Modification of Non-Linear Constrained Optimization. Ban A. Metras Dep. of Statistics College of Computer Sciences and Mathematics

Size: px

Start display at page:

Download "Modification of Non-Linear Constrained Optimization. Ban A. Metras Dep. of Statistics College of Computer Sciences and Mathematics"

Collin Simon
5 years ago
Views:

1 Raf. J. Of Comp. & Math s., Vol., No., 004 Modification of Non-Linear Constrained Optimization Ban A. Metras Dep. of Statistics College of Computer Sciences and Mathematics Received: 3/5/00 Accepted: /0/00 الملخص في هذا البحث تم استحداث معلم ابتداي ي جديد يربط بين الطريقتين الخارجية المناسبة لقيود غير المساواة) والطريقة الداخلية المناسبة لقيود المساواة وغير المساواة في مجال ا لا مثلية غير الخطية المشروطة. و ا ن التقنية الجديدة تمت برمجتها لحل بعض مساي ل ا لا مثلية القياسية المعروفة وتوصلنا ا لى ا ن نتاي جها اكثر كفاءة من طريقة.Barriar -Penalty ABSRAC In this paper we have investigated a new initial parameter, the new parameter is to mae balance between interior suitable for inequality constrained eterior method suitable for equality and inequality constrained) for non-linear constrained optimization. he new algorithm is programmed to solve some standard problems in non-linear optimization method. he results are too effective when compared with Barriar Penalty algorithm. 67

2 B. A. Metras.Introduction: We shall first state the most general from of the problem we are addressing, namely Minimize f n, R ) Subject to the general possibly nonlinear) inequality constraints c j 0 j L ) and possibly nonlinear) equality constraints c j with the simple bounds 0 L j m 3) L u, i n 4) i i i Where f and c j are all assumed to be twice continuously differentiable and defined on E n, is a subset of E n, and is a vector of n components,,,, n. he above problem must be solved for the values of the variables,,, n that satisfy the restrictions and meanwhile minimize the function f. he function f is called the objective function and any of the bounded in eq.4) may be infinite. See Conn et al, 994). he eterior-point method is suitable for equality and inequality constraints. he new objective function φ, ) is defined by: φ, ) f α 5) Where is a positive scalar and the remainder of the second term is the penalty function. Interior-point method is suitable for inequality constraints. he new objective function φ, ) is defined by φ, ) f B 6) Where is a positive scalar and the reminder of the second term is the Barrier function. see Gottefred, 973). 68

3 Modification of Non-Linear Although both eterior and interior-point methods have many points of similarly, they represent two different points of view. In an eterior-point 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. he requirement that we begin at a feasible point and remain within the interior of the feasible inequality constrained region is the chief difficulty with interiorpoint 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. hus interior-point methods cannot handle equalities. We may readily handle equalities by using a mied method in which we use interior-point penalty functions for inequality constraints only. hus, if the first m constraints are inequalities and constraints m) to n are equalities, our problem becomes: Minimize φ, ) f g ) B α 7) g ) he function φ, ) is then minimized for a sequence of monotonically decreasing > 0.. Mied Eterior-Interior Methods: We can solve the constrained problem given in eq.) to eq.3) construct a new objective function φ, ) which is defined in eq.7). Now our aim is to minimize the function φ, ) by starting form a feasible point 0 and with initial value 0 and the method reducing is simple iterative method such that: 69

4 B. A. Metras, 8) 0 where is a constant equal to 0 and the search direction d in this case can be defined d g, 9) where is a positive definite symmetric approimation matri to the inverse essian matri G - and g is the gradient vector of the function φ, ). he net iteration is set to a further point λd, 0) where λ is a scalar chosen in such that f < f, we thus test c i ) to see that it is positive for all i. We find a feasible and we can then proceed with the interpolation. hen the matri is updated by a correction matri to get φ ) where φ is a correction matri which satisfies quasi-newton condition namely y ρv) where v and y are difference vector between two successive points and gradients respectively and ρ is any positive scalar. he initial matri 0 chosen to be identity matri I. is updated through the formula of BFGS update. see Bazarra et al, 000). Given some approimation to the inverse essian matri, we compute the search direction d g, and we define v and g g G ) Gv. y We now want to construct a matri ) ) ) ) where is some symmetric correction matri that ensures that v,v,,v are eigenvectors of G with unit eigenvalues. ence 70

5 Modification of Non-Linear y v his condition translates to the requirement that y v y his therefore, leads to the ran-two DFP Fletcher and Powell, 963) update via the correction term vv y y DFP 3) v y y y he Broyden updates suggest the use of the correction matri given by B where B θτ p p 4) DFP p ) v y v y and where τ Newton condition holds by virtue of τ is chosen so that the quasi- p y being zero. hen BFGS B v v y y yv v y θ ) 5) v y v y v y and terminate the method if where ε < ε 6) 3. Combined Barrier-Penalty Algorithm: Step ): Find an initial approimation 0 in the interior of the feasible region for the inequality constraints i.e. g i 0 )<0. Step ): Set and 0 is the initial value of. Step 3): Set φ, ) f B α. Step 4): Set d - g Step 5): Set λd, where λ is a scalar. 7

