Constrained optimization

Fig. 8 Ab initio ground and excited state potential curves for a dAdT Watson/Crick nucleoside pair along the N6(A)-H stretching coordinate. At each point along the curve, the ground-state geometry was optimized constraining the sugars to their positions in a B-DNA chain.

If the solute charge density p0 is completely contained inside a cavity surrounded by the dielectric medium, which mimics the solvent, both the functionals can be variationally optimized constraining the variation of the polarization density to be on the cavity surface. 

This paper presents the application of a model based predictive control strategy for the primary stage of the freeze drying process, which has not been tackled until now. A model predictive control framework is provided to minimize the sublimation time. The problem is directly addressed for the non linear distributed parameters system that describes the dynamic of the process. The mathematical model takes in account the main phenomena, including the heat and mass transfer in both the dried and frozen layers, and the moving sublimation front. The obtained results show the efficiency of the control software developed (MPC CB) under Matlab. The MPC( CB based on a modified levenberg-marquardt algorithm allows to control a continuous process in the open or closed loop and to find the optimal constrained control. 

Allanic, N., P. Salagnac, and P. Glouannec, 2009a. Optimal constrained control of an infrared-convective drying of a polymer aqueous solution. Chemical Engineering Research and Design, 87(7), 908-914. 

I.C. Trelea, G. Trystram, and F. Courtois, Optimal constrained non-linear control of batch processes Application to corn drying, J. Food Eng., 31 403-121, 1997. 

Over the next decade MNDO parameters were derived for lithium, beryllium, boron, fluorine, aluminum, silicon, phosphorus, sulfur, chlorine,zinc, germanium, bromine, iodine, tin, mercury, and lead. " In 1983 the first MOPAC program was written, containing both the MINDO/3 and MNDO mediods, which allowed various geometric operations, such as geometry optimization, constrained and unconstrained, with and without symmetry, transition state localization by use of a reaction co-... 

Retcher R (1981) Practical methods of optimization. Constrained optimization, Vol 2. Wiley, Chichester... 

Constrained optimization refers to optimizations in which one or more variables (usually some internal parameter such as a bond distance or angle) are kept fixed. The best way to deal with constraints is by elimination, i.e., simply remove the constrained variable from the optimization space. Internal constraints have typically been handled in quantum chemistry by using Z matrices if a Z matrix can be constructed which contains all the desired constraints as individual Z-matrix variables, then it is straightforward to carry out a constrained optimization by elunination. 

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... 

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

Fletcher R 1981 Practical Methods of Optimization vol. 2—Constrained Optimization (New York Wiley)... 

Baker J 1992 Geometry optimization in Cartesian coordinates constrained optimization J. Comput. Chem. 13 240... 

Baker J 1997 Constrained optimization in delocalized internal coordinates J. Comput. Chem. 18 1079... 

Muller K and Brown L D 1979 Location of saddle points and minimum energy paths by a constrained simplex optimization procedure Theor. Chim. Acta 53 75... 

Field M J 1991 Constrained optimization of ab initio and semiempirical Hartree-Fock wavefunctions using... 

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

Example Ifyou start an optim i/ation of a planar ammonia molecule and constrain it to that geometry, the calculation finds the transition state. 

If the transition state can be defined by symmetry, do a normal geometry optimization calculation with the symmetry constrained. 

Use a pseudo reaction coordinate with one parameter constrained followed by a quasi-Newton optimization. 

HyperChem does not use constrained optimization but it is possible to restrain molecular mechanics and quantum mechanics calculations by adding extra restraining forces. 

Powell, M. J. D. A Fast Algorithm for Nonlinearly Constrained Optimization Calculations, Lecture Notes in Mathematics 630 (1977). 

V L is equal to the constrained derivatives for the problem, which should be zero at the solution to the problem. Also, these stationarity conditions very neatly provide the necessaiy conditions for optimality of an equality-constrained problem. 

Each of the inequality constraints gj(z) multiphed by what is called a Kuhn-Tucker multiplier is added to form the Lagrange function. The necessaiy conditions for optimality, called the Karush-Kuhn-Tucker conditions for inequality-constrained optimization problems, are... 

The calculations begin with given values for the independent variables u and exit with the (constrained) derivatives of the objective function with respec t to them. Use the routine described above for the unconstrained problem where a succession of quadratic fits is used to move toward the optimal point for an unconstrained problem. This approach is a form or the generahzed reduced gradient (GRG) approach to optimizing, one of the better ways to cany out optimization numerically. 

Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a Rill plant simulation can be made, various alternatives can be put through the computer. Such an operation is called jlowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-HiU, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. 

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