Abstract
In this paper, we present a full-Newton step interior-point method for solving monotone Weighted Linear Complementarity Problem. We use the technique of algebraic equivalent transformation (AET) of the nonlinear equation of the system which defines the central path. The AET is based on the square root function which plays an important role in computing the new search directions. The algorithm uses only full-Newton steps at each iteration, and hence, line searches are no longer needed. We prove that the algorithm has a quadratic rate of convergence to the target point on the central path. The obtained iteration bound coincides with the best known iteration bound for these types of problems.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The Weighted Linear Complementarity Problem (WLCP) is a generalization of the linear complementarity problem in which the zero on the right side of the complementarity equation is replaced by a nonnegative weight vector. The WLCP is important because a wide range of problems from sciences, engineering and economics can be modeled as a WLCP. In [13], Potra has shown that the quadratic programming and weighted centering problem, which is a generalization of the linear programming and weighted centering problem introduced by Anstreicher [3], can be formulated as a special case of monotone WLCP. Then, he defined a smooth central path for WLCP and proposed two interior-point methods (IPMs) for solving the WLCP, both of which follow the smooth central path. Potra [14] extended the notion of sufficiency for linear complementarity problem (LCP) introduced by Cottle et al. [7] to the WLCP. He also showed that if a sufficient WLCP is strictly feasible, then it is solvable. The author uses the notion of the central path of the WLCP introduced in [13] and proposes a path-following IPM. Tang and Zhou [17] introduced a modified damped Gauss–Newton method for nonmonotone WLCP. Tang and Zhang [16] proposed a nonmonotone smoothing Newton algorithm for solving general WCPs. Tang and Zhou [18] proposed a nonmonotone Levenberg–Marquardt type method for solving the weighted nonlinear complementarity problem (WNCP) and gave its global convergence without any additional condition. Zhang [22] presented a new one-step smoothing Newton method for solving the monotone WLCP needing only to solve one linear system of equations and performing only one line search per iteration.
Full-Newton step IPMs for solving linear optimization (LO) were initiated by Roos et al. [15]. These methods have the advantage that no line searches are needed during the solution process to update the iterates. Various full-Newton step algorithms have been presented in the literature to solve different formulations of LCPs. In 2003, Darvay introduced a new technique for finding search directions for LO [8]. The key idea of this approach consists of considering an algebraic equivalent transformation on the system which defines the central path. That is, a continuously differentiable, invertible, monotone increasing function \(\psi :(\xi , \infty )\rightarrow {\textbf{R}}\), with \(0\le \xi <1\) is applied on both sides of the nonlinear equations of the central path. The new search directions are determined by applying Newton’s method to the resulting system. Then, the author presented a feasible IPM with full-Newton steps for LO. Infeasible IPMs (IIPMs) for LO and symmetric optimization (SO) based on this technique are proposed in [2, 9]. The AET technique extended to other areas, such as LCPs [1, 10, 12], semidefinite optimization [19], Second-order cone optimization (SOCO) [20] and SO [21]. In [11], an adaptive full-Newton step IIPM for solving sufficient horizontal LCP is presented and sufficient conditions are given for the superlinear convergence of the sequence of iterates. Recently, Asadi et al. [4] extended the full-Newton step IPM to the monotone WLCP and established the feasibility of the full steps and a quadratic rate of convergence to the target points on the central path. In [6], a full-modified-Newton step O(n) IIPM is presented for the special WLCP. Chi and Wang [5] proposed a full-Newton step IIPM for solving a special WLCP.
Motivated by above-mentioned results, in this paper, we extend the full-Newton step IPM based on the AET technique for the monotone WLCP. The square root function is used to obtain an equivalent form of the system defining the same central path used in [13, 14]. We apply the Newton method to the resulting system in order to get new search directions and we take full steps along these directions. We analyze the method and derive an iteration bound for WLCP having the same complexity as the one obtained for these types of problems. The method developed in [4] is a special case of our method for \(\psi (t)=t\).
