Large-update interior point algorithm for -linear complementarity problem
© Cho; licensee Springer. 2014
Received: 30 April 2014
Accepted: 5 September 2014
Published: 24 September 2014
It is well known that each barrier function defines an interior point algorithm and each barrier function is determined by its univariate kernel function. In this paper we present a new large-update primal-dual interior point algorithm for solving -linear complementarity problem (LCP) based on a parametric version of the kernel function in (Bai et al. in SIAM J. Optim. 13:766-782, 2003). We show that the algorithm has iteration complexity, where p is a barrier function parameter and κ is the handicap of the matrix. This is the best known complexity result for such a method.
where and is a -matrix and xs denotes the componentwise (Hadamard) product of the vectors x and s.
The matrix M is a -matrix if it is a -matrix for some , where , where denotes the i th component of the vector Mξ, , . Note that M is a -matrix if and only if M is positive semidefinite.
In the following, we give some examples for -matrices.
is , for all . Indeed, since , for , and . Hence for all . For , and . Then , for all . Thus, is , for all .
is , for all . . If , and . Hence , for all . If , and . , for all . Thus, is , for all .
Linear complementarity problems (LCPs) have many applications in science, economics and engineering. LCPs include linear and quadratic programming, fixed point problems and sets of piecewise-linear equations, bimatrix equilibrium points and variational inequalities . A large-update interior point method (IPM) is one of the most efficient numerical methods for various optimization problems.
Peng-Roos-Terlaky [2–4] proposed new variants of interior point methods (IPMs) based on self-regular barrier functions and achieved so far the best known complexity for large-update methods with a specific self-regular barrier function. Bai-Ghami-Roos  proposed a new primal-dual IPM for linear optimization (LO) problem based on eligible barrier functions and a unified scheme for analyzing the algorithm based on four conditions of the kernel function and Bai-Lesaja-Roos  generalized to -LCP. Cho  and Cho-Kim  extended the complexity analysis for LO problem to -LCP. Amini-Haseli  and Amini-Peyghami  introduced generalized versions of the kernel functions in  and improved the complexity results for large-update methods for LO and -LCP, respectively. Recently, Lesaja-Roos  proposed a unified analysis of the IPM for -LCP based on the class of eligible barrier functions which was first introduced by Bai-Ghami-Roos  for LO. Wang-Bai  generalized interior point algorithm for LO to P-matrix LCP over symmetric cones based on the same kernel function. Wang-Lesaja  extended the full NT-step infeasible IPM for symmetric cone LO to the Cartesian -symmetric cone LCP and the algorithm is small-update method.
The most challenging question in this research area is whether or not there exists a kernel function for which the iteration bound for large-update method is the same as or even better than currently best known bound for such methods . Bai-Ghami-Roos  proposed a new efficient large-update IPM for LO based on a barrier-type function which is not a barrier function in the usual sense since it has finite value at the boundary of the feasible region. Despite this, they obtained the best known iteration bound. Ghami  proposed various versions of interior point algorithms based on kernel functions and showed that the kernel function in  seems promising through numerical tests. Wang-Bai  proposed a generalized version of the kernel function in  which has a parameter in the growth term for -horizontal LCPs and obtained the best known complexity bound when the parameter value equals 1, i.e. the same kernel function in . This implies that the parameter in the growth term does not improve the complexity of the algorithm except 1.
Motivated by this, we introduce a parameter in the barrier term of the kernel function in  and obtained the best known complexity result for large-update methods for all parameters. Note that when the parameter in the barrier term grows, the barrier function grows faster when t approaches zero.
The paper is organized as follows: In Section 2, we introduce the generic IPM and give some examples of -matrices. In Section 3, we introduce a class of barrier functions and propose a new large-update interior point algorithm for -LCP. In Section 4, we derive the complexity results for the algorithm. Finally concluding remarks are given in Section 5.
Throughout the paper, and denote the set of n-dimensional nonnegative vectors and positive vectors, respectively. For , and denote the i th component and the smallest component of the vector x, respectively. We denote by D the diagonal matrix from a vector d and e, the n-dimensional vector of ones. The index set . For , if there exists a positive constant such that , for all , and if there exist positive constants and such that , for all . For , and , where Z is the set of integers. log denotes the natural logarithm.
