CCO Commun. Comb. Optim.


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1 Communications in Combinatorics and Optimization Vol. 3 No., 08 pp.570 DOI: 0.049/CCO CCO Commun. Comb. Optim. An infeasible interiorpoint method for the P matrix linear complementarity problem based on a trigonometric kernel function with fullnewton step Behrouz Kheirfam and Masoumeh Haghighi Department of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran Received: 6 September 06; Accepted: 0 January 08 Published Online: January 08 Communicated by Tamás Terlaky Abstract: An infeasible interiorpoint algorithm for solving the P matrix linear complementarity problem based on a kernel function with trigonometric barrier term is analyzed. Each main iteration of the algorithm consists of a feasibility step and several centrality steps, whose feasibility step is induced by a trigonometric kernel function. The complexity result coincides with the best result for infeasible interiorpoint methods for P matrix linear complementarity problem. Keywords: Linear complementarity problem, FullNewton step, Infeasible interiorpoint method, Kernel function, Polynomial complexity AMS Subject classification: 90C5, 90C. Introduction Since Karmarkar s landmark paper [4], interior point methods IPMs have became one of the most active research areas. They have been widely extended for solving linear optimization LO, linear complementarity problems LCPs and many other problems. Due to the fact that LCP is closely related to LO, several IPMs designed for LO have been extended to P LCP. Kojima et al. [] first proved the existence of the central path for P LCP and generalized the primaldual interiorpoint algorithm for LO to P LCP. The authors obtained polynomial iteration complexity for the algorithm, which is yet the best iteration bound for solving P LCP. Miao [4] gave a generalization of MizunoToddYe predictorcorrector algorithm [6] with Corresponding Author c 08 Azarbaijan Shahid Madani University. All rights reserved.
2 5 An infeasible interiorpoint method for the P LCP the P LCP assuming the existence of a strictly positive solution. Miao s algorithm uses the l neighborhood of the central path and has both polynomial complexity and quadratic convergence. Later Potra and Sheng [9] presented a new predictorcorrector algorithm for P LCPs from arbitrary positive starting points. Their algorithm is quadratically convergent for nondegenerate problems. Potra and Sheng [0] proposed a superlinearly convergent predictorcorrector method for P LCPs and improved the results in Miao [4]. Illés and Nagy [] proposed a version of the Mizuno ToddYe predictorcorrector interiorpoint algorithm for the P LCP and showed the polynomial convergence. In the above mentioned algorithms it is assumed that the starting point satisfies exactly the equality constraints and lies in the interior of the region defined by the inequality constraints. Such a staring point is called strictly feasible. All the points generated by the algorithms are also strictly feasible. However, in practice it may be very difficult to obtain feasible starting points. Numerical experiments have shown that it is possible to obtain good practical performance by using starting points that lie in the interior of the region defined by the inequality constraints but do not satisfy the equality constraints [7]. The points generated by the method will remain in the interior of the region defined by the inequality constraints but in general will not satisfy the equality constraints. These methods are referred as infeasible interiorpoint methods IIPMs, and feasibility is reached as optimality is approached. The first IIPM was proposed by Lustig [3]. Global convergence shown by Kojima et al. [0], whereas Zhang [5] and Mizuno [5] presented polynomial iteration complexity results for variants of this algorithm. In 006, Roos [] proposed a new IIPM for LO. It differs from the classical IIPMs e.g Kojima et al. [], Potra and Sheng [0], Lustig [3], Mizuno [5], Potra [8] and ect in that the new method uses only full steps, which has the advantage that no line searches are needed. Furthermore, the iteration bound of the algorithm matches the best known iteration bound for this type of algorithms. Some kernel functionbased version of the algorithm, were carried out by Liu and Sun [] to LO and Kheirfam [7, 8] to LCP and SCO. Recently, Kheirfam [5, 9] presented fullnewton step IIPMs for P horizontal linear complementarity problem HLCP and P LCP, and developed various analysis from existing methods. Motivated by the abovementioned works, we propose another extension of