An Extended Algorithm for Finding Global Maximizers of IPH Functions in a Region with Unequal Constrains

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1 Applied Mathematical Sciences, Vol. 6, 2012, no. 93, An Extended Algorithm for Finding Global Maximizers of IPH Functions in a Region with Unequal Constrains H. Mohebi and H. Sarhadinia Department of Mathematics of Shahid Bahonar University of Kerman and Kerman Graduate University of Technology, Kerman, Iran hmohebi@uk.ac.ir, sarhaddi@uoz.ac.ir Abstract In this paper, we present an extended algorithm for finding constrained global maximizers of extended real valued increasing and positively homogeneous (IPH) functions in a region with unequal constrains, which is a version of the cutting angle method. Also, we give some numerical experiments. Mathematics Subject Classification: Primary 90C, 26A48; Secondary 52A07, 49M Keywords: Global optimization, cutting angle method, increasing and positively homogeneous function, abstract convexity 1 Introduction The cutting angle method for the global minimization of non-negative valued IPH functions over the unit simplex S := {x =(x 1,...,x n ) R n + : n i=1 x i = 1} was introduced and studied in [1]. Recently, in [5] the authors presented an algorithm for finding constrained global maximizers of extended real valued IPH Functions over S A := {x =(x 1,...,x n ) R n : n i=1 a ix i = 1, x i 0, i =1, 2, n}, where A := (a 1,a 2,...,a n ), 0 <a i 1, i =1, 2,...,n, which is a version of the cutting angle method. In this paper, we present an extended algorithm for finding global maximizers of extended real valued increasing positively homogeneous (IPH) functions in a region with unequal constrains. To do this, we define S A j := {x =(x 1,...,x n ) R n : n i=1 aj i x i = 1, x i 0, i =1, 2, n}, S := {x =(x A j 1,...,x n ) R n : n i=1 aj i x i 1, x i 0, i =1, 2, n} and S := {x =(x 1,...,x Aĵ n ) R n : n i=1 aĵ i x i

2 4602 H. Mohebi and H. Sarhadinia 1, x i 0, i =1, 2, n}, where A j := (a j 1,a j 2,...,a j n ), 0 <aj i 1, i = 1, 2,...,n and Aĵ := (aĵ 1,aĵ 2,...,aĵn ), 0 <aĵi 1, i =1, 2,...,n, j, ĵ = 1, 2, 3, j ĵ. Let L 1 := {x R n : x S A 1 S A 2} and L 2 := {x R n : x S A 1 S A 3}. If our unequal constrains are as S A,S 1 A and S 2 A, we define 3 L SA := {x S A : n i=1 a1 i x i = 1, n i=1 a2 i x i 1 and n i=1 a3 i x i 1, 0 < a j i 1,x i 0, i=1, 2,,n, j =1, 2, 3}. We present an approach for global maximum of extended real valued IPH functions over L SA S A, and then we extend this algorithm for finding its global maximum over a region with unequal constrains. The cutting angle method for the global maximization such functions is reduced to the solution of the following auxiliary problem: The structure of the paper is as follows: In Section 2, we provide some definitions and preliminary results on IPH functions. In Section 3, we present an algorithm for finding global maximizers of an IPH function over a region with unequal constrains. The numerical experiments are given in Section 4. 2 Preliminaries and IPH Functions Consider n-dimensional linear space R n. We shall use the following notations: I := {1,...,n}. x i is the ith coordinate of a vector x =(x 1,,x n ) R n. R n := {x =(x 1,,x n ) R n : x i 0, i I}. R n + := {x =(x 1,,x n ) R n : x i 0, i I}. If x, y R n, then x y x i y i for all i I. If x, y R n, then x>>y x i >y i for all i I. We shall consider the following optimization problem: max p(x) subject to x L SA S A, (2.1) where p is an extended real valued IPH (increasing and positively homogeneous of degree one) function defined on R n. Recall (see [6]) that a function p : R n [, + ] is called increasing and positively homogeneous of degree one (IPH), if p is increasing (x y = p(x) p(y)) and p is positively homogeneous of degree one, that is, p(λx) =λp(x) for all x R n and all λ>0. In the sequel, we introduce the coupling function v : R n R n [, 0] defined by v(x, y) := min{λ 0: λy x}, (x, y R n ), (2.2) (with the convention min =0). For each y R n, define the function v y : R n [, 0] by v y (x) :=v(x, y) for all x R n. It is easy to see that each v y is an IPH function (for more details see [3, 4]).