The paper is organized as follows: In Sect. 2, we remember the monotone WLCP and the notion of the central path used in [13] for WLCP. In Sect. 3, we describe the AET and we obtain the scaled Newton system. A theoretical framework of the algorithm is presented in Sects. 4, 5 is devoted to the analysis of the algorithm. The iteration bound is derived in Sect. 6. Some numerical results are presented in Sect. 7. Concluding remarks are given in Sect. 8.
2 WLCP and its Central Path
The WLCP consists of finding \((x, s, y)\in {\textbf{R}}^{n}\times {\textbf{R}}^n\times {\textbf{R}}^m\) such that
where \(P, Q\in {\textbf{R}}^{(n+m)\times n}\) and \(R\in {\textbf{R}}^{(n+m)\times m}\) are given matrices, \(a\in {\textbf{R}}^{m+n}\) is a given vector, and \(w\in {\textbf{R}}^n_+\) is a given weight vector. Here, xs denotes the vector with components \(x_is_i\). It is further assumed that the matrix R has full column rank. The WLCP (1) is called monotone, if
and it is called skew-symmetric, if
We denote the set of all feasible points (1) by
and the solution set of (1) by
The set of all strictly feasible points of (1) is given by
Given a strictly feasible starting point \((x^0, s^0, y^0)\in {\mathcal {F}}^0\). We set
The central path of the WLCP (1) is the set of all points \((\mu ; x, s, y)\), with \(\mu \in [0, \mu ^0]\), satisfying
This path is well-defined; in fact, it is shown that if the WLCP is monotone and \({\mathcal {F}}^0\ne \emptyset \), then (3) has a unique solution for any \(\mu \in [0, \mu ^0]\) (see [14]). It is worth noting that by construction, \((\mu ^0; x^0, s^0, y^0)\) belongs to the central path, because it follows from \((x^0, s^0, y^0)\in {\mathcal {F}}^0\) that \(Px^0+Qs^0+Ry^0=a\), moreover, we have \(x^0s^0=c=(1-\frac{\mu ^0}{\mu ^0})w+\frac{\mu ^0}{\mu ^0}c=w(\mu ^0)\). Note that the central path converges to a solution of (1) if \(\mu \downarrow 0\).
3 A New Search Direction for WLCP
Based on Darvay’s technique for LO [8], we introduce a method for finding new search directions for IPMs in WLCP. Let us define the continuously differentiable function \(\psi :{\textbf{R}}_+\rightarrow {\textbf{R}}_+\) and assume that the inverse function \(\psi ^{-1}\) exists. In this way, we can write system (3) in the following equivalent form:
We assume that \(Px+Qs+Ry=a\) for a triple (x, s, y) such that \(x>0\) and \(s>0\). Applying Newton’s method to system (4) and linearizing it, we obtain the following system for the search directions \(\varDelta x, \varDelta s\), and \(\varDelta y\):
This system can be rewritten as follows:
The new iterates are then given as:
indicating that a full-Newton step is taken along the search directions. Now we introduce the following notations:
Then, the system (6) becomes as follows:
where \({{\tilde{P}}}=PV^{-1}X,~{\tilde{Q}}=QV^{-1}S,~V=\textrm{diag}(v),~ X=\textrm{diag}(x), S=\textrm{diag}(s),\) and
It is worth noting that for different functions \(\psi \), we obtain different new search directions.
4 A New Interior-Point Algorithm for WLCP
We mention that \(\psi (t)=t\) yields \(p_v=v^{-1}(\frac{w(\mu )}{\mu }-v^2)\), and we obtain the search direction introduced in [4]. Here, we consider \(\psi (t)=\sqrt{t}, t>0\) based on Darvay’s technique for LO [8]. For this function, we have
To measure the proximity of any approximation (x, s, y) of \((x(\mu ), s(\mu ), y(\mu ))\) to the central path (4), we define a proximity function as follows:
Note that for any \((x, s, y)\in {\mathcal {F}}\), we have
Let \(q_v=d_x-d_s\). One can easily find
and
Since the WLCP is monotone, we have \(\varDelta x^T\varDelta s\ge 0\), and this implies
We now outline the method.