Roos algorithm to P LCP. The main iteration of the algorithm consists of a feasibility step and a few centering steps, whose feasibility step is induced by a kernel function with trigonometric barrier term. We used a normbased proximity to measure distance of the iterates from the central path, and derive the currently best known iteration bound for P LCPs. To our knowledge, this is the first infeasible interiorpoint algorithm for P LCP based on the kernel function with fullnewton step. The paper is organized as follows: In Sect. we recall basic concepts and the notion of the central path. We review some results that provide the local quadratic convergence of the fullnewton step. Sect. 3 contains an extension of Roos infeasible interiorpoint algorithm for P LCP, whose feasibility step is induced by a kernel function. In Sect. 4 we analyze the feasibility step, and then derive the iteration bound for the algorithm. In Sect. 5 are reported some numerical results. Finally, the concluding
3 B. Kheirfam, M. Haghighi 53 remarks are drawn in Sect. 6.. FullNewton Step Feasible IPM The P κlcp requires the computation of a vector pair x, s R n R n satisfying Mx + s = q, xs = 0, x, s 0, where q R n and M R n n is a P κmatrix. The class of P matrices was introduced by Kojima et al. [] and it contains many types of matrices encountered in practical applications. Let κ be a nonnegative number. A matrix M is called a P κmatrix iff it satisfies the following condition: + 4κ x i Mx i + x i Mx i 0, x R n, i I + i I where I + = {i : x i Mx i 0} and I = {i : x i Mx i < 0} are two index sets. The class of all P κmatrices is denoted by P κ, and the class P is defined by P = κ 0 P κ, i.e., M is a P matrix iff M P κ for some κ 0. Obviously, P 0 is the class of positive semidefinite matrices. The concept of the central path plays a critical role in the development of IPMs. Kojima et al. [] first proved the existence and uniqueness of the central path for P κlcp. Throughout the paper, we assume that P κlcp satisfies the interiorpoint condition IPC, i.e., there exists a pair x 0, s 0 > 0 such that s 0 = Mx 0 + q, which implies the existence of a solution for P κlcp []. The basic idea of the pathfollowing IPMs is to replace the second equation in, the socalled complementarity condition for P κlcp, by the relaxed equation xs = µe with µ > 0. Thus, we consider the system Mx + s = q, xs = µe, x, s 0. Since M is a P κmatrix and the IPC holds, the system has a unique solution for each µ > 0 cf. Lemma 4.3 in []. This solution is denoted as xµ, sµ and is called the µcenter of P κlcp. The set of µcenters gives a homotopy path, which is called the central path of P κlcp. If µ 0, then the limit of the central path exists and yields a solution for P κlcp Theorem 4.4 in []. A promising way to define a search direction is to follow Newton s approach and linearize the second equation in, which leads to the system M x s = 0, x s + s x = µe xs. 3 Since M is a P κmatrix, the system 3 uniquely defines x, s for any x > 0 and s > 0. For ease of analysis, we define xs v := µ, d x := v x x, d s := v s s. 4
4 54 An infeasible interiorpoint method for the P LCP This enables us to rewrite the system 3 as follows: Md x d s = 0, d x + d s = v v, 5 where M := DMD and D := diag x s. Since M is P κmatrix, it follows that M is also P κmatrix. Thus, the system 5 has a unique solution. The new search directions d x and d s are obtained by solving 5 so that x and s are computed via 4. The new iterate is obtained by taking a fullnewton step according to x + := x + x, s + := s + s. For the analysis of the algorithm, we define a normbased proximity measure as follows: δv := δx, s; µ := v v. 6 Here, we recall some results which are needed for the analysis of the algorithm. Lemma. Lemma II.6 in [3] Let δ := δx, s; µ. Then ρδ vi ρδ, i =,,..., n, where ρδ := δ + + δ. Lemma. Lemma 3 in [5] Let δ := δx, s; µ < is strictly feasible and δx +, s + ; µ + κδ + κδ.. Then, the fullnewton step +κ Corollary. If δ := δx, s; µ +κ, then δx +, s + ; µ + κδ, which shows the local quadratic convergence of the fullnewton step. 3. FullNewton Step IIPM In the case of an IIPM, we call the pair x, s an ɛsolution of P κlcp iff max{ s Mx q, x T s} ɛ. As usual for IIPMs, we assume that the initial iterate x 0, s 0 is as follows x 0, s 0 = ρ p e, ρ d e and µ 0 = ρ d ρ p, 7
