The statistical disclosure limitation (control) is the term that describes the theory and methods of protecting sensitive information when releasing statistical microdata or tabular data. An up-to-date overview of theory and methods of this field can be found in the monograph [19] and, for tabular data only, in the survey [8]. An excellent reference is also [27].

Minimum-distance controlled tabular adjustment (CTA) methodology was first introduced in [7,15]. As indicated in [4] CTA can be formulated as the following problem: Given a table with sensitive cells, compute the closest safe table in which sensitive cells are modified to avoid recomputation, and the remaining cells are minimally adjusted to satisfy the table equations. 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.

In general, CTA is Mixed Integer Optimization Problem (MIOP) which is a difficult problem to solve especially for the large dimension problems. A priori fixing the values of binary variables reduces the problem to the continuous optimization problem which is easier to solve, however, the quality of the solution may be reduced. In addition, the values of the binary variables have to be assigned carefully otherwise the problem may become infeasible [12, 13].

The objective function in continuous CTA is based on either the ℓ₁-norm or ℓ₂-norm. The formulation of ℓ₂ -CTA leads to the Quadratic Programing (QP) problem, while ℓ₁-CTA can be formulated as the Linear Programming (LP) problem. However, the resulting LP has the number of variables that is twice the number of cells of the table as opposed to ℓ₂-CTA where the resulting QP problem has a number of variables equal to the number of cells. In general, the QP of ℓ₂-CTA is usually more efficiently solved than the LP of ℓ₁-CTA [4,7].

In [4] the Pseudo-Huber regularization of the ℓ₁-CTA is proposed. The Pseudo-Huber approximation of the ℓ₁-norm objective function leads to the convex optimization problem. However, the advantage is that the number of variables in Pseudo-Huber formulation of the ℓ₁-CTA remains the same as the number of cells. In [4] it is shown that Pseudo-Huber-CTA can be more efficiently solved than LP ℓ₁-CTA for certain types of tables and using an appropriate method that takes into account the structure of the problem.

All these models are solved using appropriate versions of the Interior-Point Method (IPM). These methods have been developed in recent years to efficiently solve different types of, often large, nonlinear (convex) optimization problems. It has been shown both theoretically and numerically that IPMs perform better on problems that have a certain structure, such as Conic Optimization (CO) problems, which are LP problems where variables are elements of cones. Most common cones are non-negative orthant, second order (quadratic) cone and semidefinite cone [2, 3, 25].

Hence, motivated by the above comment, in this paper we develop a new Second Order (Quadratic) Cone (SOC) formulation of the ℓ₁ and Pseudo-Huber-CTA. It is shown on the small example of a two-dimensional table that SOC CTA models are more efficiently solved than the original models. It is expected that the same will be the case for larger and more complex tables. Extensive numerical testing on various types of tables is beyond the scope of this paper; however, it is needed and it is forthcoming as a part of future research.

The paper is organized as follows. In Section 2 the general MIOP and then continuous CTA are formulated. Then the ℓ₁ and ℓ₂ continuous CTA are derived. The Pseudo-Huber-CTA formulation is considered in Section 3. The new SOC formulations of both Pseudo-Huber and ℓ₁ CTA are developed in Section 4. In Section 5 the SOC CTA models are applied to the small example of two-dimensional table and these instances are solved using MOSEK SOC solver. The concluding remarks are given in Section 6.

The following CTA formulation is given in [4]: Given the following set of parameters:

A set of cells a_i, i∈N={1,…,n}. The vector a = (a₁, . . . , a_n)^T satisfies certain linear system Aa = b where A ∈ ℝ^m×n is an m × n matrix and and b ∈ ℝ^m is m-vector.

A CTA problem is a problem of finding values z_i, i∈N, to be released, such that z_i, i∈S are safe values and the weighted distance between released values z_i and original values a_i, denoted as ‖z − a‖_l(w), is minimized, which leads to solving the following optimization problem (1)minz‖z−a‖l(w)s.t.Az=b,lai≤zi≤uai,i∈N,zi,i∈Saresafevalues.

As indicated in the assumption (iv) above, safe values are the values that satisfy (2)zi≤ai−lpliorzi≥ai+upli,i∈S.

By introducing a vector of binary variables y ∈ {0,1}^s the constraint (2) can be written as (3)zi≥−M(1−yi)+(ai+upli)yi,i∈S,zi≤Myi+(ai−lpli)(1−yi),i∈S, where M ≫ 0 is a large positive number. Constraints (3) enforce the upper safe value if y_i = 1 or the lower safe value if y_i = 0.