\({{\mathbf{Algorithm~}:\mathrm{Full-Newton~step~IPM ~for~ WLCP }}}\) | |
---|---|
Input: Required parameter \(\varepsilon >0\); threshold parameter \(0<\tau <1\); | |
barrier update parameter \(0<\theta <1\); | |
An initial point \((x^0, s^0, y^0)\in {\mathcal {F}}^0\) with \(\delta (x^0, s^0; \mu ^0)\le \tau \), where \(\mu _0=\frac{(x^0)^Ts^0}{n}\); | |
Set \(k=0\); | |
while \(\Vert \sqrt{w}-\sqrt{x^ks^k}\Vert >\varepsilon \) do; | |
Set \(\mu ^{k+1}=(1-\theta )\mu ^k\); | |
Obtain the search direction \((\varDelta x^k, \varDelta s^k, \varDelta y^k)\) by solving (8) and using (7); | |
Set \((x^{k+1}, s^{k+1}, y^{k+1})=(x^{k}, s^{k}, y^{k})+(\varDelta x^k, \varDelta s^k, \varDelta y^k)\); | |
Set \(k:=k+1\); | |
end while. |
5 Analysis of the Algorithm
In the next lemma, we give a condition which guarantees the strict feasibility of the full-Newton step.
Lemma 5.1
A new iterate \((x^+, s^+, y^+)\) is strictly feasible if
Proof
Consider a step length \(0\le \alpha \le 1\) and define
Therefore,
Using (7), the second equation of (8) and (10), we get
The last equality holds, since
Furthermore, for \(0\le \alpha \le 1\), we have
where the third inequality is due to (11). From this inequality, we have
which shows that
Thus for all \(0\le \alpha \le 1\), we have
Now, it follows from (12) that \(x(\alpha )s(\alpha )>0\) for all \(0\le \alpha \le 1\). This shows that the components of \(x(\alpha )\) and \(s(\alpha )\) cannot vanish. Since \(x>0\) and \(s>0\) and \(x(\alpha )\) and \(s(\alpha )\) are linear functions in \(\alpha \), we have \(x(\alpha )>0\) and \(s(\alpha )>0\) for all \(0\le \alpha \le 1\). Hence, \(x(1)=x^+>0\) and \(s(1)=s^+>0\). This proves the lemma. \(\square \)
Lemma 5.2
Let \(\delta =\delta (x, s; \mu )\). After a full-Newton step
where \(v^+=\sqrt{\frac{x^+s^+}{\mu }.}\)
Proof
By setting \(\alpha =1\) in (12), we get
Now, from (13) and the definition of \(v^+\), we have
Taking the square root on both sides of the above inequality leads to the desired inequality and the proof is complete. \(\square \)
Lemma 5.3
Let \(\delta :=\delta (x, s; \mu )<\sqrt{\gamma }\). Then, we have
Thus, \(\delta ^+\le \frac{1}{\sqrt{\gamma }}\delta ^2\), which shows the quadratic convergence of the algorithm.