5 B. Kheirfam, M. Haghighi 55 where ρ p and ρ d are positive numbers such that x ρ p, { s, ρ p Me, q } ρ d, 8 for some solution x, s. The initial value of the residual vector is denoted as r 0 := s 0 q Mx 0. In general, r 0 0, i.e., the initial iterate is not feasible. However, a sequence of perturbed problems is generated below in such a way that the initial iterate is strictly feasible for the first perturbed problem in the sequence. For this purpose, for any ν with 0 < ν, the perturbed problem pertaining to P κlcp is given by s q Mx = νr 0, x, s 0. P ν It is obvious that x, s = x 0, s 0 is a strictly feasible solution of P ν when ν =. This means that if ν =, then P ν satisfies the IPC. Lemma 3. Lemma 3. in [9] Let the problem is feasible and 0 < ν. Then, the perturbed problem P ν satisfies the IPC. Let the problem be feasible and 0 < ν. Then Lemma 3 implies that the problem P ν satisfies the IPC, and therefore its central path exists. This means that the system s q Mx = νr 0, xs = µe, x, s 0, 9 has a unique solution xµ, ν, sµ, ν, for every µ > 0 that is called a µcenter of the problem P ν. In the sequel, the parameters µ and ν always satisfy the relation µ = νµ 0 = νρ p ρ d. It is also worth noting that, according to 7, x 0 s 0 = ρ p ρ d e = µ 0 e. Hence x 0, s 0 is the µ 0 center of the perturbed problem P ν for ν =. In other words, xµ 0,, sµ 0, = x 0, s 0 and the algorithm can easily be started since we have the initial starting point that is exactly on the central path of P ν for ν =. The outline of one iteration of the algorithm is as follows. Suppose that for some ν 0, ] we have an iterate x, s which satisfies the feasibility condition, i.e., the first equation of the system 9 for µ = νµ 0, and such that δx, s; µ τ. This is certainly true at the start of the first iteration, because initially we have δx, s; µ = 0. Each main iteration of the algorithm consists of a feasibility step, a µupdate and a few centering steps. The feasibility step serves to get an iterate x f, s f that is strictly feasible for P ν + where ν + = θν with 0 < θ <, and belongs to the quadratic convergence region with respect to the µ + center of P ν + with µ + = θµ, i.e., δx f, s f ; µ + +κ. After the feasibility step, we perform a few centering steps in order to get iterates x +, s + which satisfy δx +, s + ; µ + τ.
6 56 An infeasible interiorpoint method for the P LCP 3.. The Feasibility Step Suppose that x, s is a strictly feasible solution for P ν. This means that x, s satisfies the first equation of 9. We need displacements f x and f s such that x f, s f := x + f x, s + f s 0 is feasible for P ν +, this implies that the first equation in the following system is satisfied M f x f s = θνr 0, x f s + s f x = µe xs. The system defines the feasibility iterates uniquely since the coefficients matrix of the resulting system is exactly the same as in the feasible case. We define the scaled search directions d f x := v f x x, df s := v f s, s where v is defined as in 4. The system can be expressed as follows: Md f x d f s = θνvs r 0, d f x + d f s = v v. 3 It is clear that the righthand side of the second equation in 3 coincides with the negative gradient of the logarithmic barrier function Φv = n i= v i log v i. This coincidence motivates a new feasibility step, which is defined by the following system: Md f x d f s = θνvs r 0, d f x + d f s = Ψv, 4 where Ψv is a kernel functionbased barrier function [6] as follows ψt := t + 4 π cotht, where ht = πt + t, t > 0. Since ψ t = t 4 π h t + cot ht = t 4 + t csc ht,
7 B. Kheirfam, M. Haghighi 57 the second equation in 4 can be written as follows d f x + d f s = 4e + v csc hv v. A solution of the system 4 returns d f x and d f s, and then f x and f s compute via. The new iterates are obtained by taking a full step, as given by 0. We conclude that after the feasibility step, we have iterate x f, s f that satisfies the first equation of P ν with ν replaced by ν +. In the analysis, we should also guarantee that x f and s f are positive and δx f, s f ; µ + + κ. 5 If this is satisfied then, by using Corollary, the required number of centering steps can easily be obtained. Starting at x f, s f, after k centering steps we will have the iterate x +, s + := x k, s k that is still feasible for P ν + and satisfies δx +, s + ; µ κδv k 4 [ 4 + κ + κδv k ] 4 k+ + κ δv f k k = + κδv f + κ k. + κ From this one easily deduces that δx +, s + ; µ + τ will hold after at most + log log = + log log τ + κ τ + κ 6 centering steps. 3.. Algorithm The steps of the algorithm are summarized as Algorithm. Algorithm : T he full Newton step IIP M. Input : accuracy parameter ɛ > 0; barrier update parameter θ, 0 < θ < ; threshold parameter 0 < τ <. begin x := ρ pe; s := ρ d e; µ := ρ pρ d ; ν = ; while max x T s, ν r 0 > ɛ x, s := x, s + f x, f s; µ and ν update : µ := θµ, ν := θν; while δx, s; µ > τ x, s := x, s + x, s; endwhile endwhile end.
