Rachid Benouahboun 1 and Abdelatif Mansouri 1

Transcription

1 RAIRO Operations Research RAIRO Oper. Res DOI:.5/ro:252 AN INTERIOR POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING WITH STRICT EQUILIBRIUM CONSTRAINTS Rachid Benouahboun and Abdelatif Mansouri Abstract. We describe an interior point algorithm for convex quadratic problem with a strict complementarity constraints. We show that under some assumptions the approach requires a total of O nl number of iterations, where L is the input size of the problem. The algorithm generates a sequence of problems, each of which is approximately solved by Newton s method. Keywords. Convex quadratic programming with a strict equilibrium constraints, interior point algorithm, Newton s method.. Introduction A mathematical programming with equilibrium constraints MPEC is a constrained optimization problem in which the essential constraints are defined by a complementarity system or variational inequality. This field of mathematical programming has been of much interest in the recent years. This is mainly because of its practical usage in many engineering design [3, ], economic equilibrium [6], multi-level game-theoretic and machine learning problems [4, 9]. The monograph [2] presents a comprehensive study of this important mathematical programming problem. Definition. Let M be a n nreal matrix. We say that a matrix M is copositive if for all x, Mx. Received February 27, 22. Accepted November 9, 24. Département de Mathématiques, Faculté des Sciences Semlalia, Marrakech, Maroc; mansouri@ucam.ac.ma c EDP Sciences 25 Article published by EDP Sciences and available at or

2 4 R. BENOUAHBOUN AND A. MANSOURI It is easily seen that if M is co-positive, then for x, we have XMX x T Mx, and XMx x T Mx, where X = diagx,..., x n. In this paper, we consider the following convex quadratic program with a strict linear complementarity constraint: Minimize 2 xt Gx + d T x s.t. CQPEC Ax = b, x T Qx =, x, where x and d are n-vectors, b is an m-vectors, A is an m n matrix with ranka =m<n, G is a symmetric positive semidefinite n n matrix, Q is a symmetric co-positive n n matrix and the superscript T denotes transposition. Our purpose is to construct an interior point algorithm to solve the problem CQPEC. Using Newton s direction the algorithm generates a sequence of interior points which under some conditions converges to a solution stationary point of CQPEC in a polynomial time. The algorithm has a complexity of On 3.5 L where L is the input length for the problem. The paper is organized as follows. In Section 2, we give some applications of the problem CQPEC in mathematical programming. In Section 3, we present some theoretical background. In Section 4, we present the algorithm and we prove some results related to the convergence properties of the algorithm. Finally, in Section 6 we discuss the initialization of the algorithm. To illustrate our approach we conclude the paper with some numerical results in Section Some applications of CQPEC in mathematical programming The problem CQPEC have a wide range of applications for example in economic equilibrium, multi-level game and machine learning problems. In this section, we show some applications in mathematical programming. The problems that we consider in this section, are generally NP-Complete. Nevertheless, the approach presented in this paper solves a particular cases of these problems in polynomial times. 2.. Optimization over the efficient set Consider the multiobjective linear program { }} Minimize C x s.t. x χ = {Ã x = b, x where C is an k n matrix. Recall that a point x χ is an efficient solution of if and only if there exists no x χ such that Cx Cx and Cx Cx. Let E

3 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM 5 denote the set of the efficient solutions. Consider the problem Minimise { } 2 xt G x + dt x s.t x E 2 where G is a symmetric positive semidefinite n n matrix and d is an n-vector. In the linear case this problem has several applications in multiobjective programming see [, 2]. 2 can be written in the following form Minimize 2 xt G x + dt x s.t à x = b, e T λ =, 3 à T ỹ + z C T λ =, x T z =, x, z, λ. By taking ỹ = ỹ ỹ 2,withỹ, and ỹ 2, x = x z ỹ ỹ 2 λ 3isequivalenttoCQPEC.,A= à I n à T ÃT C T, I n I n b Q = and b = 2.2. Goal programming Consider the multiobjective problem. To solve this problem, several approaches have been developed. One class of a very used methods is the Distance- Based Methods. These methods assume that the decision-maker can select a point, at each iteration, which can be considered as an ideally best point from his viewpoint. If the ideal point is not feasible, the process gives the closer efficient solution to the ideal point as possible. The research of such a solution can be summarized by the mathematical programming problem: Minimize { df, C x } s.t. x χ 4

