Optimization Methods and Programs

Optimization is used to find the values of decision variables, which minimize or maximize the value of a given objective function and also satisfy the specified constraints. Mathematically, a SOO problem can be defined as shown in Equations (4.5)-(4.8). 

x is the vector of decision variables, x and x are, respectively, vectors of lower and upper bounds on decision variables, and g and h are sets of inequality and equality constraints, respectively. An optimization problem will have different characteristics depending on  

Some problems may have multiple optima. Although a local optimum may be acceptable in some situations and industrial practice, only the global optimum is the correct solution in certain applications (for example, equilibrium calculations by free energy minimization). Hence, an optimization method should be selected carefully for solving the optimization problem based on its characteristics and need to find the global optimum. 

Methods for solving optimization problems can be classified into two main types namely, deterministic and stochastic. Deterministic methods often require derivatives of objective 

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), 

---