New complexity analysis of interior-point methods for the Cartesian -SCLCP
© Wang et al.; licensee Springer 2013
Received: 21 March 2013
Accepted: 21 May 2013
Published: 6 June 2013
In this paper, we give a unified analysis for both large- and small-update interior-point methods for the Cartesian -linear complementarity problem over symmetric cones based on a finite barrier. The proposed finite barrier is used both for determining the search directions and for measuring the distance between the given iterate and the μ-center for the algorithm. The symmetry of the resulting search directions is forced by using the Nesterov-Todd scaling scheme. By means of Euclidean Jordan algebras, together with the feature of the finite kernel function, we derive the iteration bounds that match the currently best known iteration bounds for large- and small-update methods. Furthermore, our algorithm and its polynomial iteration complexity analysis provide a unified treatment for a class of primal-dual interior-point methods and their complexity analysis.
Note that (Lemma 2.2 in ).
The SCLCP is a wide class of problems that contains linear complementarity problem (LCP), second-order cone linear complementarity problem (SOCLCP) and semidefinite linear complementarity problem (SDLCP) as special cases. For an overview of these and related results, we refer to the survey paper  and references within.
There are many solution approaches for SCLCP. Among them, the interior-point methods (IPMs) gain much more attention. Faybusovich  made the first attempt to generalize IPMs to symmetric optimization (SO) and SCLCP using the ‘machinery’ of Euclidean Jordan algebras. Potra  proposed an infeasible corrector-predictor IPM for the monotone SCLCP. Yoshise  proposed the homogeneous model for the monotone nonlinear complementarity problems (NCP) over symmetric cones (SCNCP) and analyzed IPM to solve it.
Let be a Cartesian product of a finite number of simple Euclidean Jordan algebras with dimensions and ranks for , that is, with its cone of squares , where are the corresponding cones of squares of for . The dimension of is and the rank is . Recall that a Euclidean Jordan algebra is said to be simple if it cannot be represented as the orthogonal direct sum of two Euclidean Jordan algebras.
where and are two index sets. It is closely related to the Cartesian - and P-properties which were first introduced by Chen and Qi  over the space of symmetric matrices, and later extended by Pan and Chen  and Luo and Xiu  to the space of second-order cones and the general Euclidean Jordan algebras, respectively.
The Cartesian -SCLCP is indeed the generalization of -LCP, which was first introduced by Kojima et al. . They established the existence of the central path and designed and analyzed IPMs for solving -LCP. The theoretical importance of this class of LCPs lays in the fact that this is the largest class for which polynomial convergence of IPMs can be proved without additional conditions (such as boundedness of the level sets).
Luo and Xiu  were the first to establish a theoretical framework of path-following interior-point algorithms for the Cartesian -SCLCP and to prove the global convergence and the iteration complexities of the proposed algorithms. Wang and Bai  analyzed a class of IPMs for the Cartesian -SCLCP based on a parametric kernel function different from the logarithmic kernel function. Lesaja et al.  gave a unified analysis of kernel-based IPMs for the Cartesian -SCLCP and derived the currently best known iteration bounds for large- and small-update methods for some special eligible kernel functions. Wang and Lesaja  generalize Roos’s full-Newton step feasible IPM for LO  and Gu et al. extension to SO , to the Cartesian -SCLCP. Liu et al.  proposed smoothing Newton methods for the Cartesian - and P-SCLCPs. Huang and Lu  presented a globally convergent smoothing method with a linear rate of convergence for the Cartesian -SCLCP.
This parametric kernel function also has finite value at the boundary of the feasible region and its growth terms are between linear and quadratic. They proposed a class of primal-dual interior-point algorithms for LO and the extension to SDO  based on the parametric kernel function , respectively. Meanwhile, the results for LO in [17, 20] were extended to -LCP by Wang and Bai in , again matching the best known iteration bounds for LO with the addition . An interesting question here is whether we can directly extend the interior-point algorithms for LO in  to the Cartesian -SCLCP. As we will see later, LO cannot be trivially generalized to the Cartesian -SCLCP context. The analysis of the algorithm proposed in this paper is more complicated than in the LO case mainly due to the fact that the search directions are no longer orthogonal.
In this paper, we consider a generalization of kernel-based IPMs discussed in the paper  to the Cartesian -SCLCP. The paper also extends the results of the paper  where we consider the same type of IPMs for -LCP, however, only over the nonnegative orthant. Our goal is to provide a unified analysis for both large- and small-update IPMs for the Cartesian -SCLCP based on the finite barrier. Although the proposed algorithm is an exact extension of the algorithms for LO and -LCP, the Cartesian -property makes the analysis of the method far more complicated. Furthermore, we loose the orthogonality of the scaled search directions in the Cartesian -SCLCP case. This also yields many difficulties in the analysis of the algorithm for the Cartesian -SCLCP. However, we manage to prove the same good characteristics as in the LO case. The obtained complexity results match the best known iteration bounds known for large- and small-update methods, namely and , respectively. The order of the iteration bounds almost coincides with the bounds derived for LO in , except that the iteration bounds in the Cartesian -SCLCP case are multiplied by the factor .