4 6 R. BENOUAHBOUN AND A. MANSOURI where f is the ideal point and d.,. is a distance of point in R k. Inthecaseof the L -distance and by introducing the following notation { d + f i = i c T i x, if f i ct i x otherwise { d c T i = i x fi, if f i < ct i x otherwise the problem 4 can be written as follows: Minimize k w i d + i + d i i= s.t. Ã x = b C x d + d + = f, d + T d =, d +, d, x 5 where w is a vector of weights. This problem is often called goal programming. In many applications, a mixture of the above process and the method of sequential optimization is referred as a goal programming approach or model. Obviously the problem 5 is equivalent to CQPEC. 3. Preliminaries The algorithm that we consider in this paper is motivated by the application of the mixed penalty technique to problem CQPEC. The mixed penalty consists of examining the family of problems { P Minimize f x = x 2 xt Gx + d T T Qx } 2 n x + logx i : s.t x S 4 where S = {x : Ax = b, x > }, all x S is called an interior point of the problem CQPEC, and > is the penalty parameter. This technique is well-known in the context of general constrained optimization problems. One solves the penalized problem for several values of the parameter, with decreasing to zero, and the result is a sequence of interior points, called centers, converging to a stationary solution of the original problem [5]. Our aim is to construct a polynomial algorithm. Since, generally, it is impossible to solve P exactly in a finite time, we must renounce to the determination of centers, and work in their neighborhood. In other words, we can generate a sequence of interior points { x k} close to the path of centers such that x k T Qx k and converges to a stationary solution of the problem CQPEC. i=

5 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM 7 The KKT conditions applied to CQPEC implies that if x is a stationary, then there exist y R m, β R and z R n, such that: A T y Gx βqx + z = d, Ax = b, x T Qx =, x T z =, x, z. Let T = { x, y, z, β : Ax = b, A T y Gx βqx + z = d, x, z, β }, and consider the merit function defined for all x, y, z, β T,by x, y, z, β =x T Qx + x T z. With this merit function, the above stationary condition system becomes { x, y, z, β T, x, y, z, β =. For the time being, our objective is to construct a sequence { x k,y k,z k,β k } T such that x k is an interior point for all k. with an upper bound for the merit function x k,y k,z k,β k = x k T Qx k +x k T z k, is driven to zero at a fixed rate of σ n, where σ is a given constant. For this, we make the following assumptions. Assumptions. a There exists w R m such that A T w>andb T w>. b If x is feasible for CQPEC, then we have for all i {,..., n}, x i > if and only if [Qx] i =. c For all x, y, z, β T, we have β ρ, where ρ is a given constant. Remark. i The assumption a is used, in section 6, to transform the problem CQPEC on an equivalent problem for which an initial interior point x S is known in advance. As in [7], we show that there exists y such that: X A T y = e. ii The assumption b implies that the condition of Constraint Qualification holds for the problem CQPEC see [2]. Remark also that this assumption is satisfied for the problems described in Section 2. iii We give an upper bound for ρ in the next section. Throughout this paper, we use the following notation. When x is a lower case letter denotes a vector x =x,..., x n T, then a capital letter will denote the diagonal matrix with the components of the vector on the diagonal, i.e., X = diag x,..., x n, and. denotes the Euclidean norm.

6 8 R. BENOUAHBOUN AND A. MANSOURI 4. The algorithm Given ancqpec problem in standard form, in the Section 6, we discuss how to transform this problem into a form for which an initial interior point x S such that there exists y R m satisfying X A T y = e, 6 is given. For the moment we can suppose that a such initial point is known in advance. For a current iterate x S, let x denote the next iterate. The direction d x, chosen to generate x is defined as the Newton direction associated with the penalized problem P. The Newton direction at the point x is the optimal solution of the quadratic problem: QP Minimize 2 dt x 2 f xd x + f x T d x s.t. Ad x =. Let M denote the Hessian of the penalty function f at the point x. The following lemma show that if x T Qx <, then the matrix M is positive definite and by consequence the problem QP has an unique solution. Lemma. If x T Qx <, then the matrix M = G+ x T Qx Q+ QxxT Q+X 2 is positive definite. Proof. We have M = X XGX + x T Qx XQX + 2 XQxx T QX + I X. 2 2 XQX x T Qx 2 <, and so x T Qx On the other hand we have I + x T Qx XQX is positive definite. Thus M is also positive definite. 2 By the KKT condition, d x equations: is determined by the following system of linear [ G + x T Qx x T Qx Ad x =. Q + QxxT Q + X 2 ] d x + Gx + d+ Qx X e = A T ŷ, 7

