Abstract
This study presents a new search direction for the horizontal linear complementarity problem. A vector-valued function is applied to the system of , which defines the central path. Usually, the way to get the equivalent form of the central path is using the square root function. However, in our study, we substitute a new search function formed by a different identity map, which obtains the equivalent shape of the central path using the square root function. We get the new search directions from Newton’s Method. Given this framework, we prove polynomial complexity for the Newton directions. We show that the algorithm’s complexity is , which is the same as the best-given algorithms for the horizontal linear complementarity problem.
Introduction
Karmarkar (1984) found the first method of the interior point algorithm, so linear programming appeared as a dynamic field of research. Soon after, the interior point algorithm was able to resolve linear programming problems and other optimal problems such as semi-definite programming problems, high-order conic programming problems, and linear and nonlinear complementarity problems.
Then, Nestrov and Nemirovskii (1994) imported a new concept of self-concordant barrier functions to define the interior point method for solving the convex programming problem. In addition, Vieira (2007) proposed a different interior point algorithm using the kernel function.
It showed that linear complementarity problems have more significant adhibition in the economic field; the most significant model is the equilibrium model of the Arrow–Debreu market. (Kojima et al. (1992) proved that linear complementarity problems are equal to some models of equilibrium market, but that is not necessarily sufficient. Hence, Illés et al. (2010) analyzed the general linear complementarity problems’ solvability.
Some special search directions play an important role in analyzing interior point algorithms.
A basic idea of primal-dual inter-point algorithms is to go through the central path to get the optimal solution. Later, Peng et al. (2002a) verified that the essence of Karmarkar’s algorithm was just a special classical barrier function, which is a polynomial time algorithm. Later, Peng et al. (2002b) proposed a self-regular function and got the best iteration bound for a large-update algorithm for linear programming problems.
Moreover, Peng et al. (2002b) presented a new method for getting search directions called full-Newton methods; the new algorithm transformed the center equation using a function and then got the new search direction from Newton’s method.
Because linear complementarity problems are closely related to linear programming problemsKarimi and Tuncel, 2020; Yamashita et al., 2021; Yang, 2022; Zhang et al., 2022a; 2022b), many interior-point algorithms (Mansouri et al., 2015) are designed from linear programming to linear complementarity problems, and all got polynomial time numerical results.
Furthermore, Wang and Bai (2009) and Wang and Bai (2012) proposed the second-order cone programming using a new full Nesterov–Todd step of the primal-dual method. Scheunemann et al. (2021) presented a barrier term for the infeasible primal-dual interior algorithm of small strain single crystal plasticity. Lu et al. (2020) proposed a two-step method for horizontal linear complementarity problems, and Asadi et al. (2019) presented a large-step infeasible algorithm for horizontal, linear complementarity problems.
The above-mentioned studies almost used the square root function, which obtained a form of the central path. The basic idea of the new function is named the difference of identity. In this study, we use the new square root function to define the search direction to solve horizontal linear complementarity problems and give the complementarity problems and give the complexity of the algorithm.
The interior algorithm of HLCP
Two square matrices are given, and is a vector. The horizontal linear complementarity problems finds a pair of , such that
In this section, we study the horizontal linear complementarity problems (HLCP) based on the central path method to get the search directions.
We assume that (1) meets the need of the following two assumptions (Darvay, 2003).
Interior point condition
There are two vectors such that
The monotonic property
There are two matrices (N, M) such that
From the above two assumptions, we can conclude that there is a solution for HLCP. We find an approximate solution by solving the following system:
Using the path-following interior algorithm replaces the second equation of Eq. 2 with the parameterized equation ; then, we get the following system:With , we can get the unique solution from system (3), and we call the -center of horizontal linear programming problem. With running through all positive numbers and when , the central path exists and we get a solution for the horizontal linear programming problems (Kheirfam and Haghighi, 2019).
