A full-newton step infeasible interior-point algorithm for linear complementarity problems based on a kernel function

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1 Algorithmic Operations Research Vol7 03) 03 0 A full-newton step infeasible interior-point algorithm for linear complementarity problems based on a kernel function B Kheirfam a a Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, IR Iran Abstract In this paper, we first present a brief infeasible interior-point method with full-newton step for solving linear complementarity problem LCP) The main iteration consists of a feasibility step and several centrality steps First we present a full Newton step infeasible interior-point algorithm based on the classic logarithmical barrier function After that a specific kernel function is introduced Then the feasibility step is induced by this kernel function instead of the classic logarithmical barrier function The results of complexity coincides with the best bound known for infeasible interior-point methods for LCP Key words: Interior-point methods, Linear complementarity problem, Full-Newton step, Complexity, Kernel functions Introduction The linear complementarity problem LCP) is to find a vector x,s) R n such that P) s = Mx+q, x,s 0, xs = 0, where q R n, and M R n n is positive semidefinite It is well known that many important problems in economics, control and game theory can be formulated as LCP For a comprehensive study, the reader is referred to [] There are a variety of solution approaches for LCP which have been studied extensively Among them, the interior-point methods IPMs) gained much attention for LCP than other methods For a comprehensive learning about IPMs, we refer to [,8,0] A close look at the IPM litterarure tells us that the first IPM for LCP was due to Kojima et al [5] and their algorithm originated from the primal-dual IPMs for linear optimization LO) Later on, Kojima et al [6] set up a framework of IPMs for tracing the central path of a class of LCPs One may distinguish between feasible IPMs and infeasible IPMs IIPMs) Feasible IPMs start with a strictly feasible interior point and maintain feasibility during the solution process IIPMs start with an arbitrary positive point, and feasibilty is reached as optimality is approached Potra [4] analyzed a generalization to LCPs of the Mizuno-Todd-Ye predictor corrector method [0] for infeasible starting points, B Kheirfam [bkheirfam@azarunivedu] and other literature about IIPMs for LCPs can be found in [4,9] In Roos [7], a full Newton-step infeasible interior-point algorithm for LO was proposed and he also proved that the complexity of the algorithm coincides with the best known iteration bound for IIPMs Some extensions on LO were carried out by Liu and Sun[7], Mansouri and Roos [9], on LCP by Mansouri et al [8], and on semidefinite optimization by Kheirfam [3,4] Recently, Peng et al [ 3] proposed a new variant of IPMs based on self-regular kernel functions for LO and proved the best complexity for large-update methods with some specific self-regular kernel function Motivated by this series of work, we consider full Newton-step IIPM for LCP based on a specific kernel function In our algorithm, we use a barrier function based on the simple kernel function φt) = t ), ) instead of classical logarithmic barrier function to calculate the search directions The complexity result shows that the full Newton-step IIPM for LCP based on this kernel function enjoys the best known iteration bound for LCP The paper is organized as follows: In the next Section, we present briefly primal-dual infeasible interior-point algorithm with a proximity of iterates x,s) to the µ- center of the perturbed problem In Section 3, we give some technically results Section 4 is devoted to the analysis of the new feasibility step, which is the main c 03 Preeminent Academic Facets Inc, Canada Online version: All rights reserved

