimal in the sense of having the smallest number of functions per atom required to describe the occupied atomic orbitals of that atom. This is not exactly true, since one usually considers Is, 2s, and 2p, i.e., five functions, to construct a minimal basis set for Li and Be, for example, even though the 2p orbital is not occupied in these atoms. The 2sp (2s and 2p), 3sp, 4sp, 3d,. .., etc. orbitals are always lumped together as a shell , however. The minimal basis set thus consists of 1 function for H and He, 5 functions for Li to Ne, 9 functions for Na to Ar, 13 functions for Kand Ca, 18 functions for Sc to Kr,. .., etc. Because the minimal basis set is so small, it generally can not lead to quantitatively accurate results. It does, however, contain the essentials of chemical bonding and many useful qualitative results can be obtained. 

The equilibrium problem for the plate can be formulated as variational, namely, it corresponds to the minimum of the functional H over the set of admissible displacements. To minimize the functional H over the set we can consider the variational inequality... 

Actually, by considering the problem (4.140)-(4.142) we have in mind the minimization of the functional... 

This section is concerned with an extreme crack shape problem for a shallow shell (see Khludnev, 1997a). The shell is assumed to have a vertical crack the shape of which may change. From all admissible crack shapes with fixed tips we have to find an extreme one. This means that the shell displacements should be as close to the given functions as possible. To be more precise, we consider a functional defined on the set describing crack shapes, which, in particular, depends on the solution of the equilibrium problem for the shell. The purpose is to minimize this functional. We assume that the... 

Pure dry reactants are needed to prevent catalyst deactivation effective inhibitor systems are also desirable as weU as high reaction rates, since many of the specialty monomers are less stable than the lower alkyl acrylates. The alcohol—ester azeotrope (8) should be removed rapidly from the reaction mixture and an efficient column used to minimize reactant loss to the distillate. After the reaction is completed, the catalyst may be removed and the mixture distilled to obtain the ester. The method is particularly useful for the preparation of functional monomers which caimot be prepared by direct esterification. 

Deamidation of soy and other seed meal proteins by hydrolysis of the amide bond, and minimization of the hydrolysis of peptide bonds, improves functional properties of these products. For example, treatment of soy protein with dilute (0.05 A/) HCl, with or without a cation-exchange resin (Dowex 50) as a catalyst (133), with anions such as bicarbonate, phosphate, or chloride at pH 8.0 (134), or with peptide glutaminase at pH 7.0 (135), improved solubiHty, whipabiHty, water binding, and emulsifying properties. 

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