7 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM 9 Via a simple calculation, we obtain for d x, x, ŷ, ẑ and β : [ d x = M M A T AM A T ] AM ϑ x = x + d x ŷ = AM A T AM ϑ 8 β = xt Qx ẑ = G x + d + βq x A T ŷ where ϑ = X e Gx d x T Qx Qx. We are now ready to describe the algorithm. Let δ, η, ρ and σ be constants satisfying: δ δ δ +δ η η η<δ,ρ<min, 9 +δ δθ 2 + θ δθ θ 2 σ <Min n θ ρ 2 θ δ ρ + δ + θ 2 δ +δ θ 2 η, ρ δθ θ 2 η where θ =+δ and θ 2 = δ + n. The following lemma gives some results, related to theses constants, that will be useful to prove Lemma 6. Lemma 2. Let δ, η, ρ and σ be constants satisfying 9 and. Then: i ii iii iv ρθ θ 2 2 σ <ρθ θ 2. n α = σ n ρθ θ 2 >. [ ] ηθ + δ 2 + ρ 2 θ θ θ 2 ρθ 2 + σ δα. 2 ρθ θ 2 σ δ η. n Proof. σ i From we obtain: n δ < ρ η θ θ 2 < ρθ θ 2, which implies ρθ θ 2 2 σ n ρθ θ 2. ii From it follows that σ n >ρθ θ 2. Thus α = σ n ρθ θ 2 >.

8 2 R. BENOUAHBOUN AND A. MANSOURI iii By we have ] σ n θ [ρ 2 δ θ ρ δ +δ η θ 2 + δ + θ 2 [ ] [ ] σ n θ 2 ρ 2 θ θ θ δ 2 ρθ 2 ρθ θ 2 δ + θ δ +δ η σ n δ + [ ] n ρ 2 θ θ θ 2 ρθ 2 + δ δ 2 ηθ ρθ θ 2 δ [ ] σ n δ + σ + δ 2 + ηθ + ρ 2 θ θ θ 2 ρθ 2 δ ρθ θ 2 δ [ ] δ 2 + ηθ + ρ 2 θ θ θ 2 ρθ 2 + σ δ n σ ρθ θ 2 [ ] δ 2 + ηθ + ρ 2 θ θ θ 2 ρθ 2 + σ δα. The third inequality follows from the fact that θ =+δ and θ 2 = δ + n. iv By we have σ n ρ δθ θ 2, then we obtain ρθ θ 2 η η σ n δ Hence 2 ρθ θ 2 σ δ η. n We now state the algorithm. ALGORITHM Step.: letx S be a given interior point which satisfies 6, for an initial penalty parameter >, and ρ, δ, η and σ a given positive constants satisfying 9 and. Let ε> be a tolerance for the merit function. Set k :=. Step.: compute d k x,yk+,β k+ and z k+ by using 8. Step.2: setx p+ := x p + d k x and k+ := k σ n Set k := k +. Step.3: compute the merit function: x k,y k,z k,β k := x k T Qx k +x k T z k. If x k,y k,z k,β k ε, stop. In the next section, we prove that all points generated by the algorithm lie in the set T and that they remain close to the central path T c. We also show that the algorithm terminates in at most O nl iterations. This fact will enable us to show that the algorithm performs no more than On 3.5 L arithmetic operations until its termination.

9 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM 2 5. Convergence We begin this section by stating the following main results. Theorem 3. Let ρ, δ, η and σ be positive constants satisfying relations 9 and. Assume that x and d x satisfy X d x δ and XQxx T Qd x η 2. Let = σ/ n and consider the point x, ŷ, ẑ, β R n R m R n R given by 8. Then, we have: a x, ŷ, ẑ, β T ; b c x T ẑ θ 2 δ + η + n and xt Q x ρ σ/ n < ; X d x δ and XQ x x T d x η 2. This theorem show that if, the current iterate is close to the central path, so it is for the next iterate. Proof. To prove this theorem, we use the following Lemmas 4, 5 and 6 see the annex for the proofs of those lemmas. Lemma 4. Under the assumptions of the above theorem, we have i x, ŷ, ẑ, β T and x T Qx ρ; ii x T ẑ θ 2 δ + η + n and xt Q x ρ σ/ n The following lemma will be used in the proof of Lemma 6. Lemma 5. Let x, d x, and be as above. If XQxx T Qd x η 2 and X d x δ, then, we have i XX I δ,and XQ x ρθ θ 2 ; [ ] ii xt Q x xt Qx θ ρ 2 σ/ n ; iii xt Q x d x T Qd x ρθ θ X 2 2 d x ; iv XQxx T Qd 2 x ηθ. Now, we prove that if X d x δ and XQxx T d x η 2 are small, then they are also for the next iteration. Lemma 6. Let δ, η and σ be as above. If X d x δ, and XQxx T d x η 2, then we have X d x δ and XQ x x T d x η 2. θ 2 2 θ 2 2

