Penalty function method

Nonlinear Systems and Optimization for Chemical Engfoeer Solving Numerical Problems, 

First Edition. Guido Buzzi-Ferraris and Flavio ManentL 

Many variants have been proposed for Wh and Wp. The pros and cons of this 

1) A dedicated program is not required to solve constrained optimization problems. A good unconstrained optimization program will suffice. 

2) The function is not modified within the feasible region, although we are very close to one or more constraints. 

---

Fiacco, A. V. Sensitivity Analysis of Nonlinear Programming Using Penalty Function Methods. Math Program 10 287-311 (1976). 

The classical penalty-function methods have now finally become part of history, the early promise of the augmented Lagrangian approach has faded, and there has been a coalescence of the approach used in the projection methods with the exact penalty-function approach. 

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 form C (x) = 0. In the penalty function method, additional terms of... 

Classical Unconstrained Results 2546 4.7.1. Penalty Function Methods 2560... 

Barrier function methods are very similar to penalty function methods except that they start at an interior point of the feasible region and set a barrier against leaving the feasible region. In this case, the feasible region must have an interior, so this method is generally restricted to inequality constraints. Consider the nonlinear problem with inequality constraints. 

To illustrate the penalty function method, we consider the following optimal control problem. 

The penalty function method addresses this difficulty by prescribing... 

Following is the computational algorithm to solve this problem using the penalty function method ... 

Figure 7.7 shows the convergence of the penalty function method. At each outer iteration r, the plotted objective functional value is the optimal with corresponding constraint violation quantified as q. It was 17.2 at the very beginning (corresponding to the initial controls) and dropped finally to 5.9 x 10 at the convergence, which was attained in 11 outer iterations. 

Figure 7.8 shows the optimal states obtained at the convergence of the penalty function method. The states correspond to the optimal controls, which are shown in Figure 7.9. The final time was reduced from 60 to 42.8 min.

Applying the penalty function method developed in Section 7.2.1 (p. 201),... 

To handle the inequalities, the penalty function method is slightly modified as follows. At any time, the multipliers corresponding to the inequalities are prescribed as... 

Develop a computational algorithm based on the penalty function method to solve Example 8.4 (p. 254) subject to the following constraints ... 

Penalty function methods, barrier function methods, and hybrid penalty-barrier function methods are discussed in Chapter 12 they are implemented and adopted... 

---