3 Extended algorithm for finding global maximizers 4603 We will use the vector e A 1 m := (0,, 0, a m, 0,, 0) R n, where A := (a 1,a 2,...,a n ), 0 < a m 1 for each m I. Note that e A m S A for all m I. Clearly, I (e A m) = {m}, and for the vector y := we have ea m p(e A m ) v y (x) = a m x m p(e A m ) for each x = (x 1,,x n ) R n. We define αxk := (α 1 x k 1,α 2 x k 2,...,α n x k n), where α =(α 1,α 2,...,α n ) R n + and x k =(x k 1,x k 2,,x k n) R n. Now, we present an algorithm for the search for a global maximizer of a finite valued IPH function p over L SA S A. Recall that a finite valued IPH function p defined on R n is non-positive valued because p(x) p(0) = 0 for all x R n. We assume that p(x) < 0 for all x S A. It follows from the non-positivity of p that I (y) =I (x) for all x S A, and y = Algorithm 1 x. p(x) Step 0: (initialization) (a) Take points x m := e A m for m =1,,n, and construct the basis vectors y m := xm (m =1,,n). p(x m ) (b) Define the function h n (x) := min v yj(x) = min a jx j p(e A j ), x =(x 1,,x n ) j n j n S A. (c) Set k := n. Step 1: Find x := arg[max h k (x)]. x S A Step 2: Set k := k +1, and x k := x. Step 3: Choose α k = α =(α 1,α 2,...,α n ) R n + such that αx k L SA S A and for k>n+1,p( α k x k ) p( α k 1 x k 1 ) ( and hence n i=1 a i(α i x k i )= 1). Compute y k := αxk. Define the function p(αx k ) h k (x) := min j k v y j(x) = min(h k 1 (x), max i I (y k ) x i = min j k max i I (y j ) y j i x i ) yi k, x =(x 1,,x n ) L SA S A. Go to Step 1. The convergency of the Algorithm 1 has been proved in [5]. Also, the following result has been proved in [5], and therefore we omit its proof. Theorem 2.1. Let x<<0 be a local maximizer of the function h k over the set ril SA (the relative interior of L SA ) such that h k (x) < 0. Let α R n + be such that αx k ril SA. Then there exists a subset {y j 1,,y jn } of the set {y 1,,y k } such that (1) x =(y j 1 1,,yn jn )d A, where

4 4604 H. Mohebi and H. Sarhadinia d A := h k (αx k )= (2) max j k min i I (y j ) y j i i y j i P 1 a i α i y j i i i I =1. and A =(a 1,a 2,...,a n ), 0 <a i 1, i I. (3) If j m n (m I), then j m = m, and if j m n +1, then y j i i I, i m. <y jm i, i Remark 2.1. Consider the set of k vectors Λ k := {y 1,..., y k } generated by Algorithm 1. Every local maximizer x of h k in ril SA corresponds to a combination of n vectors L = {y j 1,..., y jn } which satisfy the following conditions: (I) For all i, r I, i r, we have y j i i <y jr i. (II) For each y r Λ k \ L, there exists i I such that y j i i y r i. To illustrate the above conditions (I) and (II), visualize L as an n n matrix, whose rows are y j 1,y j 2,..., y jn : y j 1 1 y j y j 1 n y j 2 1 y j y j 2 n... y jn 1 y jn 2.. y jn n Condition (I) implies that the diagonal of L is dominated by their columns, and condition (II) implies that the diagonal of L is not dominated by any other vector y r, not already in L (diag(l) is dominated by y r, means that diag(l) <y r ). The location of the local maximum x max and its value d(l) =h k (x max ) can be found from the diagonal of L:. x max = diag(l) trace(l), and d(l) =h k (x max )= 1 trace(l). Recall that the function h k is continuous on the compact set S A. In order to find the global maximum of the function h k at Step 1 of Algorithm 1, we need to examine all its local maxima, and hence all combinations of L of the n vectors which satisfy the conditions (I) and (II). In view of h k (x) = min(h k 1 (x), v y k(x)),

5 Extended algorithm for finding global maximizers 4605 if we have already computed all combinations of n vectors out of k 1 vectors satisfying the conditions (I) and (II) (i.e. all candidates for local maxima of the auxiliary function h k 1 (x)), at the previous iteration, we only need to compute those combinations that have been added by aggregation of the last vector y k, that is, those combinations of L that include vector y k. Suppose we already know the set V k 1 of combinations of k 1 vectors satisfying (I) and (II). We need to update V k 1 to V k (i.e. all possible combinations of n vectors out of k vectors satisfying (I) and (II)). Algorithm 2 (Update of the set V k 1 to V k ) Input: the set V k 1 ; the new vector y k. Output: the set V k. Step 1: Set V k =. Step 2: Test all elements L of V k 1 against condition (II), with y r = y k. Put those L that fail the test into Temp and those that pass into V k. Step 3: For every L in Temp, form n copies of it, and replace row i in the ith copy with y k. Test condition (I). If test passed, add this modified copy to V k, otherwise discard it. 1 Step 4: Calculate d(l) = for all elements L of V k and sort V k with trace(l) respect to d(l) in ascending order and choose the largest d := d(l). Algorithm 3 Step 0: (a) Evaluate the objective function p(x) in the vertices of S A and form the matrix L root = {y 1,..., y n }. (b) Set α = (1, 1, 1) and Calculate d A = ( i I a i α i y i i ) 1, where A = (a 1,a 2,...,a n ), 0 <a i 1, i I. (c) Set k = n, Λ k = {y 1,..., y n } and V k = {L root }. Step 1: (a) Select L = Head(V k ) with the biggest d A (the global maximum of h k (x) exception case k = n). (b) Form x = diag(l), and evaluate p trace(l) best = p(x ). Step 2: Set k = k +1, and set x k = x. Choose α k = α =(α 1,α 2,,α n ) R n + such that αx L SA and for k>n+1,p( α k x k ) p( α k 1 x k 1 ). Form y k =( α 1x 1, α 2 x p(αx 2,, α nx ) p(αx n ), and ) p(αx ) set Λ k =Λ k 1 {y k }. Test if αx L SA and yi k = yi i for some i I, then print p(αx is a global maximizer of p) and stop, else, call Algorithm 2 (V k 1,y k,v k ).

