In this paper, we consider the monotone Linear Complementarity Problem (LCP) in the standard form: Given a positive semi-definite matrix M∈Rn×n and a vector q∈Rn, find a pair of vectors (x,s)∈R2n such that (1) −Mx+s=qx,s≥0xs=0, where xs denotes the component-wise product of vectors x and s. Monotone LCPs are commonly used for both theoretical and practical purposes. The importance of LCPs stems from the fact that Karush-Kuhn-Tucker (KKT) optimality conditions of important optimization problems such as Linear Optimization (LO) and Quadratic Optimization (QO) can be formulated as LCPs (see [21]). In addition, many important practical problems in game theory, engineering, economics, and optimal control can be formulated as LCPs, see [16, 20].

Due to the theoretical and practical importance of LCPs, efficient methods for solving LCPs are of significant interest. The Interior-Point Methods (IPMs) have revolutionized optimization theory and practice in the last three decades and have been proven to be quite efficient in solving different classes of LCPs that were inaccessible or difficult to solve using simplex-type pivoting methods. The path-following IPMs are based on following a central path trajectory of the LCP approximately by staying within a predetermined neighborhood of the central path towards a solution of LCP. The main tool in this process is Newton’s method which is used to determine the search direction at each iteration.

There are two types of path-following IPMs, IPMs with short-step updates at each iteration, and long-step updates at each iteration. Long-step IPMs take more aggressive steps that can deviate farther away from the central path, while short-step IPMs take more conservative steps staying closer to the central path. Most of the IPMs are based on the classical logarithmic barrier function [22] that is stated in the sequel in (7). The long-step versions of these IPMs have Onlognϵ iteration bound while short-step versions have Onlognϵ iteration bound which is for order of magnitude better. However, in practice, the situation is the opposite, long-step IPMs perform much better than short-step IPMs. This is considered the irony of IPMs, the term coined by Renegar [32]. Throughout the years a considerable effort has been devoted to designing long-step IPMs with improved theoretical iteration bounds to reduce, or even close, the gap between the theoretical complexities of short- and long-step IPMs. Several approaches can be observed.

The first approach was to consider higher-order methods based on the classical logarithmic barrier function [31]. The second approach was the IPM developed by Ai and Zhang [1] which was based on two key ideas: a new wide neighborhood and the decomposition of Newton’s directions into two components. In both of these approaches, the gap between short- and long-step versions of the IPMs was closed on the expense of more computational work.

The third approach was a fruitful idea of kernel-based IPMs introduced by Peng, Roos, and Terlaky [29] that led to long-step IPMs with an improved iteration bound. They considered the class of self-regular kernel functions (SRKFs). Subsequently, Bai, Roos, and El Ghami [7] introduced a different class of kernel functions, a class of eligible kernel functions (EKFs), and obtained a similar improvement for long-step IPMs for LO. These results were extended to P*(κ)-LCPs in [30] for SRKFs and in [22] for EKFs. For some instances of SRKFs and EKFs they obtained Onlognlognϵ iteration bound for long-step IPMs which is a significant improvement when compared with Onlognϵ for long-step IPM based on the classical logarithmic kernel function.

In this paper, we introduce a new kernel function which is a modification of the classical logarithmic kernel function and show that it is EKF and SRKF. The general convergence analysis of the IPMs based on the general EKFs presented in [22] is used to show the convergence of the IPM based on the new EKF. The iteration bounds for short- and long-step versions of the IPM are derived, which turns out to be as good as for the classical logarithmic kernel function.

It is also shown that the IPM developed in this paper provides an alternative efficient method to solve the Controlled Tabular Adjustment (CTA) problem [24] which is usually solved using the simplex method.

CTA is a method of Statistical Disclosure Limitation (SDL) that was first introduced in [10, 17]. The SDL is an increasingly important area of research and practice for the statistical agencies that collect data from individuals or enterprises and then release it to the public, researchers, and policymakers for statistical analysis and research. Prior to such a release, the collected data have to undergo some SDL procedure in order to guarantee the privacy and confidentiality of data providers. The goals of such procedures are two-fold: minimize the risk of disclosure of confidential information about data providers and, at the same time, maximize the amount of released information, that is, maximize the utility of the data for the legitimate data users. These are conflicting goals and therefore SDL practice as a whole can be thought of as a search for the solution of complex and multifaceted optimization problem: maximize the utility of the released data, subject to some upper bound on disclosure risk. The way utility and risk are formulated depends on the scenario of data release and on the data format.