10 22 R. BENOUAHBOUN AND A. MANSOURI Let x S be an initial interior point which satisfies 6. The following lemma shows how to choose >, such that the Newton step d x, at x, satisfies X Qx x T Qd x η 2 and X δ. Lemma 7. Let δ, η, and ρ satisfy 9, : X Gx + d [ δ ρ 2 + δ ] 2 and x T Qx ρ. 3 Then X δ and X Qx x T Qd x η 2. Proof. From 22 we have X 2 d x T Gx + d x T Qx d x T Qx On the other hand, we have Then 4 becomes ρ 2 X x T Qx d x T Qd x d x T X e. 4 x T Qx d x T Qd x x T Qx X QX X x T Qx 2 X d 2 x ρ 2 X 2. d 2 x 2 d x T Gx + d x T Qx d x T Qx d x T X e. From Ad x =, we have d x T A T y =, then ρ 2 X 2 d x T Gx + d x T Qx d x T QX d x T X [ e X A T y ].

11 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM 23 Using 6, we conclude that ρ 2 X d 2 x d x T X [X Gx + d + x T Qx ] X Qx [ X X d x Gx + d x T Qx + X Qx ] [ X X Gx + d x T Qx 2 ] + X X [[ δ ρ 2 + δ ] + ρ 2] ρ 2 δ. Hence ρ 2 X d 2 x X which implies X d x δ. For the last inequality of the lemma, we have X Qx x T Qd x 2 ρ 2 δ X Qx 2 X x T Qx 2 X ρ 2 δ. By we have ρ 2 η δ and hence X Qx x T Qd x 2 η δ δ = η2. As a consequence of Theorem, we have the following result: Corollary 8. Let x S be an initial point which satisfies 6, be an initial penalty parameter, and δ, η, σ and ρ constants satisfying the conditions of Lemmas 2 and 5. Then the sequence { } x k,y k,z k,β k generated by the algorithm satisfies for all k i Xk Qx k x k T Qd k x η 2 k and X k dk x δ; ii x k,y k,z k,β k T with x k T z k k δ + nδ + η + n; iii x k T Qx k k ρ; where k = σ/ n k. Proof. This result follows trivially from Theorem. We now derive an upper bound for the total number of iterations performed by the algorithm.

12 24 R. BENOUAHBOUN AND A. MANSOURI Proposition 9. The total number of iterations performed by the algorithm is no greater than k = n σ log [ ε [ σ/ n ρ +δ + nδ + η + n] ], where ε> denotes the tolerance for the merit and is the initial penalty parameter. Proof. The algorithm terminates whenever [ ] k δ + n δ + η + n + σ/ n ρ ε. Thus, it is enough to show that k satisfies this inequality. By the definition of k, we have log ε = kσ n +log [ [ σ/ n ρ +δ + nδ + η + n ]] k log σ/ n +log [ [ ]] σ/ n ρ +δ + nδ + η + n k [ ] log [ ] σ/ n σ/ n ρ +δ + nδ + η + n log [ [ ]] k σ/ n ρ +δ + nδ + η + n. The second inequality is due the fact that log x x, for all x<. Therefore k satisfies k [δ + nδ + η + n+ σ/ n ρ] ε. If we define L to be the number of bits necessary to encode the data of problem CQPEC, then the following corollary clearly holds. Corollary. If log 2 =OL, log 2 ε = OL and the hypothesis of the last theorem holds, then the algorithm terminates in at most O nl iterations and with On 3.5 L arithmetic operations. Proof. From log 2 = OL andlog 2 ε = OL, it is clear that we have k = O nl. Since for each iteration we have to solve one linear system, which requires On 3 arithmetic operations, the algorithm converges in at most On 3.5 L. 6. Initialization of the algorithm Consider the convex quadratic problem with a strict equilibrium constraint CQPEC Minimize 2 xt G x + dt x à x = b, x T Q x =, x,