Proof
Using Lemma 5.2, (13) and (11), we have
which completes the proof. \(\square \)
Lemma 5.4
After a full-Newton step, one has
Proof
From (13), we obtain
as desired. \(\square \)
Lemma 5.5
Let \(\delta :=\delta (x, s; \mu )<\sqrt{\gamma }\) and \(\mu ^+:=(1-\theta )\mu \), where \(0<\theta <1\). Then, we have
Proof
Let \({{\bar{v}}}^+:=\sqrt{\frac{x^+s^+}{\mu ^+}}\). From the definition of \(w(\mu )\), we have
Therefore, using (11), (13) and Lemma 5.2, we have
The proof is complete. \(\square \)
6 Iteration Bound
We obtain a worst-case upper bound on the number of iterations of the proposed algorithm. Before doing so, we determine the values for the barrier parameter \(\theta \) and the threshold parameter \(\tau \) guaranteeing that the algorithm is well-defined, that is, if \(\delta :=\delta (x, s; \mu )\le \tau \) then \(\delta (x^+, s^+; \mu ^+)\le \tau .\) From Lemma 5.5, using \(\delta \le \tau \), the condition \(\delta (x^+, s^+; \mu ^+)\le \tau \) holds if
For any given \(\tau \), this inequality determines how big the updates of the barrier parameter \(\theta \) are allowed to be. Let
Clearly, we have \(\tau =\sqrt{\frac{\gamma }{2}}<\sqrt{\gamma }\). Thus, from Lemma 5.1, the full-Newton step will be feasible. Note that \(\theta \le \frac{1}{3}\), which yields \(\frac{1}{\sqrt{1-\theta }}\le \sqrt{\frac{3}{2}}\). Moreover, using the values of \(\theta \) and \(\tau \) given in (15), we obtain
Thus, the condition (14) is satisfied. Hence, Algorithm is well-defined.
Theorem 6.1
If WLCP is monotone and \(\theta \) and \(\tau \) are defined as in (15), then Algorithm finds an \(\varepsilon \)-approximate solution \((x, s, y)\in {\mathcal {F}}^0\) such that
in at most
iterations, where \(\beta =\max \big \{\big \Vert \sqrt{c-w}\big \Vert , \big \Vert \frac{c-w}{2\sqrt{w}+e}\big \Vert \big \}\) and \(e=(1, \ldots , 1)^T.\)
Proof
We have
On the other hand, from the definition of \(w(\mu )\), we obtain
Consider the following cases:
Case 1. \(c-w\ge 0\). We have
Case 2. \(c-w<0\). We have
Using the above two inequalities, we have
Substituting the above bound into (16), we obtain the first inequality below and, after k iterations, we obtain second inequality below.
Using the definition of \(\gamma \) from (2), we deduce that \(\Vert \sqrt{w}-\sqrt{xs}\Vert \le \varepsilon \) is satisfied if
Taking logarithms of both sides and using the inequality \(\log (1+\theta )\le \theta \) for \(\theta >-1\), we obtain
which completes the proof. \(\square \)
7 Numerical Results
In this section, we present computational results of an implementation of our algorithm in MATLAB R2009a on some randomly generated WLCPs. We generate the matrices R and Q with entries randomly selected from the interval \([-6, 6]\). To build the matrix P, we first constructed matrix \(A\in {\textbf{R}}^{n\times n}\) with elements randomly from \([-6, 6]\). Then, we considered \(P=-Q(A^TA)\). Elements of w were chosen randomly from [0.2, 0.8]. We used the value of the accuracy parameter \(\varepsilon =10^{-8}\) and we reduced the value of the barrier update parameter \(\mu \) by the factor \(1-\theta \) when \(\theta =0.25\). Table shows the number of iterations (iter) as well as the value of the complementarity gap \(\mu \) at the \(\varepsilon \)-approximate solutions for the WLCP problems.
8 Concluding Remarks
In this paper, we presented a full-Newton step IPM based on the AET for solving monotone WLCP. The square root function is used to obtain an equivalent form of the system defining the central path. The Newton method is applied to the resulting system in order to obtain new search directions. By appropriately selecting the barrier and threshold parameters, the iterates are shown to be in a certain neighborhood \(\delta (x, s; \mu )\le \tau \) of the central path . It is also shown that the proximity measure reduces quadratically at each iteration. The derived iteration bound coincides with the iteration bounds obtained for these types of problems [13, 14]. Some numerical results illustrate the efficiency of the method for solving monotone WLCPs.