2 04 B Kheirfam A full-newton step infeasible interior-point algorithm part of the paper and then, iteration bound is derived Finally, we end the paper in section 5 where v = xs µ The statement of the algorithm Without loss of generality [6], we assume that LCP satisfies the interior point condition IPC), ie, there exists a vector x 0,s 0 ) R n such that; s 0 = Mx 0 +q, x 0 > 0, s 0 > 0 We assume that ρ p and ρ d are such that x ρ p, max{ s,ρ p Me, q } ρ d, for some optimal solution x,s ) of the problem P) and start the algorithm with x 0 = ρ p e, s 0 = ρ d e, µ 0 = ρ p ρ d For any ν with 0 < ν, we consider the perturbed problem where, P ν ) s Mx q = νr 0, x,s) 0, r 0 = s 0 Mx 0 q Note that if ν =, then x,s) = x 0,s 0 ) yields a strictly feasible solution of P ν ) We conclude that if ν =, then P ν ) satisfies the IPC More generally, one has the following result see [8], Lemma 4) Lemma If the original problem P) is feasible then the perturbed problem P ν ) satisfies the IPC for each 0 < ν Assuming that P) is feasible, it follows from Lemma that the problem P ν ) satisfies the IPC, for each 0 < ν Then, its central path exists, meaning that the system s Mx q = νr 0, x 0,s 0, ) xs = µe, 3) has a unique solution, for anyµ > 0 For0 < ν and µ = νµ 0, we denote this unique solution in the sequel by xν),sν)), where is the µ-center of P ν ) In this notation, if we takeν =, thenx),s)) = x 0,s 0 ) We measure the proximity of iterate x,s) to the µ- center of the perturbed problem P ν ) by the quantity δx,s;µ) = v v, 4) Initially, we have δx,s;µ) = 0 In the sequel, we assume that at start of each iteration, δx,s;µ) τ with τ > 0 This certainly holds at the start of the first iteration We now describe one main iteration of the algorithm given by Mansouri et al [8] The algorithm begins with an infeasible interior pointx,s) such that x,s) is feasible for the perturbed problemp ν ), x T s n+δ )n and δx,s;µ) τ, where µ = νµ 0 Each main iteration consists of one so-called feasibility step, a µ- update, and a few centering steps First we find a new point x f,s f ) which is feasible for the perturbed problem with ν + := θ)ν Then µ is decreased to µ + := )µ Generally, there is no guarantee that δx f,s f ;µ + ) τ So a limited number of centering steps are applied to produce a new point x +,s + ) such that δx +,s + ;µ + ) τ, where µ + = ν + µ 0 This process is repeated until the algorithm terminates We now describe the search directions used in the feasibility and centering steps For the feasibility step, the search direction f x, f s) defined by the system M f x f s = θνr 0, 5) s f x+x f s = µe xs, 6) where, θ 0,) If x,s) is feasible for the perturbed problem P ν ), then after the feasibility step the iterates satisfies the affine equation in ), with ν = ν + Assuming thatδx,s;µ) τ holds before the step, and by taking θ small enough, it can be guaranteed that after the step the iterates x f = x+ f x, s f = s+ f s, 7) are positive and δx f,s f ;µ + ) In the centering step, starting at the iterate x,s) = x f,s f ) and targeting at the µ-center, the search direction x, s) is the usual primal-dual Newton direction, defined by s = M x, s x+x s = µe xs 8) Denoting the iterate after a centering step by x + and s +, we recall the following result from [8] Lemma [8], Lemma 35, corollary 36 ) If δ := δx,s;µ) <, then x + and s + are positive, x + ) T s + n + δ )µ Moreover, if δ, then δx +,s + ;µ) δ

3 B Kheirfam Algorithmic Operations Research Vol7 03) Define d f x := v f x x, df x := v f s, 9) s where, v defined as 4) By using 9), we can rewrite 5)-6) as follows MS Xd f x d f s = θνvs r 0, d f x +df s = v v, 0) where X = diagx), S = diags) Note that the right-hand-side in the second equation of system 0), v v, equals the negative gradient of the classical logarithmic barrier function Ψv) := ψv i ), v i = whose kernel function is ψt) = t ) logt xi s i µ, The main contribution of this paper is a modification of the feasibility step We present a slightly algorithm, obtained by changing the definition of the feasibility step via replacing the second equation of 0) by d f x +df s = φv), where the kernel function of φv) defined as ) Therefore, the system of the new feasibility step becomes MS Xd f x df s = θνvs r 0, d f x +df s = φv) Since φ t) = t, the second equation in the system can be written as d f x +df s = e v ) We define σx,s;µ) := σv) := e v ) It is obvious that σv) = 0 if and only if v = e, thus σv) is also a suitable proximity We now give a more formal description of Algorithm below Algorithm :Primal Dual InfeasibleIPM Input : Accuracy parameterǫ > 0; barrier update parameter θ, 0 < θ < ; feasible x 0,s 0 ) with x 0 ) T s 0 = nµ, δx 0,s 0 ;µ) < τ = begin x := x 0 ; s := s 0 ; µ := µ 0 ; while maxnµ, r ) > ǫ do begin feasibility step :x,s) := x,s)+ f x, f s); µ update :µ := )µ; centering steps : while δx,s;µ) τ do begin x,s) := x,s)+ x, s); end end end 3 Technical results We now give some lemmas which are used in the analysis later Lemma 3 Lemma 3 in []) Suppose that ψx) = ψ x)+ψ x), bothψ x) andψ x) are strictly monotone increasing in a given interval The roots ofψ x) = 0 and ψ x) = 0 are x and x respectively Then the root x of ψx) = 0 satisfies that x min{x,x } Lemma 4 [9], Lemma A) For i =,,,n, let f i : R + R denote a convex function Then, for any nonzero vector z R n +, the following inequality f i z i ) e T z z j f j e T z)+ ) f i 0) i j holds Lemma 5 Lemma 56 in [8]) The iteratesx f,s f ) are certainly strictly feasible if qδ) v i qδ), i =,,,n 3) where qδ) = δ + δ +