Search directions for HLCP
Considering the continuously differentiable and the inverse function , then (2.3) can be transformed into the following form:Applying Newton’s method yields new search directions. Let
Let ; then,
From (5) and (6), (4) can be written in the formAt this time,
We get different values for the from the function and obtain the search directions.
Now, for , we choose ; then, from the new function, we get a new direction, and
We define from
and monotonicity
Furthermore, let ; then,
Primal-dual interior-point algorithm for HLCP
1) Let be the accuracy parameter, the update parameter, Assume a strictly feasible point (x0, y0), s.t. .
2) If , then stop; otherwise, go to the next step.
3) According to (4), find (4) and . We get . Then, turn to step 2.
Convergence analyses
Lemma 4.1. Let (dx, dy) be a solution of (7). Then, we have
Proof. Because the pair [N, M] is in the monotone HLCP, we conclude that
That is,
Lemma 4.2. Let and . Then, .
Proof. Let
Therefore,
Furthermore, from (8),
From (11), we get
From , we get .
From (13), we obtain
Using .
Then,
Therefore, we get a conclusion that, for any , the inequality holds, which signifies that the signs of and do not change on the interval [0,1]. Hence, leads to .
Lemma 4.3. Let be a decreasing function, where
Furthermore, let such that . Then,
Proof.
Lemma 4.4. Let . Then,
Proof. From Lemma 4.2, we get
Due to (4.4), as , we get
From and , that is , then
By using the function for any t 0.5, f’(t) 0, f is monotone decreasing.
From Lemma 4.3,
Substituting and making reductions, we get
We have for all .
Moreover, .
Thus,
Furthermore,
Let .
For and then , we obtain
A simple calculus yields
We have .
Lemma 4.5. Let and suppose that the vectors x+ and y + are obtained using a full-Newton step. Thus, . We get .
If , then we obtain .
Lemma 4.6. Let
then and .
If , we have .
Proof. from Lemma 4.4
Consider ; we get
For h’(t) < 0, for h’(t) < 0, we get that h is a decreasing function.
Using (4.9), we have
Using , we have
This implies that g is decreasing.
We get .
Lemma 4.7. We assume that the () is strictly feasible and , and assume that the two vectors and are obtained by the algorithm; then, after k iterations k and .
Proof. From lemma 4.5,
Taking logarithms on two sides, then we get
From , we obtain
Because the self-dual embedding allows us to propose without any loss of generality that , we have .
Theorem 4.1. Suppose that x0 = y0 = e. If we consider the default values for and , we get that the algorithm just requires no more than interior-point iterations. The conclusion satisfies .
Conclusion and future works
This study proposed a primal-dual path-following algorithm for the horizontal linear complementarity problem based on a new search direction, which differs from those available. We analyzed this algorithm and illustrated that the proposed algorithm has iteration complexity bound. Some interesting topics remain for future research. Firstly, we can extend the algorithm to linear complementarity problems over symmetric cones. Secondly, we can develop the infeasible interior point algorithm based on the method given in this study.
Statements
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors without undue reservation.
Author contributions
XG: algorithm analysis; LX: astringency; BY: feasibility study.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
linear complementarity, interior-point method, full-Newton step, complexity, HLCP
Citation
Gong X, Xi L and Yuan B (2023) A new search direction of IPM for horizontal linear complementarity problems. Front. Energy Res. 10:977448. doi: 10.3389/fenrg.2022.977448
Received
24 June 2022
Accepted
18 July 2022
Published
05 January 2023
Volume
10 - 2022
Edited by
Bin Zhou, Hunan University, China
Reviewed by
Yuanzheng Li, Huazhong University of Science and Technology, China
Jian Zhao, Shanghai University of Electric Power, China
Yingjun Wu, Hohai University, China
Updates
Copyright
© 2023 Gong, Xi and Yuan.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Xiaoyu Gong, 39191267@qq.com ; Lei Xi, xilei2014@163.com
This article was submitted to Process and Energy Systems Engineering, a section of the journal Frontiers in Energy Research
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.