Global optimization problem

The formulation without operating variables is qualitatively easier to solve as one level of optimization is eliminated. More specifically, finding v for the optimization problem defined by Eq. (2) is a nondifferentiable global optimization problem that is extremely difficult to solve rigorously in the general case. It is therefore important to consider the pros and cons of using operating variables carefully. 

In putting everything together, we have to face these algorithmic problems (1) determine the optimal parameters 0 by maximizing the probability p(O,j 0) -this is a nonlinear global optimization problem-, (2) determine the optimal sequence of hidden metastable states j = jt) for given optimal parameters, and (3) determine the number of important metastable states which we, up to now, simply assumed to be identical with the number of hidden states. 

NMR structure determination can be viewed as a global optimization problem for a target function in the conformational space to determine three-dimensional coordinates. The target function is normally a hybrid potential between the NMR-derived structural information and empirical steric conditions. The conformational space means the total degrees of freedom for the atomic positions, typically more than a thousand. The conformational space search for the highly complicated system has been a challenging computational target. 

The methods described in Section 12.2 can only locate the nearest minimum, which is normally a local minimum, when starting from a given set of variables. In some cases, the interest is in the lowest of all such minima, the global minimum in other cases it is important to sample a large (preferably representative) set of local minima. Considering that the number of minima typically grows exponentially with the number of variables, the global optimization problem is an extremely difficult task for a multidimensional function. It is often referred to as the multiple minima or combinatorial explosion problem in the Uterature. 

Parsopoulos, K.E. and Vrahatis, M.N. (2002) Recent Approaches to Global Optimization Problems Throu Particle Swarm Optimization, Kluwer Academic Publishers, Netherlands. 

In this chapter, we consider a multiobjective formulation for the global optimization problems arising from integrated design and control, and we present two alternative methods... 

In order to apply the aBB algorithm to (17), we must reformulate it as a global optimization problem. This is accomplished by introducing a slack variable s and minimizing its value over an augmented variable set (x,s) subject to a set of relaxed constraints ... 

The corresponding global optimization problem is obtained by introducing a slack variable s and minimizing s subject to the constraints... 

The problem of finding all local minimum energy conformations can also be formulated as a single global optimization problem, which can be deterministically solved using the aBB algorithm [23]. This method stems from the idea that all stationary points (i.e., minima, maxima, and transition states) of the energy hypersurface satisfy the constraint V (0) = 0. This can be written as ... 

Both experimental and theoretical methods exist for the prediction of protein structures. In both cases, additional restraints on the molecular system can be derived and used to formulate a nonconvex optimization problem. Here, the traditional unconstrained problem was recast as a constrained global optimization problem and was applied to protein structure prediction using NMR data. Both the formulation and solution approach of this method differ from traditional techniques, which generally rely on the optimization of penalty-type target function using SA/MD protocols. 

According to the aBB algorithm, the original problem (88) is first reexpressed as a global optimization problem by introducing a slack variable ... 

We formulated and solved the problem of obtaining the best fit of two protein structures as a global optimization problem. The best rotation and translation matrices that minimize the fitting distance for the two protein structures are... 

In SAR research, this directed stochastic search makes genetic algorithms a very robust and universal tool for global optimization problem of conformation search, which can be expressed in a reasonably small set of parameters. 

K. E. Parsopoulos and M. N. Vrahatis, "Recent approaches to global optimization problems through Particle Swarm Optimization, Natural Computing, vol. l,2002,pp. 235-306. 

The local unconstrained optimization problem in the Euclidean space SR" can be stated as in equation (1) for x e I> C SR" where V is region in the neighborhood of the starting point, xq. The global optimization problem requires X> to be the entire feasible space. 

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