4 06 B Kheirfam A full-newton step infeasible interior-point algorithm Lemma 6 Lemma 58 in [8]) Letx,s) be feasible for the perturbed problem P ν ) and x 0,s 0 ) = ρ p e,ρ d e) Then x +qδ) ) nρ p, where, qδ) is defined in Lemma 5 The following lemma shows the effect on the proximity measure if v is replaced by ṽ := v Lemma 7 Let x,s) be a primal-dual Newton step and µ > 0 such that x T s n + δ )µ Moreover let δv) := δx,s;µ) and ṽ := v Then δṽ) )δ + θ θ) nn+δ ) Proof By using v n+δ and Holder inequality, one has v = v i n vi) nn+δ ) By following definition δv) and the clear inequality, t t t t, for t > 0 we have δṽ) = v v = v v ) θv = ) v v + θ v θ v ) v Tv )δv) + θ nn+δ ) ) +θ v n )δv) + θ nn+δ ) nn+δ ) +θ ) n )δ + θ θ) nn+δ ) 4 Analysis Letx,s) denote the iterate at start of an iteration with x T s µn+δ ) and δx,s;µ) τ Recall that at the start of the first iteration this certainly holds, because δx,s;µ) = 0 4 Effect of the feasibility step According to Lemma, we need to show that δx f,s f ;µ + ) after the feasibility step, that is, that the new iterates are within the region, where the Newton process targeting at the µ + -center of P ν +) is quadratically convergent Using 9), ) andxs = µv, we obtain x f s f = xs+s f x+x f s)+ f x f s = µv +µvd f x +df s )+µdf x df s = µv +d f xd f s) 4) The next lemma gives conditions for strict feasibility of the full Newton-step Lemma 8 The iterates x f,s f ) are strictly feasible if and only if v +d f xd f s > 0 Proof Note that if x f and s f are positive then 4) makes clear that v + d f xd f s > 0 For the proof of the converse, introduce a step lengthαwith0 α and define x α = x+α f x, s α = s+α f s We thus have x 0 = x,x = x f,s 0 = s and s = s f Note that x 0 s 0 = xs > 0 We may write x α s α =x+α f x)s+α f s) =xs+αs f x+x f s)+α f x f s 5) From 9), we deduce f x f s = µd f xd f s Using this and x f s + s f x = µvd f x + df s ) = µve v) and µv = xs, we obtain x α s α = µ α)v +αv +αd f xd f s) ) 6) If v + d f x df s > 0, then df x df s > v Substituting into 6), we get x α s α > µ α)v +αv αv ) = µ α)v +αv) Since µ α)v +αv) 0, it follows that x α s α > 0 for 0 α Hence, none of the entries of x α and s α vanish for 0 α Since x 0 = x > 0 and s 0 = s > 0, and x α and s α depend linearly on α, this implies that x α > 0 and s α > 0 for 0 α Hence x and s must be positive, and the proof is complete In the sequel, we denote w i := w i v) := d f x i + d f s i,

5 B Kheirfam Algorithmic Operations Research Vol7 03) and w := wv) := w,w,,w n ) These imply, by Cauchy-Schwartz inequality, d f x )T d f s df x df s df x + d f s ) w, d f x i d f s i = d f x i d f s i d f x i + d f s i ) wi w, i =,,,n Theorem 9 Ifv+d f xd f s > 0 and qδ)w > 0, then δv f ) )δ + w + )w qδ) w qδ) + θ θ) nn+δ ) Proof Using 4), after division of both sides byµ + = )µ we obtain v f ) = µ ) v +d f xd f s µ + = v +df x df s Hence δv f ) = Define = ) v f i ) +v f i ) vi +d f x i d f s i vi +w i f i w i) = v i +w i + ) v i +d f x i d f s i + v i w i ) + v i wi, i =,,,n Using Lemma 5 and the hypothesis qδ)w > 0, we obtain v i w i > 0 This implies that f i wi ) is convex in w i Therefore, by Lemma 4, δv f ) f j wj) w wj f j w ) + ) f i 0) i j = w wj [ +w + v j w ) + v i i j + ] ) v i Using Lemma 7, we obtain v i + ) = v i i j vj + )δ + θ θ) vj + Therefore, [ δv f ) w wj +w ] + ) + w w j vi + ) v i ) nn+δ ) ) )δ + θ θ) nn+δ )) [ = w w j +)δ + θ θ) = w n +) wj + w ) w + w nn+δ ) ) w +)δ + θ θ) nn+δ ) w qδ) +)w w qδ) +)δ + θ θ) nn+δ ) This completes the proof We conclude this section by presenting a value that we do not allow w to exceed Needing δv f ), it follows from Lemma 9 that it is sufficient to have )δ + w + )w qδ) w qδ) + θ θ) nn+δ ) Now, ]