Replacing the last constraint in the CTA model (1) with (3) leads to a mixed integer convex optimization problem (MIOP) which is in general a difficult problem to solve; however, it provides solutions with high data utility [11]. The alternative approach is to fix binary variables up front which leads to a CTA that is acontinuous convex optimization problem. The continuous CTA may be easier to solve; however, the obtained solution may have a lower data utility. Furthermore, a wrong assignment of binary variables may result in the problem being infeasible. Strategies on how to avoid this difficulty are discussed in [12, 13].

In this paper we consider a continuous CTA where binary variables are fixed and vector z is replaced by the vector of cell deviations (4)x=z−a.

The CTA (1) with constraints (3) reduces to the following convex optimization problem: (5)minx‖x‖l(w)s.t.Ax=0,l≤x≤u, where upper and lover bounds for x_i, i∈N are defined as follows: (6)li={upliifi∈Sandyi=1lai−aiif(i∈N\S)or(i∈Sandyi=0) (7)ui={−lpliifi∈Sandyi=0uai−aiif(i∈N\S)or(i∈Sandyi=1).

The two most commonly used norms in problem (5) are the ℓ₁ and ℓ₂ norms. For the ℓ₂-norm the problem, (5) reduces to the following ℓ₂-CTA model which is a QP problem: (8)minx∑i=1nwixi2s.t.Ax=0,l≤x≤u.

For the ℓ₁-norm the problem, (5) reduces to the following ℓ₁-CTA model: (9)minx∑i=1nwi|xi|s.t.Ax=0,l≤x≤u.

The above ℓ₁-CTA model (9) is a convex optimization problem; however, the objective function is not differentiable at x = 0. Since most of the algorithms, including IPMs, require differentiability of the objective function, problem (9) needs to be reformulated. The reformulations that have been considered in [4] are reviewed in the next section.

The ℓ₂-CTA model (8) is a standard QP problem that can be efficiently solved using IPM or other methods. However, as noted at the end of the previous section, the ℓ₁-CTA model (9) needs reformulation in order to be efficiently solved by IPM or some other method. The standard reformulation is the transformation of model (9) to the following LP model: (10)minx−,x+∑i=1nwi(xi++xi−)s.t.A(xi+−xi−)=0,l+≤x+≤u+,l−≤x−≤u−, where (11)x+={xifx≥00ifx<0,x−={0ifx>0−xifx≤0, and lower and upper bounds for xi− and xi+, i∈N are as follows: (12)li+={upliifi∈Sandyi=10if(i∈N\S)or(i∈Sandyi=0)ui+={0ifi∈Sandyi=0uai−aiif(i∈N\S)or(i∈Sandyi=1)li−={lpliifi∈Sandyi=00if(i∈N\S)or(i∈Sandyi=1)ui−={0ifi∈Sandyi=1ai−laiif(i∈N\S)or(i∈Sandyi=0).

Problem ℓ₁-CTA (10) is an LP problem; however, it has twice the number of variables as the QP problem (8) and twice the number of box constraints. As indicated in [4], the splitting of the variables x = x⁺−x⁻ and the increased dimension of the model may cause problems. In order to overcome these difficulties in [4] it was suggested to use a regularization of problem (9) by approximating absolute value with the Pseudo-Huber function that has the same number of variables as in the QP formulation (8).

The original Huber function φ_δ : ℝ → ℝ₊ is defined as (13)φδ(xi)={xi22δ|xi|≤δ|xi|−δ2|xi|≥δ. It approximates |x_i| for small values of δ > 0; the smaller the δ, the better the approximation.The Huber function is continuously differentiable; however, the second derivative is not continuous at |x_i| = δ which may cause problems when this function is used in second order optimization algorithms, such as IPMs. Hence, it is better to consider the Pseudo-Huber function ϕ_δ : ℝ → ℝ₊ (14)ϕδ(xi)=δ2+xi2−δ whose first and second derivatives are bounded and Lipschitz continuous [17]. Again, the smaller the δ the better the approximation.

Now, the ℓ₁-CTA problem (9) can be approximated by the following convex optimization problem (15)minx∑i=1nwiϕδ(xi)s.t.Ax=0,l≤x≤u.

The advantage of the Pseudo-Huber-CTA model (15) is that it has the same number of variables as ℓ₂ - CTA and the same feasible region, the only difference is that the quadratic objective function is replaced by a strictly convex function.

Optimization problems (8), (10) and (15) can be solved with appropriate versions of the Interior-Point Methods (IPM). Since IPMs are the methods of choice to solve different CTA models, in the rest of the section we describe the main ideas of IPMs, only on a conceptual level, and then we discuss their application on given CTA models.