8 58 An infeasible interiorpoint method for the P LCP 4. Analysis of the Algorithm Let x and s denote the iterates at the start of an iteration such that δx, s; µ τ. Recall that at the start of the first iteration this is certainly true, because δx 0, s 0 ; µ 0 = The Effect of the Feasibility Step Recall that the feasibility step generates new iterates x f, s f that satisfy P ν with ν = ν +, except possibly the positive conditions. A crucial element in the analysis is to show that after the feasibility step δx f, s f ; µ + +κ and xf and s f are positive. Note that, by using the second equation in 4 we have x f s f = xs v v + df xv + d f s = µ v + vd f x + d f s + d f xd f s = µ 4ve + v csc hv + d f xd f s. 7 The lemma below provides a sufficient condition for the strict feasibility of the feasibility step x f, s f. Lemma 4. The new iterate x f, s f is strictly feasible if and only if 4ve + v csc hv + d f xd f s > 0. Proof. We introduce a step length α with 0 α, and we define xα := x + α f x, sα := s + α f s. We hence have x0 = x, s0 = s, x = x f, s = s f and x0s0 = xs > 0. On the other hand, we have xαsα = µv + αd f xv + αd f s = µ v + αvd f x + d f s + α d f xd f s = µ v + αv 4e + v csc hv v + α d f xd f s > µ αv + 4αve + v csc hv + α 4ve + v csc hv = µ αv + 4α αve + v csc hv 0. Hence, none of the entries of xα and sα vanishes, for 0 < α. Since x0 and s0 are positive and xα and sα depend linearly on α, this implies that xα > 0 and sα > 0 for 0 < α. Hence, x and s are positive and this completes the proof.
9 B. Kheirfam, M. Haghighi 59 Lemma 5. For π t π, one has cost π π π t + 4 π π t. π Proof. For π t π, we have 0 π t π. Using the inequality cosz 4 π π z π π z, for 0 z π see Remark. in [], we get cost = cosπ t 4 π π π t π π π t. This implies the desired result. Lemma 6. For t > 0, one has + t csc ht 0. Proof. If 0 < t, then we have 0 < ht π. In this case, by using the inequality sinz 4 π z 4 π z for 0 < z π see Remark. in [], we obtain + t csc ht = + t sin ht + t 4 π ht 4 π h t = + t. If t >, to prove the statement, we need to show ft := + t sin ht 0. As π < ht < π, for t >, then we have f t = 4 t sinht πt cosht + t 4 sin 3. ht By using the following inequality sinz 4π π 3 z 3 π π z + π 4 z + 8 3π π
10 60 An infeasible interiorpoint method for the P LCP for π z π see Remark.6 in [], we obtain 4π t sinht t π 3 h 3 t + and from Lemma 5 we have π 4 ht + 8 3π π π h t π πt + πt + 8 3π = t + t, 8 π πt cosht πt π π ht + 4 π π π ht t + πt + π = πt + t. 9 Substituting 8 and 9 in f t, we get f t 4 πt 3 + π 5πt π π t 3π t 6 sin 3 ht = πt + π 4πt + 3π 8 + t 6 sin 3 ht which implies that ft is increasing for t >, i.e., ft f = 0. This completes the proof. 0, Lemma 7. For t > 0, the following inequality holds. + t csc ht t. t Proof. From Lemma 6 it suffices to prove + t csc ht t. t We consider two cases: If 0 < t, then we have 0 < ht π. Using sinz 3 π z 4 π z 3 for 0 < z π 3 see Remark.4 in [], we get 3 sin ht π ht 4 t 3 + 6t t π 3 h3 t = + t 6. The above inequality implies that + t csc ht = + t sin ht 4 + t 4 t3 + 6t t. 0