Without loss of generality, we assume that (1.1) satisfies the interior point condition (IPC), i.e., there exists a such that . Since M is a -matrix for some and (1.1) satisfies the IPC, the system (2.1) has a unique solution for . We denote the solution of (2.1) by which is called the μ-center for . The set of μ-centers is called the central path of (1.1). Since the limit of the μ-centers satisfies (1.1) as , it yields the solution for (1.1) . IPMs follow the central path approximately and approach the solution of (1.1) as .
where and , for . We call the kernel function of the classical logarithmic barrier function .
3 New algorithm
From (3.3), is strictly convex and has a minimum value 0 at .
For (ii), , .
For (iii), it is clear from (3.2). □
Corollary 3.2 Let . By Lemma 3.1(i) and Lemma 2.1.2 in , we have , i.e., is exponentially convex, for all .
Remark 3.3 By Lemma 3.1(ii), (iii), and Lemma 2.4 in , , , .
Proof Let for . Then . Using the first inequality in Lemma 3.4, we have . Then we have . □
Lemma 3.6 Let be the inverse function of for . Then , , , .
Proof Let , for . By the definition of ρ, , for and . By (3.2) and , . Hence . □
Note that . In this paper, we replace the right-hand side of (2.5), , by as in (3.4). This defines a new search direction and proximity function.
In the following we compute upper bound of proximity function during an outer iteration.
Lemma 3.7 Let and be defined as in (3.5) and (3.4), respectively. Then , , .
Hence we have . □
Lemma 3.8 Let and . If and , then .
Proof If , then . Suppose that . Let . Since , , i.e., . This implies that , . Let . Then is monotone decreasing in σ. Since and , .
Let . Then and hence . Let . is monotone increasing in p and L, respectively. Since and , . Hence . □
4 Complexity analysis
In this section we give the iteration complexity of the algorithm for large-update methods. For complexity analysis of the algorithm we follow a similar framework to the one defined in  for LO problems. In the following we compute the bound of the growth of the barrier function during an outer iteration of the algorithm.
Using Lemma 3.1(ii), (iii), and Theorem 3.2 in , we obtain the following lemma.
In the following we compute the upper bounds of when we update the barrier parameter μ.
We will use for the upper bounds of for large-update methods.
Remark 4.3 Let . Without loss of generality, we can assume that . Indeed, when , and , if . In the algorithm we take .
Remark 4.4 For large-update method with and , we have and .
For notational convenience, we denote , , and . To estimate the bound for and , we need the following technical lemma.
Lemma 4.5 (Modification of Lemma 4.1 in )
Lemma 4.6 (Modification of Lemma 4.2 in )
, and .
Using (4.5) and Lemma 4.6, we have the following lemma.
Lemma 4.7 (Modification of Lemma 4.3 in )
Using (4.5) and Lemma 4.7, we have the following lemma.
Lemma 4.8 (Modification of Lemma 4.4 in )
Lemma 4.9 (Modification of Lemma 4.5 in )
Lemma 4.10 (Modification of Lemma 4.6 in )
Using Lemma 4.6, Lemma 3.8, and (4.10), ≥ ≥ ≥ ≥ > , for all . In the same way, , for all . Hence we can use the exponential convexity of .
Lemma 4.11 (Lemma 1.3.3 in )
Using , (4.5), , and Lemma 4.11, we have the following lemma.
Using Lemma 4.12, (4.10), and Lemma 3.7, we have the following theorem.
Theorem 4.13 For as in (4.10), .
Proposition 4.14 (Proposition 1.3.2 in )
where and . Then .
We define the value of after the μ-update as and the subsequent values in the same outer iteration are denoted as , . Then we have . If we let K be the number of inner iterations per an outer iteration, then we have , . In the following theorem we give the bound for the total number of iterations.
Proof Using Proposition 4.14 with and , we obtain the number of inner iterations . If the central path parameter μ has the initial value and is updated by multiplying , , then after at most iterations we have . For the total number of iterations, we multiply the number of inner iterations by that of the outer iterations. Hence the total number of iterations is bounded by . □
5 Concluding remarks
Wang-Bai  defined a parametric version of the kernel function in , the parameter is in the growth term of the kernel function, and generalized the algorithm for LO to -LCPs based on this kernel function. Ghami-Roos-Steihaug  extended the algorithm for LO to semidefinite optimization based on the kernel function in . However, they obtained the best known complexity bound for large-update methods when the kernel function takes the form in .
Motivated by this, we consider a parametric version of the kernel function in  with parameters in the barrier term of the kernel function. For large-update methods, by taking and , we obtained , for , which is the best known complexity bound for such a method.
Further research will be concerned with a numerical test and extension to general problems.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2010094).
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