The paper is organized as follows. In Section 2, we briefly describe some concepts, properties, and results from Euclidean Jordan algebras. In Section 3, we provide and develop some useful properties of the finite kernel function and the corresponding barrier function . In Section 4, we mainly study primal-dual IPMs for the Cartesian -SCLCP based on the finite kernel function. The analysis and complexity bounds of the algorithm are presented in Sections 5 and 6, respectively. Finally, some conclusions and remarks follow in Section 7.
Notations used throughout the paper are as follows. , , and denote the set of all vectors (with n components), the set of nonnegative vectors, and the set of positive vectors, respectively. The largest eigenvalue of x and the smallest eigenvalue of x are defined by and . The Löwner partial ordering ‘’ of defined by a symmetric cone is defined by if . The interior of is denoted as , and we write if . Finally, if is a real-valued function of a real nonnegative variable, the notation means that for some positive constant , and that for two positive constants and .
In what follows, we assume that the reader is familiar with the basic concepts of Euclidean Jordan algebras and symmetric cones. The detailed information can be found in the monograph of Faraut and Korányi  and in [1, 14, 24–29] as it relates to optimization.
is the identity element in .
It enables us to extend the definition of any real-valued, continuous univariate function to elements of a Euclidean Jordan algebra, using the eigenvalues. In particular this holds for the finite kernel function.
It should be noted that , which does not mean that the derivative of the vector-valued function defined by (12) is just a vector-valued function induced by the derivative of the function .
Furthermore, we have the following important result.
Lemma 2.1 (Lemma 14 in )
respectively, where , , and , , are orthogonal projection operators that appear in the Peirce decomposition of with respect to the Jordan frame .
3 Properties of the finite kernel (barrier) function
It follows from (30) that is strictly convex and is monotonically decreasing in .
Lemma 3.1 (Lemma 2.4 in )
Note that is exponentially convex, whenever . The following lemma makes clear that when v belongs to the level set , for some given , the exponential convexity is guaranteed and it is proved that the value of σ is large enough.
Lemma 3.2 (Lemma 2.5 in )
Let and . If , then .
This means that the derivative of the barrier function in essence coincides with the vector-valued function defined by (14) and (15).
As the consequence of Lemma 3.1, we have the following theorem, which is indeed a slight modification of Theorem 4.3.2 in . Thus, we omit its proof.
Proof From Taylor’s theorem and the fact that , the inequality is straightforward. □
This implies the desired result. □
The following lemma can be directly obtained from Lemma 2.5 in , which provides the lower and upper bounds of the inverse function of the finite kernel function for .
We can conclude that and if and only if .
Clearly, and depend only on the eigenvalues of the symmetric cone for each j, . The following theorem gives a lower bound on in terms of , which is precisely the same as its LO counterpart (cf. Theorem 4.8 in ).
This completes the proof of the corollary. □
In what follows, we consider the derivatives of the function with respect to t, where with and . For more details, we refer to .
where and .
which bounds the second-order derivative of with respect to t.
4 Interior-point algorithm for the Cartesian -SCLCP
In this section, we first introduce the central path for the Cartesian -SCLCP. Next, we mainly derive the new search directions induced by the finite kernel function . Finally, we present the generic polynomial interior-point algorithm for the Cartesian -SCLCP.
4.1 The central path for the Cartesian -SCLCP
where is a parameter. The parameterized system (43) has a unique solution for each . This solution is denoted as and we call the μ-center of the Cartesian -SCLCP. The set of μ-centers (with μ running through all positive real numbers) gives a homotopy path, which is called the central path of the Cartesian -SCLCP. If , then the limit of the central path exists and since the limit points satisfy the complementarity condition , the limit yields a solution for the Cartesian -SCLCP (see, e.g., ).
4.2 The new search directions for the Cartesian -SCLCP
To obtain the search directions for the Cartesian -SCLCP, the usual approach is to use Newton’s method and to linearize the system (43). In what follows, we briefly outline the details.
Due to the fact that x and s do not operator commute in general, i.e., , this system does not always have a unique solution. It is well known that this difficulty can be solved by applying a scaling scheme. This goes as follows.
Lemma 4.1 (Lemma 28 in )
By choosing u appropriately, this system can be used to define the commutative class of search directions (see, e.g., ). In the literature the following three choices are well known: , , and , where w is the NT-scaling point of x and s. The first two choices lead to the so-called xs-direction and sx-direction, respectively. In this paper, we consider the third choice that is called NT-scaling scheme and the resulting direction is called NT search direction. This scaling scheme was first proposed by Nesterov and Todd for self-scaled cones [30, 31] and then adapted by Faybusovich [1, 26] for symmetric cones.
Lemma 4.2 (Lemma 3.2 in )
The transformation also has the Cartesian -property (cf. Proposition 3.4 in ).
Since the linear transformation has the Cartesian -property, the system (51) has a unique solution .
Since the system (52) has the same matrix of coefficients as the system (51), also the system (52) has a unique solution.a
Hence, the value of can be considered as a measure for the distance between the given iterate and the μ-center .