References
Achache, M.: Complexity analysis and numerical implementation of a short-step primal-dual algorithm for linear complementarity problems. Appl. Math. Comput. 216, 1889–1895 (2010)
Ahmadi, K., Hasani, F., Kheirfam, B.: A full-Newton step infeasible interior-point algorithm based on Darvay directions for linear optimization. J. Math. Model. Algorithms Oper. Res. 13, 191–208 (2014)
Anstreicher, K.M.: Interior-point algorithms for a generalization of linear programming and weighted centering. Optim. Methods Softw. 2, 605–612 (2012)
Asadi, S., Darvay, Zs., Lesaja, G., Mahdavi-Amiri, N., Potra, F.A.: A full-Newton step interior-point method for monotone weighted linear complementarity problems. J. Optim. Theory Appl. 186, 864–878 (2020)
Chi, X.N., Wang, G.Q.: A full-Newton step infeasible interior-point method for the special weighted linear complementarity problem. J. Optim. Theory Appl. 190, 108–129 (2021)
Chi, X.N., Wan, Z.P., Hao, Z.J.: A full-modified-Newton step \(O(n)\) infeasible interior-point method for the special weighted linear complementarity problem. J. Ind. Manag. Optim. 18, 2579–2598 (2022)
Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, San Diego, CA (1992)
Darvay, Zs.: New interior-point algorithm in linear programming: Adv. Model. Optim. 5, 51–92 (2003)
Kheirfam, B.: A new infeasible interior-point method based on Darvay’s technique for symmetric optimization. Ann. Oper. Res. 211, 209–224 (2013)
Kheirfam, B.: A predictor-corrector interior-point algorithm for \(P_*(\kappa )\)-horizontal linear complementarity problem. Numer. Algorithms 60, 349–361 (2014)
Lesaja, G., Potra, F.A.: Adaptive full Newton-step infeasible interior-point method for sufficient horozontal LCP. Optim. Methods Softw. 34, 1014–1034 (2018)
Mansouri, H., Pirhaji, M.: A polynomial interior-point algorithm for monotone linear complementarity problems. J. Optim. Theory Appl. 157, 451–461 (2013)
Potra, F.A.: Weighted complementarity problems- a new paradigm for computing equilibria. SIAM J. Optim. 2, 1634–1654 (2012)
Potra, F.A.: Sufficient weighted complementarity problems. Comput. Optim. Appl. 64, 467–488 (2016)
Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization. An Interior-Point Approach. John Wiley & Sons, Chichester, UK (1997)
Tang, J.Y., Zhang, H.C.: A nonmonotone smoothing Newton algorithm for weighted complementarity problems. J. Optim. Theory Appl. 189, 679–715 (2021)
Tang, J.Y., Zhou, J.C.: A modified damped Guass–Newton method for non-monotone weighted linear complementarity problems. Optim. Methods Softw. 37, 1145–1164 (2022)
Tang, J.Y., Zhou, J.C.: Quadratic convergence analysis of a nonmonotone Levenberg-Marquardt type method for the weighted nonlinear complementarity problem. Comput. Optim. Appl. 80, 213–244 (2021)
Wang, G.Q., Bai, Y.: A new primal-dual path-following interior-point algorithm for semidefinite optimization. J. Math. Anal. Appl. 353, 339–349 (2009)
Wang, G.Q., Bai, Y.: A primal-dual interior-point algorithm for second-order cone optimization with full Nesterov–Todd step. Appl. Math. Comput. 215, 1047–1061 (2009)
Wang, G.Q., Bai, Y.: A new full Nesterov–Todd step primal-dual path-following interior-point algorithm for symmetric optimization. J. Optim. Theory Appl. 154, 966–985 (2012)
Zhang, J.: A smoothing Newton algorithm for weighted linear complementarity problem. Optim. Lett. 10, 499–509 (2016)
Acknowledgements
The author would like to thank the editor and anonymous reviewers for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Goran Lesaja.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kheirfam, B. Complexity Analysis of a Full-Newton Step Interior-Point Method for Monotone Weighted Linear Complementarity Problems. J Optim Theory Appl 202, 133–145 (2024). https://doi.org/10.1007/s10957-022-02139-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-022-02139-3
Keywords
- Weighted linear complementarity problem
- Interior-point methods
- New search direction
- Polynomial complexity