Constrained optimization penalty functions

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... 

The penalty function approach adds a tenn of tire type k r — ro) to the function to be optimized. The variable r is constrained to be near the target value ro, and the force constant k describes how important the constraint is compared with the unconstrained optimization. By making k arbitrary large, tire constraint may be fulfilled to any given... 

Another useful program (E04HAA) provides constrained optimization with bounds for each parameter using a sequential penalty function technique, which effectively operates around unconstrained minimization cycles. 

Some well-known stochastic methods for solving SOO problems are simulated annealing (SA), GAs,DE and particle swarm optimization (PSO). These were initially proposed and developed for optimization problems with bounds only [that is, unconstrained problems without Equations (4.7) and (4.8)]. Subsequently, they were extended to constrained problems by incorporating a strategy for handling constraints. One relatively simple and popular sdategy is the penalty function, which involves modifying the objective function (Equation 4.5) by the addition (in the case of minimization) of a term which depends on constraint violation. Eor example, see Equation (4.9),... 

Theoretical studies (Nocedal and Wright, 2000 Bazaraa et al, 2006) and many practical applications have demonstrated that the following function allows constrained optimization to be solved using smaller weights than the ones required by a quadratic penalty Junction ... 

The penalty function is often used as the function to be minimized not only in unconstrained optimization programs but also as the check function in programs that use other constrained optimization strategies. 

Penalty function methods are the most popular methods used in the GAs for constrained optimization problems. These methods transform a constrained problem into an unconstrained one by penalizing infeasible solutions. Penalty is imposed by adding to the objective function/(x) a positive quantity to reduce fitness values of such infeasible solutions ... 

The CE method is directed towards the solution of unconstrained optimization problems. Thus, in order to find X, it is necessary to pose the problem in Equation 4 (constrained optimization problem) in an alternative way. This is done by incorporating the constraint as a penalty term in the objective function, i.e. ... 

This is a simple method for solving an optimal control problem with inequality constraints. As the name suggests, the method penalizes the objective functional in proportion to the violation of the constraints, which are not enforced directly. A constrained problem is solved using successive applications of an optimization method with increasing penalties for constraint violations. This strategy gradually leads to the solution, which satisfies the inequahty constraints. 

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. 

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