Data can be released in two basic formats: microdata - a collection of individual records, and tabular data - a table of cumulative data that is obtained from cross-tabulations of attributes from microdata. CTA is a perturbative method of protecting tabular data when a specified subset of its cells, called sensitive cells, must be modified to avoid the re-identification of an individual respondent. The goal of CTA is to guarantee that the modified value of a sensitive cell is outside of the disclosure interval- an interval that is determined by the data protector (usually a statistical agency). The remaining cells are minimally adjusted to satisfy table equations which usually represent the requirement that the sum of elements in each row and column should be constant and remain unchanged. Hence, the goal of CTA is to find the closest safe table to the original table with respect to the constraints outlined above. The closeness of the original and modified table is measured by the weighted distance between the tables with respect to a certain norm. Most commonly used norms are ℓ1 and ℓ2 norms. Thus, the problem can be formulated as a minimization problem with the objective function being a particular weighted distance function and constraints being table equations and lower and upper bounds on the cell values. The ℓ2-CTA reduces to a Quadratic Optimization problem while the ℓ1-CTA is a convex but nonsmooth problem that can be reformulated as a Linear Optimization (LO) problem. However, the number of variables and inequality constraints doubles. Alternatively, in [24] a novel second-order cone (SOC) reformulation of ℓ1-CTA is proposed that does not increase the dimension of the problem as much. As it is shown in [24], conic reformulation of ℓ1-CTA is a viable alternative to LO reformulation of the problem.

For the sake of completeness, the general formulations of CTA and its various reformulations are given in the Numerical Results section of the paper. The IPM developed in this paper is applied to an LO formulation of a CTA problem that appears as a test problem in [24] and several other papers. It is shown to work well confirming the fact that it is a viable alternative for solving continuous CTA. We also apply the algorithm to the small set of randomly generated monotone LCPs showing that the initial implementation performs well on these instances of LCPs. However, this limited numerical testing is done for illustration purposes. More sophisticated implementation, extensive numerical testing, and comparison to other methods are needed to draw more definite conclusions about the practical performance of the method and are subject of future research.

The outline of the paper is as follows. The generic barrier-based IPM is presented in Section 2. The definitions of the kernel function, EKFs, and SRKFs are presented in Section 3.1. The new kernel function and corresponding barrier function are discussed in Section 3.2. The analysis of the algorithm based on the new kernel function and derivation of the iteration bounds for short- and long-step variants of the method are presented in Section 4. The application of the method to a CTA problem and preliminary numerical results are presented in Section 5 and the concluding remarks are outlined in Section 6.

As already indicated in the previous section, in this paper we consider the monotone LCP in the standard form (1). We assume that the LCP satisfies the interior-point condition (IPC), which means that there exists a point x0>0 such that s0=Mx0+q>0, indicating that the interior of the feasible region is not empty. The IPC can be assumed without loss of generality, since the original LCP can be embedded in the larger LCP with the known starting point as described in Section 5.1 in [21].

Most IPMs are based on the use of Newton’s method. However, it is well-known using the direct application of Newton’s method to the system (1) will most likely fail at the second equation called the complementarity equation. Hence, the idea is to replace the complementarity equation in (1) with a parametrized equation xs=μe with parameter μ>0 and e being a vector of all ones. Therefore, we consider the parametrized system (2) −Mx+s=q,xs=μe,x,s>0.

Since we assume that IPC holds and M is a positive-semidefinite matrix, it is well known that the parametrized system (2) has a unique solution, for each μ>0; see Lemma 4.1 in [21]. This solution is denoted as (x(μ),s(μ)) and it is called the μ-center of the LCP. The set of μ-centers, with μ running through all positive real numbers, forms a homotopy path, which is called the central path of the LCP. Under the above assumptions, if μ→0, the limit of the central path exists and it is a solution of LCP (1).

The limiting property of the central path mentioned above leads naturally to the main idea of the IPMs for solving LCP: Trace the central path while reducing μ at each iteration. However, tracing the central path exactly would be computationally too costly and inefficient. It has been shown that it is sufficient to trace the central path approximately within a certain neighborhood of the central path. The neighborhood of the central path is determined using a certain proximity measure to the central path which in this paper is determined by the kernel function and the corresponding barrier function.

Suppose an iterate (x, s) of the IPM in the neighborhood of the central path is known. We then decrease μ to μ:=(1−θ)μ, for some value of the barrier parameter θ∈(0,1). Applying Newton’s method to (2) leads to the following Newton’s system for the search directions (Δx, Δs): (3) −MΔx+Δs=0,sΔx+xΔs=μe−xs.

Since M is a positive semidefinite matrix, the above system has a unique solution for any (x,s)>0.

For the analysis of IPMs it is important to associate any pair (x,s)≥0 and μ>0 with the variance vector (4) v:=xsμ.

Note that the pair (x,s) coincides with the μ-center (x(μ),s(μ)) if and only if v=e. Furthermore, it is also important to introduce the following scaled search directions dx and ds (5) dx:=vΔxx,ds:=vΔss, where the operations are component-wise product and division. Using (4) and (5) the system (3) can be rewritten as (6) −M‾dx+ds=0,dx+ds=v−1−v, where M‾:=DMD, with D:=X1/2S−1/2, X:=diag(x), S:=diag(s).

A crucial observation is that the right-hand side of the second equation of (6) equals minus the gradient of the function (7) Ψcv:=∑i=1nvi2−12−logvi, where vi represents the i-th component of the variance vector v. In other words, (8) dx+ds=−∇Ψcv.