6 08 B Kheirfam A full-newton step infeasible interior-point algorithm ψt)= t + )qδ) t +)δ qδ)t + θ θ) nn+δ ), and ψ t) = t +)δ + θ θ) nn+δ ), qδ) ψ t) = )t tqδ) Note thatψt) = ψ t)+ψ t), and that bothψ t) and ψ t) are monotonically increasing in t By Lemma 3, the roott ofψt) = 0 satisfiest min{t,t }, where t and t are the roots of ψ t) = 0 and ψ t) = 0, respectively Since ψ t ) = 0, t = )δ θ θ) nn+δ )), and from ψ t ) = 0, At this stage, we choose t = 4)qδ) +qδ) τ = 8, θ = α 8n, α Then for n and δ τ, it can easily be verified that t 5 3 α + δ 8 n + α n+δ 3 n , 7) t 4 3 8) Using 7) and the assumption that qδ)w > 0, it is easily verified that if w min{ 5 56, 4 3, 4 9 } = 5 56, 9) then δv f ) 4 Upper bound for d f x + d f s In this section, we obtain an upper bound for d f x + d f s, which enables us to find a default value for θ We consider the following system: MS Xd f x df s = θνvs r 0 d f x +d f s = e v, 0) wherex = diagx) ands = diags) By eliminating d f s from 0), we have d f x = I +MS X ) e v +θνvs r 0) ) Since I +MS X is positive definite, it follows d f x e v +θνvs r 0 ) Hence, by using 0), Cauchy-Schwartz inequality and positive semidefiniteness of MS X we have d f x + d f s = d f x +d f s d f x) T d f s = e v Using ), we get d f x + d f s d f x )T MS Xd f x θνvs r 0 ) = e v d f x )T MS Xd f x +θνdf x )T vs r 0 e v + d f x θνvs r 0 e v + e v +θνvs r 0) θνvs r 0 e v + e v + θνvs r 0 ) θνvs r 0 δ 3θ ) 3θ + δ + x x ρ p v min ρ p v min By using Lemmas 5 and 6, we obtain d f x + d f s δ +6nθ δ +3nθqδ) +qδ) )) qδ) +qδ) ) 43 Value for θ At this stage, we choose τ = 8 3) Since δ τ = 8 and qδ) is monotonically increasing in δ, d f x + d f s δ +6nθ δ +3nθqδ) +qδ) )) qδ) +qδ) ) 8 ) +6nθ 8 +3nθq 8 ) +q 8 ))) q 8 ) +q 8 )) ) nθ 09nθ Using θ = α, the above inequality reads 8n