11 B. Kheirfam, M. Haghighi 6 Now, for 0 < t, we have 4 + t 4 t3 + 6t t t t = t t5 3t t t + 3t + t3 + 6t t 0. Using the above inequality, 0 and t t = t t for 0 < t, we obtain 4 + t 4 + t csc ht t3 + 6t t t t = t. t If t >, then we have π ht < π. In this case, by using the inequality sinz 4 π z 3 3 π z + 9 π z, for π z π see Remark.6 in [], we have sin ht which implies that 3 + t 3 t + t + 9t + t = 3t + 6t + t 6, + + t csc t4 ht = + t sin ht 3t + 6t. On the other hand, we have + t 4 3t + 6t t t = tt5 + 3t t t 3t + t3t + 6t 0. From the above inequality, and t t = t t for t, it follows that This completes the proof. + t csc ht + t4 3t + 6t t. t Lemma 8. For t > 0, one has t 4 + t csc ht + t t. t Proof. If 0 < t, then by using the inequality sinz 3 π z 4 π 3 z 3 for 0 < z π see Remark.4 in [], we have 4 + t csc ht t + t t t 4 + t 3 π ht 4 π 3 h 3 t = 4 + t 4 t 3 + 6t t t + t t + t t t t t 0,
12 6 An infeasible interiorpoint method for the P LCP where the last inequality follows from fact that the lefthand side of the inequality is monotonically increasing for t. If t >, then t 4 +t csc ht 0 and t t 0. In this case, we have t t csc ht = t + cot + t ht t 4 + t + t The above two inequalities prove the desired result. t. t Lemma 9. For t > 0, one has + t csc ht 4 π t + π t 36. Proof. As 0 < ht < π, for t > 0, then by using the inequality cscz > z + π z 6π z for 0 < z < π see Theorem in [], we have + t csc ht = + t + t ht + + t + πt π ht 6π h t πt + t 6 + t = 4 π t + tπ t t 4 π t + π t 36, where the last inequality follows from the fact that gt := π t ++ +t is nonnegative and decreasing for t > 0, so gt lim gt = π. Therefore, the proof is complete. t 4 In the sequel, we denote w := d f x + d f s, which implies d f xd f s d f x d f s df x + d f s = w, d f xi df si df xi + d f si w, i =,,, n. 3 In what follows, we denote δx f, s f ; µ + also briefly by δv f, where v f := x f s f µ +.
13 B. Kheirfam, M. Haghighi 63 Lemma 0. Let the iterate x f, s f be strictly feasible. Then, we have v f min 3 0ρδw 5 θρδ, where v f min = min i n {vf i }. Proof. Using 7, after dividing both sides by µ + = θµ, we have v f = xf s f µ + = 4ve + v csc hv + d f xd f s. 4 θ Therefore, using 3, Lemma 9 and Lemma, we have v f min = θ min 4v i + v i csc hv i + d f xi i df si min 4vi + v i csc hv i + min d f xi θ df si i i min 4vi + v i csc hv i w θ i 4 min θ i π v i + π v i w 36 4 θ π + π v max 36 v min w 4 θ π ρδ + π 36ρδ w 3 θ 5ρδ w = 3 0ρδw. θ 5ρδ This implies the desired result. Lemma. Proof. The following inequality holds. e v f θ n + δv + w. θ Using 4, the triangular inequality, and Lemma 7, we have e v f = e 4ve + v csc hv + d f xd f s θ = θe 4ve + v csc hv d f θ xd f s θ n + e 4ve + v csc hv + w θ θ n + θ v v + w θ n + δv + w. θ
14 64 An infeasible interiorpoint method for the P LCP This completes the proof. Lemma. If 4ve + v csc hv + d f xd f s > 0, then δv f 5ρδ θ n + δ + w θ 3 0ρδw. Proof. We may write, using 6, δv f = v f v f = v f e v f v f e v f = e v f. v f min Using the obtained bounds in Lemma 0 and Lemma the lemma follows. In what follows, we want to choose 0 < θ <, as large as possible, and such that x f, s f lies in the quadratic convergence neighborhood with respect to the µ + center of P ν +, i.e., δv f +κ. According to Lemma, it suffices to have At this stage, we choose 5ρδ θ n + δ + w θ 3 0ρδw + κ. 5 τ = 6 + κ, θ 5n + κ. 6 Since the lefthand side of 5 is monotonically increasing with respect to w, then, for δ τ, one can verify that for w 3 + κ 7 the inequality 5 holds. 4.. Upper Bound for w We start by finding some bounds for the unique solution of the linear system 4. Lemma 3. Corollary. in [3] Let x, s, a, r be four ndimensional vectors with x > 0 and s > 0, and let M R n n be a P κmatrix. Then the solution u, v of the linear system Mu v = b, Su + Xv = a, 8