13 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM 25 where x and d are vectors of length ñ, b is a vector of length m, à is an m ñ matrix, G is a symmetric positive semidefinite ñ ñ matrix, Q is a symmetric co-positive ñ ñ matrix. We assume that Assumption. A There exists w R m such that: à T w > and b T w [ >. ] A 2 For all feasible point x, we have: x i > if and only if Q x =. i A 3 For all x, v, w, β Ϝ we have β ρ, where Ϝ = with ρ satisfies 9. { u, w, v, β : à T w G x β Q x + v = d, } x, v, β The aim of this section is to transform this problem into a convex quadratic problem with equilibrium constraints that satisfies assumptions a, b andc and has an interior point x which satisfies 6. The approach that we propose is one that has been suggested by numerous authors for transforming linear and convex quadratic programs into a form suitable for interior point algorithms. Let N be a large positive constant. For all i we tack x i = [ÃT w ] i and λ = x T à T w bt w = ñ bt w 5 As in [8,] it is clear that the following convex quadratic problem with equilibrium constraint Minimize 2λ xt G x + dt x + α N à x +λ b à x α = λ b, CQPEC α + α 2 =2 α α 2 = λ xt Q x =, x,α,α 2, where x = λ x, canbewrittenintheformofcqpewith x = λ x [ α à λ b à x,a= α 2 G λ Q λ G =,Q=,d= ],b= [ λ b 2 d N. ],

14 26 R. BENOUAHBOUN AND A. MANSOURI Let x = x [ w and y = problem CQPEC. Using 5 we obtain ]. It is clear that x isafeasiblepointforthe à T w X A T y = X λb T w x T à T w + à T w = X X à T w = = ẽ = e. Then x is an initial interior point for the problem CQPEC which satisfies 6. On the other hand, we have A T y > andb T y >. Thus assumption a is verified for CQPEC. The augmented set T is of the form { T = x, y, z, β, τ,τ 2 : à x +λ b à x α = λ b, α + α 2 =2, à T y λ G x β λ Q x + z = d, λ b à x T y + u τα2 + v = N, u τα + v 2 =, x,α,α 2, z, v,v 2, β, τ }. It is easily seen that if x, α,α 2,y,u,z,v,v 2,β,τ T, then we have x λ,y,z,β Ϝ and by the assumption A 3 we obtain β ρ. Thus the augmented problem CQPEC satisfies assumption c andbya 2 this problem satisfies also the assumption b. Then we can apply the algorithm to CQPEC for a large enough N. If x = x, α,α 2 is an optimal solution of CQPEC, then we have: If α =, then x = x λ is an optimal solution of CQPEC. Else, CQPEC has no solution.

15 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM Numerical example To illustrate the result of the paper, the progress of the algorithm is presented on a numerical example. The example is a - convex quadratic problem. Minimize x 2 +2x2 2 3x x 2 s.t..25x +.25x x +.375x x +.5x x +.25x x +.25x x.25x x +.75x x +.25x x i {, }. Adding the necessary surplus variables, this problem becomes Minimize 2 xt Gx + d T x s.t Ax + z = b x i {, } z. Converting this system of constraints to the form required by our proposed approach, results in augmented system given by Minimize 2 xt G x + dt x s.t à x = b x T Q x = x, where x = x z R 2 A I G, à =, G = I I e x Q = I, b b =. e I The augmented system shown above has the initial interior point: x = T.

16 28 R. BENOUAHBOUN AND A. MANSOURI Table. iteration x x 2 merit d T x Figure. By starting the algorithm with this initial solution with σ =. and =.5, the process terminates at the solution: x = T. A summary of the first 3 iterations is given in Table. The trace of the interior solution trajectory is shown in Figure. 8. Conclusion We have established polynomial method for convex quadratic programming with strict equilibrium constraints. To our knowledge this is the first time an interior