This equation is called the scaled centering equation. Its importance arises from two facts, first, it essentially defines the search direction, and second, it defines a proximity measure to the central path.

One may easily verify that ∇2Ψc(v)=diage+v−2. Since this matrix is positive definite, Ψc(v) is strictly convex. Moreover, since ∇Ψc(e)=0, it follows that Ψc(v) attains its minimal value at v=e, with Ψc(e)=0. Thus, Ψc(v) is nonnegative everywhere and vanishes if and only if v=e, that is, if and only if (x,s) is a μ-center, i. e., x=x(μ) and s=s(μ). Hence, we see that the μ-center (x(μ),s(μ)) can be characterized as the minimizer of the function Ψc(v). Thus, the second important feature of the function Ψc(v) is that it essentially serves as a ’proximity’ measure of closeness for (x, s) with respect to the μ-center.

The above observations regarding the function Ψc(v) lead to an obvious generalization: We can replace Ψc(v) by any strictly convex function Ψ(v), v∈R++n, such that Ψ(v) is minimal at v=e and Ψ(e)=0. Thus, the new scaled centering equation becomes (9) dx+ds=−∇Ψv.

The function Ψ(v) is called a (scaled) barrier function. Hence, different barrier functions lead to different Newton’s directions, as they are calculated from Newton’s system (10) −M‾dx+ds=0,dx+ds=−∇Ψv.

Since Ψv=0⇔∇Ψv=0⇔v=e, the function Ψ(v) still serves as a proximity measure of closeness to the μ-center (x(μ),s(μ)) on the central path.

Introducing a parameter τ>0 as a threshold value, the inequality Ψ(v)≤τ defines a τ-neighborhood of the central path.

In the sequel we also use norm-based proximity measure, which is defined by (12) δv:=12‖∇Ψ(v)‖.

The function δ(v) is often called proximity function and it is easy to see that (13) δ(v)=0⇔v=e.

From the above discussion, the main idea of the IPM based on the appropriate barrier function follows. Given the current iterate (x,s), we start the new outer iteration by targeting the new μ+-center obtained by reducing μ by a factor 1−θ with 0<θ<1, that is, μ+=(1−θ)μ. This implies the change in the variance vector v, v+=v1−θ. Most likely, after this step, the value of the barrier function will exceed the threshold value, that is, Ψv+>τ, and hence, the algorithm enters the inner iteration loop. The inner iteration loop consists of computing the scaled search directions dx and ds at the current iterate and the current value of μ+ from Newton’s system (10), which we rename to be a new μ. Next, we compute the corresponding search directions Δx and Δs from dx and ds by using (5). Then, the new inner iterate is calculated as (14) x+=x+αΔx,s+=s+αΔs with the appropriately calculated step size α. If necessary, the procedure is repeated, however, with keeping μ fixed, until the iterate belongs to the τ-neighborhood (11) of the current μ-center (x(μ),s(μ)), that is, until Ψ(v)≤τ. This process is repeated until μ is small enough, which is usually measured by nμ<ϵ. At this stage, we have found an ϵ-approximate solution of LCP (1).

Note that the algorithm can be started, since, as we discussed at the beginning of this section, we may assume that a strictly feasible point (x0,s0) is given, and this point can be chosen such that Ψv0≤τ which means it is in the τ-neighborhood of the μ-center.

The generic form of the barrier-based IPM for monotone LCP is given in Figure 1.

The parameters τ, θ and the step size α in the algorithm should be tuned in such a way that the number of iterations required by the algorithm is as small as possible.

Two main types of the algorithm can be distinguished: short-step method and long-step method, according to the value of the barrier-update parameter θ. The long-step method is characterized by the fact that θ is a fixed constant (θ∈(0,1)), independent of the dimension n of the problem, whereas the short-step method uses a value of θ that depends of the dimension of the problem, with θ=O1n.

The resulting iteration bound depends on a careful selection of these parameter values. It also heavily depends on the choice of the barrier function.

In this section, we analyze the algorithm presented in Figure 1 for the specific barrier function (19) derived from the new EKF (18). In the sequel, we refer to the algorithm in Figure 1 with this specific barrier function as simply the Algorithm.

We first give an overview of the analysis of the algorithm with general EKF. The analysis is performed in the following four major steps.

We will not repeat this general analysis here, instead, we refer the reader to [22] where the details of the analysis and the proofs can be found. We will especially use the Scheme from [22] which streamlines the analysis of the generic algorithm and calculations of iteration bounds for short- and long-step versions of the method. However, to better follow the process, we first state the definition of the following two functions that play a crucial role in the analysis of the generic algorithm and derivation of the iteration bounds and are frequently used in the Scheme.

Numerical experiments were implemented in Python and carried out on a regular PC operating on a 64-bit Operating System Windows 10 machine, lntel(R) Core(TM) i7–10700K CPU @ 3.80GHz, 16.0 GB RAM.