4.3 The generic interior-point algorithm for the Cartesian -SCLCP
It is clear from the above description that the closeness of to is measured by the value of , with as a threshold value. If , then we start a new outer iteration by performing a μ-update, i.e., ; otherwise, we enter an inner iteration by computing the search directions using (52) and (50) at the current iterates with respect to the current value of μ and apply (53) to get new iterates. If necessary, we repeat the procedure until we find iterates that are in the τ-neighborhood of the central path. Then μ is again reduced by the factor with , and we apply inner iteration(s) targeting at the new μ-centers, and so on. This process is repeated until μ is small enough, say until . At this stage, we have found a ε-approximate solution of the Cartesian -SCLCP.
5 Analysis of the algorithm
In this section, we first discuss the growth behavior of the barrier function during an outer iteration. Next, we choose the default step size and obtain an upper bound for the decrease of the barrier function during an inner iteration. Finally, we show that the default step size yields sufficient decrease of the barrier function value during each inner iteration.
5.1 Growth behavior of the barrier function during an outer iteration
It should be mentioned that during the course of the algorithm the largest values of occur just after the update of μ. So, next we derive an estimate for the effect of a μ-update on the value of .
which means that depends only on the eigenvalues of the symmetric cone for each j, . The growth behavior of the proximity is precisely the same as its LO counterpart (cf. Theorem 3.2 in ).
Proof With and , the corollary follows immediately from Theorem 5.1. □
5.2 Choice of the default step size
which means that gives an upper bound for the decrease of the barrier function . Furthermore, we can conclude that .
Hence, we can conclude that is monotonically decreasing in a neighborhood of .
Contrary to the LO case, the vectors and are not necessarily orthogonal any more. However, the Cartesian -property of SCLCP still allows us to find a good lower bound of the inner product .
This completes the proof of the lemma. □
The key steps in the analysis of the algorithm are based on the effort to find upper bounds on and in terms of the proximity measure δ. The following lemma yields their upper bounds.
This implies the inequalities in the statement of the lemma. □
The last inequality holds due to the fact that (62). This completes the proof of the lemma. □
From this point on, the analysis of the algorithm follows almost completely the similar analyses in [17, 22] with straightforward modifications that take into account the Cartesian -property. Therefore, the intermediate results are omitted and only main results are mentioned without the proofs.
and we will use as the default step size. It is obvious that .
5.3 Decrease of the value of during an inner iteration
In what follows, we will show that the barrier function in each inner iteration with the default step size , as defined by (65), is decreasing. For this, we need the following technical result.
Lemma 5.6 (Lemma 3.12 in )
As a consequence of Lemma 5.6 and the fact that , which is a twice differentiable convex function with , and , we can easily prove the following lemma.
The following theorem states the results which show that the default step size (65) yields sufficient decrease of the barrier function value during each inner iteration.
This completes the proof of the theorem. □
6 Complexity of the algorithm
In this section, we first derive an upper bound for the number of the iteration bounds by our algorithm. Then we obtain the iteration bounds that match the currently best known iteration bounds for large- and small-update methods, respectively.
6.1 Iteration bound for a large-update method
where and .
Lemma 6.1 (Lemma 14 in )
where and . Then .
Combining Lemma 6.1 and (66), we can easily verify the following main result.
From the above expression with and , and also applying Lemma 3.2, we can conclude that .
After some elementary reductions, we have the following theorem, which gives the currently best known iteration bound for the large-update method.
iterations. The output gives a ε-approximate solution of the Cartesian -SCLCP.
6.2 Iteration bound for a small-update method
It is not hard to show that if the above analysis is used for a small-update method, the iteration bound would not be as good as it can be for these types of methods. For the analysis of the iteration bound of a small-update method, we need to estimate the upper bound of more accurately. It should be noted that the following analysis only holds for .
After some elementary reductions, we have the following theorem, which gives the currently best known iteration bound for a small-update method.
iterations. The output gives a ε-approximate solution of the Cartesian -SCLCP.
7 Conclusions and remarks
In this paper, we have shown that primal-dual IPMs for LO  and -LCP  based on the finite barrier can be extended to the context of the Cartesian -SCLCP. The iteration bounds for large- and small-update methods are obtained, namely and , respectively. In both cases, we were able to match the best known iteration bounds for these types of methods. Moreover, this unifies the analysis for the -LCP, the Cartesian -SOCLCP, and the Cartesian -SDLCP.
Some interesting topics for further research remain. One possible topic is to investigate whether it is possible to replace NT-scaling scheme by some other scaling schemes and still obtain polynomial-time iteration bounds. Another worthwhile direction for further research may be the development of infeasible kernel-based IPMs for SCLCP.
a It may be worth mentioning that if we use the kernel function of the classical logarithmic barrier function, i.e., , then , whence , and hence system (52) then coincides with the classical system (51).
This work was supported by the National Natural Science Foundation of China (No. 11001169), China Postdoctoral Science Foundation funded project (No. 2012T50427) and Connotative Construction Project of Shanghai University of Engineering Science (No. NHKY-2012-13).
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