Investigation on Self-Scaling of Mixed Interior-Exterior Method for Non-Linear Optimization

Received on: 22/05/2005 Accepted on: 29/06/2005 ABSTRACT In this paper, we have investigated self-scaling sequential unconstrained minimization techniques (SUMT). Our new modified version on CG-method and QNmethod shows that it is too effective when compared with other established algorithms to solve standard constrained optimization problems.


1-General Introduction to Nonlinear Constrained
The general constrained minimization problem minimize ( ) * Solve a constrained optimization problem by solving a sequence of unconstrained optimization problems, and in the limit, the solutions of the unconstrained problems will converge to the solution of the constrained problem. * Use an auxiliary function that incorporates the objective function together with "penalty" terms that measure violations of the constraints. INT [2]

3-Classical SUMT:
Two groups of classical methods: Barrier methods: impose a penalty for reaching the boundary of an inequality constraint. Penalty methods: impose a penalty for violating a constraint.

4-Exterior Point Methods (Penalty function):
is called a penalty function for eq.(1) satisfies

4-1 General Type of Penalty Function Methods
There are several types of penalty function method with the inequality constrained which has the following two terms: (1967)

loss function)
Investigation on self-scaling… 75 or with the equality constraint which has the following two forms 1- Hence ,our objective function may be defined by

5-Interior Point Methods (Barrier Function):
Definition: A barrier function for eq(1) is any function ( ) for all The idea in a barrier method is to dissuade points x from ever approaching the boundary of the feasible region. We consider solving

6-Mixed Exterior-Interior Methods:
we consider some method, which can be used to solve a general class (equality and inequality of problem) thus, the new problem can be converted into an unconstrained minimization problem by constructing a function of the form. (Fiacco & Mc Cormick, 1968a, 1968b Although both exterior and interior-point methods have many points of similarity, they represent two different points of view. In an exteriorpoint procedure, we start from an infeasible point and gradually approach feasibility. While doing so, we move away from the unconstrained optimum of the objective function. In an interior-point procedure, we start at a feasible point and gradually improve our objective function, while maintaining feasibility. The requirement that we begin at a feasible point and remain within the interior of the feasible inequality constrained region is the chief difficulty with interior-point methods. In many problems we have no easy way to determine a feasible starting point, and a separate initial computation may be needed. Also, if equality constraints are present, we do not have a feasible inequality constrained region in which to maneuver freely. Thus interior-point methods cannot handle equalities. We many readily handle equalities by using a "mixed" method in which we use interior-point penalty functions for inequality constraints only. Thus, if the first m constraints are inequalities and constraints (m+1) to n are equalities, our problem becomes: The function ) , is then minimized for a sequence of monotonically decreasing 0   . We can solve the constrained problem given in eq.  (6) where  is a scalar chosen in such that . We thus test ci(xk+1) to see that it is positive for all i. We find a feasible xk+1 and we can then proceed with the interpolation. Then a correction matrix to get updates the matrix where k  is a correction matrix which satisfies quasi-Newton condition namely ) (    Step3: Set  is a scalar. Step6: Update H by correction matrix defined in eq.(7)-(11). Step7: Check for convergence i.e. if eq.(12) is satisfied then stop.

7-New Self-Scaling Variable Metric Methods:
In order to eliminate the truncation and rounding errors, the new scalar parameter  is added to make the sequence and efficiency as problem dimension increase. The poor-scaling is an imbalance between the values of the function and change in x. The function values may be changed very little even though x is changing significantly. This difficulty can sometimes be removed by good scaling factor for the updating H and the performance of self-scaling methods is undoubtedly favorable in some cases especially when the number variables are large (Scales, 1985). An idea is multiplying part of BFGS by scaling factor  before the update takes place. The original motivation for self-scaling method arises from the analysis of quadratic objective function, and the main results also assume that exact line searches are performed.
Many authors have proposed a special scaling as follows: The above suggestion will be true if we prove that: Hence, the new formula (20)satisfied the QN-condition. Our last inquiry: Does formula (20) generates mutually conjugate gradient search direction?To answer this equation, follow this new theorem

3-1 New theorem:
The new formula (18) generates mutually conjugate gradient search direction Proof: Let  7-2 Outlines of the New Self-Scaling method: Step 1: Find an initial approximation x0 in the interior of the feasible region for the inequality constraints i.e. ( ) 0  x c i .
Step 2: Set 1 = i and 1 0 =  is the initial value of 0  .
Step 3: Set Step 4: Set , where  is a scalar.

8-Numerical Results:
Several standard non-linear constrained test functions were minimized to compare the new algorithms with standard algorithm see (Appendix), with 10  All the results are obtained using Pentium 4. All programs are written in FORTRAN language and for all cases the stopping criterion taken to be 5 10 − =  All the algorithms in this paper use the same ELS strategy which is the quadratic interpolation technique directly adapted from (Bunday, 1984).
The comparative performance for all of these algorithms are evaluated by considering NOF, NOI, NOG and NOC, where NOF is the number of function evaluations and NOI is the number of iterations and NOG is the number of gradient evaluations and NOC number of constrained evaluations.
In table (1) we have compared our new algorithm with the standard algorithm Table (1