IPMs have in many ways revolutionized the optimization theory and practice in the past three decades since the appearance of the Karmarkar’s breakthrough paper [20]. Since then, the field of IPMs has been a very active area of research with literary thousands of papers published as well as numerous excellent monographs and textbooks. The general theory of IPMs for convex optimization problems can be found in the seminal monograph of Nesterov and Nemirovskii [26]. In addition to this monograph, and without any attempt to be complete, we mention a few other relevant references [29, 28, 22]. The reason for such an interest is that IPMs have proven to be very efficient in solving large linear and non-linear (convex) optimization problems which were previously hard to solve. Now-days almost every relevant optimization software, whether commercial or open source, contains an IPM solver which is capable of solving at least LP problems and in many cases QP problems, and, less frequently, conic optimization problems. In the case of LP there are plenty of numerical studies showing that IPMs are at least as efficient, if not more, as the classical Simplex Method (SM) on large scale LP problems.

The basic idea of path-following IPMs, that are most commonly used and studied, is centered around approximately following the parametric trajectory that is called central path which leads to the solution of the problem when a parameter is approaching zero. The points on the central path are called μ-centers and are obtained as solutions of the Karush-Kuhn-Tucker (KKT) optimality conditions of the problem where a (the) complementarity equation(s) is (are) perturbed by a positive parameter μ > 0. In particular, the perturbed KKT system for Pseudo-Huber-CTA is explicitly listed in [4].

The solution of the problem, which is obtained when μ = 0, is found by tracing the central path while gradually reducing μ to zero. However, tracing the central path exactly would be prohibitively inefficient. The main achievement of IPMs have been to show that it is sufficient to trace the central path approximately; as long as the iterates are in the certain neighborhood of the central path, it is still possible to prove global convergence and, moreover, show that the -approximate solution of the problem, according to the appropriate proximity measure, can be obtained in polynomial number of iterations with the best theoretical upper bound being O(nlognϵ), where n represents the number of variables of the problem at hand.

However, practical behavior of IPM heavily depends on many factors, such as the structure of the problem, the starting point, the accuracy needed, etc. As reported in [4], Pseudo-Huber-CTA (15) can be difficult to solve with a general convex optimization solver even for small instances if the solver is not ‘appropriately tuned’. However, for problems that exhibit a special structure such as 3-D tables whose constraints have a block-angular structure, the specialized block-angular IPM of J. Castro [5, 9, 10] solves Pseudo-Huber-CTA more efficiently than ℓ₁-CTA while ℓ₂-CTA has by far the best CPU time. Hence, Pseudo-Huber-CTA is a viable option for solving ℓ₁-CTA; however, the IPM have to be implemented with care and, in addition, the specialized IPM may not work efficiently for other types of tables. As indicated in [4], modifications and tuning of the Block-angular IPM so it can handle large and complex tables of different types is a direction for future research.

Another direction in searching how to efficiently solve Pseudo-Huber-CTA and ℓ₁-CTA is to investigate whether these models can be transformed into the conic optimization (CO) problems. The motivation for such investigation comes from the fact that it has been established both theoretically and numerically that IPMs perform better on the well structured problems such as CO problems than on general convex optimization problems [2, 3, 25]. CO problems are LP problems over cones, that is, variables belong to certain types of cones. Most common cones are either non-negative orthant, second-order (quadratic) cone or semidefinite cone definitions; of which are listed in the next section. Thus, formulating Pseudo-Huber and ℓ₁-CTA as CO problems would be advantageous. In the next section we develop SOC formulation of both Pseudo-Huber and ℓ₁ CTA.

In this section we investigate how Pseudo-Huber and ℓ₁ CTA can be formulated as SOC models. The CO problems can be formulated as (16)minxcTxs.t.Ax=b,x∈K, where K is a cone of the following three types:

The linear cone or non-negative orthant: K=ℝ+n:={x∈ℝn:xi≥0,i=1,…,n}.

More generally, K can be a Cartesian product of the above mentioned cones. It is also worth mentioning that the cones defined above are examples of symmetric cones, thus problem (16) can be considered in a more general framework of Symmetric Optimization (SO) problems, see [16,18, 24] and references therein.

In what follows, we present a reformulation of Pseudo-Huber-CTA problem (15) as a SOC problem. Consider Pseudo-Huber Function (14) ϕδ(xi)=δ2+xi2−δ. Let’s define (17)ti:=δ2+xi2andyi:=δ,i=1,…,n. Hence, we have ti=xi2+yi2 which is the boundary of the second-order (quadratic) cone Ki={(xi,yi,ti)∈ℝ3:ti≥xi2+yi2}.

Now, the reformulation of the Pseudo-Huber-CTA (15) as a SOC problem follows (18)maxx∑i=1nwi(ti−yi)s.t.Ax=0,yi=δ;i=1,…,n,(xi,yi,ti)∈Ki;i=1,…,n,l≤x≤u.