6 4606 H. Mohebi and H. Sarhadinia Step 3: (Stopping Criterion) If αx L SA and k<k max and abs(d p best ) <ε. Go to Step 1. 3 An Extended Algorithm for Solving Problem (2.1) Algorithm 4 Step 0 : (a): Eliminate all redundant constrains. (b): Set k =1. Choose an arbitrary small real number η as length step and choose an arbitrary small positive real number ɛ. Set η = η. (c): Assign m coefficient matrices A j 1 to A jm for m constrains s and s A j Aĵ, j and ĵ {1, 2,,m}, respectively. Step 1: (a): If k =1, then use Algorithm 3 with one arbitrary A j from one constrains s A and find x on the constrain L j SA, where j {1, 2,,m}. j (b) If k >1, use Algorithm 3 from step 1 on the L SA, where j {1, 2,,m} j and find its x. Step 2 : (a) For chosen j {1, 2,,m}, construct L SA k and assign all vertices points (x v ) ij of the set L J 1 S and all vertices points (xˆv ) ij of the set LĴ2 S (i j, i, j =1,,n). (b): If (x v ) ij and (xˆv ) ij exist on the plane x i x j (i j, i, j =1,,n) such that (x v ) ij (xˆv ) ij = max((x v q) ij (xˆv q) ij ) ɛ (q =1,,n) for some i, j = 1,,n, i j, then set η = η, else η =0. (c): Assign x i = e i =(0,..., 0, 1 a i + k η, 0,..., 0) and ith coordinate of coefficient matrix A k for k iteration as a k i = 1 ( 1 a i + k η). (d): Set A j = A k and set k = k +1. (f) If k =1, set x best = x and go to step 1, else assign (x best ) k+1 = opt((x ) k,x best ). If (x best ) k+1 =(x ) k, then A opt = A k, else A opt = A j. Go to Step 1.

7 Extended algorithm for finding global maximizers 4607 Note that if we start from one constrain S Aĵ, we must change Step 2 (c) as the following: Assign x i = e i =(0,..., 0, 1 a i k η, 0,..., 0) and ith coordinate of coefficient matrix A k for k iteration as 1 a k i = ( 1 a i k η). 4 Numerical Experiments Problem 3.1 f(x 1,x 2,x 3 ) := max{a i x i : i =1, 2, 3} + min{b j x j : j =1, 2, 3}, where a i =2+0.5i (i =1, 2, 3),b j =(j + 2)(n j +2)(j =1, 2, 3) x i R (i =1, 2, 3) such that 0.2x x x 3 1 x x x x x x 3 1 x i R (i =1, 2, 3). The global maximizer over this region is x =( , , ) with p(x )= Problem 3.2 f(x 1,x 2,x 3 ) := min(({ 3 x 1 x 2 x 3 + min{x 1 +2x 3, 2x 1 + x 2 }), (max{a i x i : i =1, 2, 3} + min{b j x j : j =1, 2, 3})) such that 0.2x x x x x x x x x 3 1 x i R (i =1, 2, 3). The global maximizer over this region is x = ( 0.05, 0.59, 0.72) with p(x )= Conclusion: From the above examples we conclude that Algorithm 4 is applicable for solving extended real value increasing and positively homogeneous (IPH) functions over a region with unequal constrains. Acknowledgments: This research was supported partially by Kerman Graduate University of Technology and Mahani Mathematical Research Center.

8 4608 H. Mohebi and H. Sarhadinia References [1] A. Bagirov and A. M. Rubinov, Global minimization of increasing positively homogeneous functions over the unit simplex, Annals of Operations Research, 98 (2000), [2] L. M. Batten and G. Beliakov, Fast algorithm for the cutting angle method of global optimization, Journal of Global Optimization, 24 (2002), [3] H. Mohebi and H. Sadeghi, Monotonic analysis over ordered topological vector spaces: I, Optimization, 56 (2007), No. 3, [4] H. Mohebi and A. R. Doagooei, Abstract convexity of extended real valued increasing and positively homogeneous functions, Journal of DCDIS-B, 17 (2010), [5] H. Mohebi and H. Sarhadinia, An algorithm for finding constrained global maximizers of extended real valued IPH functions in unit simplex, to appear. [6] A. M. Rubinov, Abstract convexity and global optimization, Kluwer Acadamic Publishers, Boston, Dordrecht, London, Received: April, 2012