7 B Kheirfam Algorithmic Operations Research Vol7 03) d f x + d f s +08+3α)6α 4) 3 From 9), we know that w 5 56 is needed to guarantee that δv f ) By 4), this holds if 08+ 3α)6α 036 This means if we take α = 05, 5) then it is guaranteed that δv f ) 44 Complexity analysis We have seen that if the iterates satisfyδx,s;µ) τ at the start of an iteration, withτ as defined in 3), then after the feasibility step, with θ = α and α as defined 8n in 5), the iterates satisfy δx f,s f ;µ + ) After the feasibility step, we preform some centering steps in order to make the iterate x +,s + ) that satisfies δx +,s + ;µ + ) τ, where τ is smaller than This process is repeated until the maximum of the norm of the residual and x + ) T s + is less than ǫ The next theorem gives an upper bound for the total number of iterations, which this bound coincides with the currently best bound of infeasible IPMs for LCP except a factor [5,6] Theorem 0 If P) has optimal solution x,s ) such that x ρ p and s ρ d, then after at most 8nlog max{x0 ) T s 0, r 0 }, ǫ iterations the algorithm finds an ǫ-optimal solution of P) Proof Let k denote the number of centering steps By Lemma, after k centering steps, the iterate x +,s + ) is still feasible for P ν +) and satisfies δx +,s + ;µ + ) ) k From this inequality, one can easily deduces that δx +,s + ;µ + ) τ will hold after at most log log τ ) = log log 64) = 3, 6) centering steps So each iteration consists of at most 4 so-called inner iterations, in each of which we need to compute a new search direction In each main iteration both the value of x T s and the norm of residual 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 } ǫ By multiplying the number of inner iterations by the number of main iterations, the total number of iterations is bounded above by 4 θ log max{x0 ) T s 0, r 0 }, ǫ whereθ = 3n, by 5), and hence we obtain the upper bound 8nlog max{x0 ) T s 0, r 0 } ǫ 5 Conclusion In this paper we extended the full Newton-step infeasible interior-point algorithm to LCP After that we used a new kernel function to induce the feasibility step and we analyzed the algorithm based on this kernel function The same results of complexity are obtained References [] Cottle, R W, Pang, J S and Stone, R E, The linear complementarity problem, Academic Press, San Diego, CA, 99) [] Karmarkar, N, New polynomial-time algorithm for linear programming, Combinattorica, 4 984) [3] Kheirfam, B, Simplified infeasible interior-point algorithm for SDO using full Nesterov-Todd step, Numer Algorithms, 59 0) [4] Kheirfam, B, A full NT-step infeasible interior-point algorithm for semidefinite optimization based on a selfregular proximity, ANZIAM J, 53, 0) [5] Kojima, M, Mizuno, S and Yoshise, A, A polynomialtime algorithm for a class of linear complementarity problems, Math Program, ) -6 [6] Kojima, M, Megiddo,N and Noma, T, A unified approach to interior point algorithms for linear complementarity problems, Lecture Notes in Computer and Science, Springer, [7] Liu, Z, Sun, W and Tian, F, A full-newton step infeasible interior-point algorithm for linear programming based on a kernel function, Appl Math Optim, ) 37-5

8 0 B Kheirfam A full-newton step infeasible interior-point algorithm [8] Mansouri, H Zangiabadi, M and Pirhaji, M, A full- Newton step On) infeasible-interior-point algorithm for linear complementarity problems, Nonlinear Analysis: Real World Applications, Article in Press [9] Mansouri, H, and Roos, C, Simplified OnL) infeasible interior-point algorithm for linear optimization using full- Newton step, Optim Methods Softw, 007) [0] Mizuno, S,Todd, MJ and Ye, Y, On adaptivestep primal-dual interior-point algorithms for linear programming, Math Oper Res, 8 993) [] Peng, J, Roos, C, and Terlaky, T, Self-regular functions and new search directions for linear and semidefinite optimization, Math Program, 93 00) 9-7 [] Peng, J, Roos, C and Terlaky, T, Primal-dual interiorpoint methods for second-order conic optimization based on self-regular proximities, SIAM J Optim, 3 00) [3] Peng, J, Roos, C and Terlaky, T, Self-Regularity, A new paradigm for primal-dual interior-point algorithms, Princeton University Press, 00 [4] Potra, FA, An infeasible-interior-point predictorcorrector algorithm for linear programming, SIAM J Optim, 6 996) 9-3 Received 9-0-0; accepted --03 [5] Potra, FA, An OnL) infeasible-interior-point algorithm for LCP with quadratic convergence, Ann Oper Res, 6 996) 8-0 [6] Sheng, R and Potra, FA, A quadratically convergence infeasible-interior-point algorithm for LCP with polynomial complexity, SIAM J Optim, 7 997) [7] Roos, C, A full-newton step OnL) infeasible interiorpoint algorithm for linear optimization, SIAM J Optim, 6 006) 0-36 [8] Roos, C, Terlaky, T and Vial, J-Ph, Theory and Algorithms for Linear Optimization An Interior-Point Approach, John Wiley and Sons, Chichester, UK, 997, nd Ed, Springer, 006 [9] Wright, S, An infeasible-interior-point algorithm for linear complementarity problems, Math Program, ) 9-5 [0] Wright, S J, Primal-dual interior-point methods, SIAM, 997 [] Zhongyi, L and Wenyu, S, An infeasible interior-point algorithm with full-newton step for linear optimization, Numer Algorithms, ) 73-88

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