This model is valid even for δ = 0. In that case we obtain a SOC formulation of the l1-CTA (9) (19)minx∑i=1nwitis.t.Ax=0,(xi,ti)∈Ki;i=1,…,n,l≤x≤u. This model could have been obtained directly from l1-CTA (9) because the absolute value has an obvious second-order cone representation since the epigraph of the absolute value function is exactly second-order cone, that is, ti=|xi|→Ki={(xi,ti)∈ℝ2:ti≥xi2}.

It is well known that the solutions of SOC problems (18) and (19) achieve solutions at the boundary of the cones, hence, equations (17) will hold at the solution [2,3]. Thus, it is not necessary to enforce these equations in SOC models; in fact, their inclusion would lead to noncovex problems that would be difficult to solve.

An IPM for SOC can now be used to find an -approximate solutions to SOC Pseudo-Huber and ℓ₁ CTA models. We have used MOSEK SOC solver [1] that is considered one of the best, if not the best, SOC solver available on the market today.

In this section an example of the small two-dimensional table stated in Figure 3 in [4] is considered. The table is listed in Figure 1 below as the table (a).

The continuous CTA model based on the table (a) is formulated in the following way:

The linear constraints are obtained from the requirement that the sum of the elements in each row (or column) remains constant and is equal to the corresponding component in the last column (or row) of table (a).

From this basic CTA model different CTA models discussed in the paper were formulated and then these models were solved using appropriate IPM solvers. The results are listed in Figure 1. Below is the summary of the IPM solvers used.

The ℓ₂-CTA (8) instance was solved in [4] using IPM based MOSEK QP solver. Table (d).

In [4] it was observed that ℓ₂-CTA had the fastest execution. Hence, we replicated the solution of the ℓ₂-CTA instance of the example and compared its performance with SOC models instances. The calculations were carried out on a Lenovo ThinkPad W530 computer with Intel(R) CORE i7–3740QM 2.70GHz processor. The results are given in Table 1.

From Table 1 we can observe that SOC versions are comparable to the ℓ₂ version both in number of iterations and CPU time; SOC ℓ₁ was slightly faster than ℓ₂ while SOC Pseudo-Huber was slightly slower, which is the expected result. Hence, the SOC models are more effective than the LP ℓ₁ and Pseudo-Huber-CTA models for this example.

Furthermore, for LP ℓ₁, Pseudo-Huber ϕ_0.001, SOC ℓ₁, and SOC Pseudo-Huber ϕ_0.001 CTA instances the optimal values of their respective objective functions are the same, namely, the value is 20, while for ℓ₂-CTA instance it is 20.69. Thus, the objective values for SOC Pseudo Huber and ℓ₁-CTA instances are the same as for the original non-SOC instances, namely 20, which was expected.

These results are in line with plenty of other evidence that it is advantageous to solve the SOC formulation of the problem by IPM, rather than using IPM to the original formulation of the problem (see for example [2, 3, 25, 23]). We are confident that the advantages of the SOC models will be even more visible when applied to larger tabular data sets. Moreover, the SOC IPM is robust and flexible enough to handle different types of tables.

The main goal of the paper is mainly theoretical, that is, to present a Second Order Cone (SOC) formulation of the Pseudo-Huber and the ℓ₁ CTA models, (18) and (19) respectively as an alternative to the original Pseudo-Huber and LP ℓ₁ CTA models, (15) and (10) respectively. The application of the SOC models to the small example in Section 5 shows promise to be an effective alternative to the application of the original models to the small example. More numerical testing is needed and is forthcoming as a future research topic where SOC models would be implemented and tested on the different types of tables of large dimensions mentioned in the Conclusion of [4].

From Figure 1, it can be observed that the resulting tables for all the models except LP ℓ₁ change most of the cells of the original table (a) that are not fixed, even the ones that are not sensitive cells. The reason lays in the nature of IPMs. In these methods, the iterates approximately follow the central path that converges to the analytic center of the optimal set which implies that most of the cells will be changed, while the IPM with crossover or alternatively the Simplex Method, for LP ℓ₁ finds the basic solution which implies fewer cells will be changed. Hence, if there is a requirement to minimize the number of non-sensitive cells that are changed, then the LP ℓ₁ models solved with SM or IPM with crossover is the right approach. However, if the number of nonsensitive cells changed is not an issue such as for certain types of magnitude tables, then the suggested approach is to use either the SOC ℓ₁ model or the ℓ₂ model because they are faster. Unless prior regularization of the ℓ₁ model is necessary, which then leads to the Pseudo-Huber model and related SOC Pseudo-Huber model, it is more efficient to use the SOC ℓ₁ model directly.

As noted in [4], it has been empirically shown that CTA in general exhibits a low disclosure risk [6] and, at the same time, high data utility [14, 13] (see also [21]). However, the study of the disclosure risk and data utility of tables protected by the Pseudo-Huber-CTA model and the SOC CTA models is lacking and is certainly an interesting future research topic.