15 B. Kheirfam, M. Haghighi 65 satisfies the following inequality: Du + D v ã + κ c + b + b ã + b + κ c, where D = X S, ã = XS a, b = D b, c = ã + b. Comparing system 8 with the system 4, which can be expressed equivalently as follows: M f x f s = θνr 0, s f x + x f s = µv Ψv 9 and considering u, v = f x, f s, b = θνr 0 and a = µv Ψv in the system 8, we obtain D f x + D f s XS µv Ψv + θνd r 0 +κ XS µv Ψv + θνd r 0 + θνd r 0 XS µv Ψv + θνd r 0 + κ XS µv Ψv + θνd r 0 = µ Ψv + θ ν D r 0 + κ θνd r 0 µ Ψv +θν D r 0 µ Ψv + θ ν D r 0 + κ θνd r 0 µ Ψv. Let x, s be the optimal solution of that satisfies 8 and suppose that the algorithm starts with x 0, s 0 = ρ p e, ρ d e. Then, we have x x 0 ρ p e, s s 0 ρ d e. 30 Also, by using 8 and 30, we get D r 0 x xr 0 = s r0 = xr 0 xs µvmin µvmin S 0 r 0 s 0 x + ρ p Me + q ρ d x 3ρ d x. 3 µvmin ρ d ρ d µvmin Lemma 4. Let δ := δv. Then, one has Ψv + ρδδ.
16 66 An infeasible interiorpoint method for the P LCP Proof. Using Lemma 8 and Lemma, we have Ψv = v 4e + v csc hv e + e + v v v = + + ρδ This completes the proof of lemma. Using 3 and Lemma 4, we get δ = + ρδδ. v v v δ v min θνd r 0 µ Ψv θν D r 0 + µ Ψv 3θνρ d x µvmin + µδ + ρδ. 3 Therefore, using bounds in 3, 3 and the equations D f s = µd f s D f x = µd f x, we obtain and d f x + d f s δ + ρδ + 8θ ν ρ d µ vmin x 3θνρd + κ x + δ + ρδ µv min + 6θνρ d µv min x δ + ρδ + 9θ ν ρ d µ vmin x + κ 3θνρd x + δ + ρδ µv min. 33 Lemma 5. Lemma in [5] Let x, s be feasible for the perturbed problem P ν and let x 0, s 0 and x, s be as defined in 7 and 8, respectively. Then, x + 4κ + ρδ nρ p. Substituting the bounds of x and v min into 33 and using µ = νρ p ρ p, we obtain d f x + d f s δ + ρδ κ θ n ρδ + ρδ +κ 3 + 4κnθρδ + ρδ + δ + ρδ κnθρδ + ρδ δ + ρδ κ θ n ρδ + ρδ + κ 3 + 4κnθρδ + ρδ + δ + ρδ. 34
17 B. Kheirfam, M. Haghighi Fixing θ and Complexity Analysis Since δ τ and the righthand side of 34 is monotonically increasing in δ, so it follows that d f x + d f s τ + ρτ κ θ n ρτ + ρτ +κ 3 + 4κnθρτ + ρτ + τ + ρτ κnθρτ + ρτ τ + ρτ κ θ n ρτ + ρτ + κ 3 + 4κnθρτ + ρτ + τ + ρτ. Using 6, we have found that δv f +κ holds if the inequality 7 is satisfied. Then, by the above inequality, 7 holds if τ + ρτ κ θ n ρτ + ρτ +κ 3 + 4κnθρτ + ρτ + τ + ρτ κnθρτ + ρτ τ + ρτ κ θ n ρτ + ρτ + κ 3 + 4κnθρτ + ρτ 4 + τ + ρτ 9 + κ. One may easily verify that, by some elementary calculation, the above inequality is satisfied if τ = 6 + κ, θ = 33n + κ 3, 35 which is in agreement with 6. Note that we have found that if at the start of an iteration, δx, s; µ τ, then after the feasibility step, δx f, s f ; µ + +κ. According to 6, at most + log log = τ + κ centering steps are needed to get iterates x +, s + such that δx +, s + ; µ + τ. In each main iteration both the duality gap and the norm of the residual vector are reduced by the factor θ. Hence, the total number of main iterations is bounded above by θ log max{x0 T s 0, r 0 }. ɛ