6 B. A. Metras Step 6): Chec for convergence i.e. if eq.6) is satisfied then stop. Step 7): Otherwise, set and tae * and set 0 and go to Step he Initial Value of the Parameter: he initial value can be important in reducing the number of iterations and the number of function calls to minimize φ, ), since the unconstrained minimization of φ, ) is to be carried out for a decreasing sequence of, it might appear that by choosing a very small value of, we can avoid an ecessive number of minimization of the function φ, ). Also as φ, ) is close to f, the method should converge more quicly. owever, such a choice can cause serious computational problems. Also if is small, the function φ, ) will be changed rapidly in the vicinity of its minimum. his rapid change in the function can cause difficulties for a gradient based on methods. see Bazaraa, 000). Al-Bayati and amed in 997) suggested a new parameter of the Barrier function. Al-Assady and amed in 00) proposed a new initial parameter of Barrier-Penalty method. 5. Development of the New Algorithm: Consider the problem stated in eq.) to eq.4). he new objective function φ, ) defined in eq.7) with a starting feasible point 0 and with an initial value 0 which is derived as φ, ) f B α 7) f [ h ]. 8) c 7

7 Modification of Non-Linear hen the gradient of φ, ) is c φ, ) f h h 9) [ c ] such that φ, ) 0 we have c f h h 0 0) [ c ] Now, since > 0, then we have c f h h 0 ) [ c ] Arranging eq.) and multiplying it by -), we have c f h h 0 ) [ c ] he optimum value of is then given by one of the following roots to eq.): c f m f ) 8h h c ) min 3) c c ) In the above suggestion corresponding to the assumption for deriving a new parameter to mae balance between the eteriorinterior point method, we have suggested the following new algorithm. 6. he Outline of the New Algorithm: Step ): Find an initial approimation 0 in the interior of the feasible region for the inequality constraints i.e. g i 0 )<0. Step ): Set and 0 is the initial value of. Step 3): Find the initial value of by using eq.3), and compute φ, ) f B α. 73

8 B. A. Metras Step 4): Set d - g Step 5): Set λd, where λ is a scalar. Step 6): Chec for convergence i.e. if eq.6) is satisfied then stop. Otherwise go to step 7. Step 7): Set 0 Step 8): Set * and set and go to step Results and Calculation: In order to test the effectiveness of the new algorithm that has been used to Barrier-Penalty function method, the comparative tests involving several well-nown test function see Appendi have been chosen and solved numerically by utilizing the new and established method. So the new algorithm has been compared with Barriar -Penalty algorithm. In table ) we have compared the new algorithm with standard Barrier-Penalty algorithm for n 3 and c i 7 using 5) nonlinear test functions. From table ) it is clear that, taing the standard Barrier- Penalty algorithm as 00%, and the new algorithm has 75%, 76.8%, and 8.9% improvements on the standard Barrier-Penalty algorithm in bout number of iterations NOI and number of function evaluations NOF. 74

9 Modification of Non-Linear able ) Comparison between Barrier-Penalty and new algorithms est Barrier-Penalty algorithm New algorithm function NOI NOF) NOI NOF). 7 6) 7). 8 4) 9 99) ) 5 7) ) 5 6) ) 9 955) ) 9 596) otal ) ) able ) Barrier-Penalty algorithm New algorithm NOI 00% 75 NOF 00% Appendi: est functions:. min f ) ) s.p7,9) 0 4. min f 8,6) min f s.p.0.9,) 75

10 B. A. Metras i 4. ) 3 ) ) min f,7) ) ) min f s.p4,3,3,3) i 6. ) 3) ) min f s.p0.,) 0 3 i

11 Modification of Non-Linear REFERECES [] Al-Assady, N.. and amed, E.. 00), Investigation A New Initial Parameter of Mied Interior-Eterior Method for Nonlinear Optimization Al-Rafideen J. No.5, Vol.4. [] Al-Bayati, A.Y. and amed, E.. 997), Some Modifications in Non-Linear Constrained Optimization Algorithms, Ms.c. hesis, Dep. of Math. College of Science, Univ. of Mosul. [3] Bazaraa, S. and Shetty, C.M. 000), Nonlinear Programming: heory and Algorithms, New Yor, Chichester, Brisbane, oronto. [4] Biggs, M.C. 989), alfield Polytechnic Numerical Optimization, he Optima Pacage, Special Communication. [5] Conn, A.R., Gould, N. and oint, Ph.L., 994), Large scale Nonlinear Constrained Optimization, A current Survey, Report 94/ IBM.J. Watson Research Center, USA. [6] Fiacco, A.V. and McCormic, G.P. 968), Etension of Sum for Non-Linear Programming Equality Constraints and Etrapolation, Management Science, Vol., No., pp [7] Fletcher, R. and Powell, M.J.D., 963), A Rapidly Convergent Descent Method for Minimization, Computer Journal, 6, pp [8] Gottefred, B.S. and Weisman, J., 973), Introduction to Optimization heory, Prentice-all, Englewood Cliffs, N.J. 77

Using "Filter" Approach to Solve the Constrained Optimization Problems Baan A. Metras College of Computer Sciences and Mathematics University of Mosul

Using Filter Approach to Solve the Constrained Optimization Problems Baan A. Metras College of Computer Sciences and Mathematics University of Mosul Raf. J. of Comp. & Math s., Vol. 7, No., 00 Using "Filter" Approach to Solve the Constrained Optimization Problems College of Computer Sciences and Mathematics University of Mosul Received on:/3/009 Accepted