17 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM 29 point algorithm solves the MPEC in the convex quadratic case at polynomial time. The complexity obtained here coincides with the bound obtained for linear programming by the most of interior point methods. Annex ProofofLemma4. i By the definition of x and ẑ we have: and x = x + d x = X e + X d x > 6 ẑ = X e QxxT Qd x X 2 d x 7 = X e 2 XQxxT Qd x X d x >. ii This means that x, ŷ, ẑ, β T. On the other hand, we have β = x T Qx and from assumption c weobtainx T Qx ρ. From 6 and 7 we have, x T ẑ e + X d x e 2 XQxx T Qd x X d x θ 2 δ + η + n. 8 Thus, using the fact that XQX x T Qx ρ we obtain x T Q x X x 2 XQX x T Qxθ 2 2 ρθ2 2. So we have x T Q x ρ θ2 2 σ/ n ProofofLemma5. i We have XX { } xi I = max i n x i the last inequality of i, we have XQ x = max i n XX XQX X x ρ + δ δ + n = ρθ θ 2. { } h i x x i δ. To prove The second inequality follows from the fact that XQX x T Qx ρ and X x = e + X h x δ + n.

18 3 R. BENOUAHBOUN AND A. MANSOURI ii Form i of the last lemma, it follows that x T Q x xt Qx = x T Qx +2 xt Qd x ρ +2ρδ + ρδ2 [ ] ρ + δ2. + dt x Qd x Thus x T [ Q x xt Qx ρ θ 2 ] σ/ n. 9 iii From ii of the above Lemma, we conclude x T Q x d x T Qd x ρ θ2 2 σ/ n X 2 d x XX 2 XQX ρ 2 θ2 2 θ2 σ/ n X 2 d x ρθ 2 θ X d x 2. iv The last inequality is obtained by i of Lemma 2. We have XQxx T 2 Qd x XX 2 XQxxT Qd x ηθ. The last inequality follows from the fact that 2 XQxx T Qd x η. ProofofLemma6. From 7 d x and d x satisfy: [ G + x T Qx Q + QxxT Q + X 2 ] d x + Gx + d x T Qx + Qx X e = A T ŷ 2 and [ x T Q x G + Q + ] Q x xt Q + X x 2 T Q x d x + G x + d + Q x X e = A T ỹ. 2

19 AN INTERIOR POINT METHOD FOR CONVEX QUADRATIC PROBLEM 3 Using the fact that Ad x =and x = x + d x, and multiplying 2 and 2 by d Ṱ x we conclude that: d Ṱ x [ ] x QxxT Q + X 2 d x + d Ṱ T x G x + dṱ x d + Qx d Ṱ x Q x dṱ x X e = and d Ṱ x [ G + x T Q x Q + ] Q x xt Q + X 2 x d x + d Ṱ T x G x + dṱ x d + Q x d Ṱ x Q x d Ṱ x X e =. 22 So, this implies that d Ṱ x [G + ] [ ] Q x xt Q + X 2 d x = d Ṱ x QxxT Q + X 2 d x x T Qx + xt Q x x d Ṱ T x Q x Q x d Ṱ x Qd x + d Ṱ X x e d Ṱ x X e. We know that d Ṱ x Q x xt Qd x = x T Qd x 2 andd Ṱ x Gd x, then we have X 2 x T Q x d x + [ d Ṱx Qd x d Ṱx X XQxx T 2 Q + XX 2 d x + x T Qx xt Q x XQx + ] e XX e. Using e = e σ n e and the identity e XX e = X d x, we can write the above expression as: X 2 x T Q x d x + d Ṱ x Qd x d Ṱ x X [ 2 XQxx T Qd x + XX I X d x + x T Qx xt Q x XQx σ ] e. n Now by iioflemma5wehave: x T Q x d x T Qd x ρθ θ X 2 2 d x.

20 32 R. BENOUAHBOUN AND A. MANSOURI Thus X 2 d x ρθ θ X 2 [ 2 d x X d x XQxx T 2 Qd x + XX I X d x + x T Qx xt Q x XQx ] + σ. Hence α X d x XQxx T 2 Qd x + XX I X d x + x T Qx xt Q x XQx + σ, with α = ρθ θ 2. By using 9, i andiii of the above lemma, we obtain: α X [ d x ηθ + δ 2 + ρ 2 θ 2 ] θ θ 2 σ/ n + σ. From i of Lemma 2, it follows that This implies σ/ n ρ + δδ + n 23 X [ [ ] ] d x ηθ + δ 2 + ρ 2 θ θ θ 2 + σ. 24 α ρθ 2 On the other hand, from iii oflemma2weconclude Now we prove that X d x δ. XQ x x T Qd x 2 η. We have XQ x x T Qd x XQ x 2 X d x,

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