Multi-minima problem

An approach to overcome the multi minima problem of proteins is simulated annealing (SA) run. Besides global molecular properties such as structural and thermal motions, functional properties of fast biological reactions can also be studied by MD. 

Local energy minimum principle. The multi-minimum problem was overcome by a energy functional of Kim, Mauri, and Galli, who made the generalization to allow the number of the localized orbitals to exceed the number of occupied states. The functional is defined for an V-electron system described by a nonself-consistent Hamiltonian h and a chemical potential p. In terms of a set of M (possibly linearly dependent) orbitals i =, ...,M, where M > N/I, it is... 

The chemical potential is chosen such that enp. < p < fw/2+i, and the energy functional (43) has a local minimum at the ground-state energy of equation (39) with respect to the variation of The functional kmg[ V ] uses more localized orbitals than occupied states and has the chemical potential p to allow the transfer of electrons. It has been shown to be fiee of the multi-minimum problem. If Af = N/2, and p is set to make the matrix (H — pS) negative definite, then kmg[ (/ ] becomes the original functional. 

For one-dimensional problems the direct TDMA algorithm is an efficient solver. In these cases the solver is computationally inexpensive and has the advantage that it requires a minimum amount of storage. For direct solvers, the number of operations to be performed to obtain the solution of a system of equations can be determined beforehand. However, for multi-dimension problems the TDMA algorithm is applied line by line on a selected plane... 

In 2003, Wu and Bogy [45] introduced a multi-grid scheme to solve the slider air bearing problem. In their approach, two types of meshes, with unstructured triangles, were used. They obtained the solutions with the minimum flying height down to 8 nm. 

In this paper, we extend the work of [10] by simultaneously considering minimization of the total utility consumption, maximization of operational flexibility to source-stream temperatures, and even minimum number of matches as multiple design objectives. The flexible HEN synthesis problem is thus formulated as the one of multi-objective mixed-integer linear programming (MO-MILP). This formulation also assumes that the feasible region in the space of uncertain input parameters is convex, so that the optimal solution can thus be explored on the basis of the vertices... 

Example 1. The 2-hot/2-cold streams example studied by [10], with problem data presented in Table 1, is illustrated. With these parameters, the multi-objective MILP formulation has 408 linear equality constraints, 760 linear inequality constraints, 12 binary variables, and 545 positive continuous variables. Notably, the restriction of MEUmax = 6 in Eq. (7) will be removed should the minimum number of matches be simultaneously taken into account as one of the design objectives. 

Table 5.9. Summary of Multi-CUT Minimum Time Optimisation Problem.

It is desirable to design a process with minimum cost to realize zero effluents. This is a synthesis problem of multi-component MEN with a rich stream, one or two lean streams. 

Shih, H. S., and Lee, E. S. (1999), Fuzzy Multi-Level Minimum Cost Flow Problems, Fuzzy Sets and Systems, Vol. 107, pp. 159-176. 

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