18 68 An infeasible interiorpoint method for the P LCP So, due to 35 the total number of inner iterations is bounded above by 99n + κ 3 log max{x0 T s 0, r 0 }. ɛ In the following we state our main result without further proof. Theorem. If the problem has a solution x, s such that x ρ p and s ρ d, then after at most 99n + κ 3 log max{x0 T s 0, r 0 }, ɛ inner iterations, the algorithm finds an ɛoptimal solution of. 5. Numerical results We test our algorithm on some instances. We write simple MATLAB codes for our Algorithm and the algorithm of Zhang et al. [4]. In our experiments, we choose x = ρ p e, s = ρ d e and µ = ρ p ρ d as the starting data. In order to guarantee the convergence property of these algorithms, we take the parameters τ and θ as τ = 6+κ, θ = 33n+κ and τ = 3 6, θ = 33n respectively. We terminate the algorithms if xt s ɛ = 0 4. Table shows the required number of iterations for P 0LCP problems corresponding to positive semidefinite matrices, with various sizes, as follows M,n =...., M 5 6 6,n = n 3 Table problem x ρ p s ρ d ρ p Me ρ d Iter. Algor. Algor. [4] M, M, M, M, M, M, M, One can easily see that the assumptions in the theoretical results are satisfied, i.e., x ρ p, { s, ρ p Me } ρ d and µ = ρ p ρ d. Based on the obtained numerical results, as is shown in Table, our proposed algorithm appears to have a competitive edge over the algorithm in [4].
19 B. Kheirfam, M. Haghighi Concluding Remarks In this paper, we have proposed a fullnewton step infeasible interiorpoint method based on a kernel function for the P matrix LCP. The iteration bound coincides with the currently best known one for IIPM. For further research, this algorithm may be possible extended to the Cartesian P matrix LCP over symmetric cones. Acknowledgements The authors are very grateful to the editors and the anonymous referees for their useful comments and suggestions which helped to improve the presentation of this paper. References [] Ch.P. Chen and F. Qi, Inequalities of some trigonometric functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat , [] T. Illés and M. Nagy, A mizunotoddye type predictorcorrector algorithm for sufficient linear complementarity problems, European J. Oper. Res , 097. [3] J. Ji and F.A. Potra, An infeasibleinteriorpoint method for the p matrix lcp, Rev. Anal. Numér. Théor. Approx. XXVII 998, no., [4] N.K. Karmarkar, A new polynomialtime algorithm for linear programming, Combinatorica 4 984, [5] B. Kheirfam, Fullnewton step infeasible interiorpoint algorithm for sufficient linear complementarity problems, TOP Revised. [6], Primaldual interiorpoint algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier term, Numer. Algorithms 6 0, no. 4, [7], A full nesterovtodd step infeasible interiorpoint algorithm for symmetric optimization based on a specific kernel function, Numer. Algebra Contr. Optim. 3 03, no. 4, [8], A fullnewton step infeasible interiorpoint algorithm for linear complementarity problems based on a kernel function, Algor. Oper. Res. 7 03, [9], A new complexity analysis for fullnewton step infeasible interiorpoint algorithm for horizontal linear complementarity problems, J. Optim. Theory Appl. 6 04, no. 3, [0] M. Kojima, N. Megiddo, and S. Mizuno, A primaldual infeasibleinteriorpoint algorithm for linear programming, Math. Program , no. 3,
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