Iteration Complexity of an Infeasible Interior Point Methods for Seconder-order Cone Programming and its Warmstarting

The following notations are used throughout the paper. $\mathcal{R}_{+}$ denotes the set of positive real numbers, $\mathcal{R}^{n}$ denotes the $n$-dimensional Euclidean space, and $\mathcal{R}^{m\times n}$ denotes the set of all $m\times n$ matrices with real entries. $\mathbf{I}_{n}$ denotes the identity matrix of order $n$. $\mathbf{G}\succ 0$ means $\mathbf{G}$ is symmetric and positive definite, and $\mathbf{G}\succeq 0$ means $\mathbf{G}$ is symmetric and positive semi-definite. For a square matrix $\mathbf{M}$ with all real eigenvalues, its largest and smallest eigenvalues are denoted by $\lambda_{\max}(\mathbf{M})$ and $\lambda_{\min}(\mathbf{M})$. For a set $\mathcal{K}$, $\text{int}(\mathcal{K})$ denotes its interior. $\text{diag}(\cdots)$ denotes a diagonal matrix using a given vector as diagonal, and $\text{blkdiag}(\cdots)$ means a block diagonal matrix with given matrix blocks. $\left\|\cdots\right\|$ without specification is used instead of $\left\|\cdots\right\|_{2}$ for simplicity. With a little abuse of notations, when concatenating column vectors $\mathbf{x}$ and $\mathbf{y}$, we use $(\mathbf{x},\mathbf{y})$ instead of $(\mathbf{x}^{T},\mathbf{y}^{T})^{T}$ for simplicity.

SOCP minimize a linear function over the intersection of an affine set and the direct product of linear cones (LC) and second-order cones (SOC).

The linear cones $\mathcal{K}_{L}^{n}$ and second-order cones $\mathcal{K}_{S}^{n}$ are defined as

The 1-dimensional SOC is defined as $\mathcal{K}_{S}^{1}=\mathcal{K}_{L}^{1}$.

Then, the standard form of SOCP is defined as

where $\mathbf{A}\in\mathcal{R}^{p\times n}$, $\text{rank}(\mathbf{A})=p\leq n$. Without loss of generality, we assume that in the solution variable the linear cone is arranged at first, followed with the SOCs. The number of cones is

where the $l$-dimensional LC $\mathcal{K}_{L}^{n}$ is viewed as $l$ 1-dimensional LCs, so that $n_{i}=1$ for $i\in\left\{1,\cdots,l\right\}$. The $i$-th cone is also mentioned as $\mathcal{K}^{i}$ for simplicity, and correspondingly $\mathcal{K}=\mathcal{K}^{1}\times\cdots\times\mathcal{K}^{k}$. For a vector $\mathbf{v}\in\mathcal{K}$, we use $\mathbf{v}^{(i)}$ to denote the subvector in the cone $\mathcal{K}^{i}$. For $\mathbf{v}\in\mathcal{R}^{n}-\mathcal{K}$, $\mathbf{v}^{(i)}$ also denotes the subvector corresponding to the position of $\mathcal{K}^{i}$.

The dual problem of SOCP Eq. (2) is

This subsection introduce some important concepts about SOCP for later use.

Firstly, we introduce the Euclidean Jordan algebra associated with the SOC (see e.g. [Faraut and Koranyi, 1994, Faybusovich, 1997a, b]), which is crucial in the IPMs for SOCP. The algebra for the SOC $\mathcal{K}^{i}$ is defined as

The Euclidean Jordan algebra for $\mathcal{K}$ is defined as

From now on, we assume that the space $\mathcal{R}^{n}$ is endowed with the Euclidean Jordan algebra.

The unit element of the algebra is $\mathbf{e}=\left(\mathbf{e}^{(1)},\cdots,\mathbf{e}^{(k)}\right)$, where $\mathbf{e}^{(i)}=(1,0,\cdots,0)^{T},i\in{1,\cdots,k}$, because $\mathbf{v}\circ\mathbf{e}=\mathbf{v}$. Then, the algebra can be written as multiplication of arrow head matrices and the unit element, as

where the arrow head matrix for $\mathbf{v}\in\mathcal{R}^{n}$ is defined as

It is easy to verify that the eigenvalues of the arrow head matrix satisfies

Consequently, we define the notation

For a vector $\mathbf{v}\in\mathcal{K}$, $\lambda_{i}^{1}(\mathbf{v})$ and $\lambda_{i}^{2}(\mathbf{v})$ are used as abbreviations of $\lambda_{\min}(\mathbf{v}^{(i)})$ and $\lambda_{\max}(\mathbf{v}^{(i)})$, where $\mathbf{v}^{(i)}$ denotes the part of $\mathbf{v}$ in the $i$-th cone.

Secondly, we introduce the homogeneous and self-dual (HSD) model and the central path. The HSD model combines the primal and dual problem, and enables to solved them simultaneously. The model is given by

where $\nu\in{{R}_{+}}$ is free, and ${{\mathbf{r}}_{p}}\in{{R}^{p}}$, ${{\mathbf{r}}_{d}}\in{{R}^{n}}$, ${{r}_{g}}\in R$, $\beta\in R$ are residuals defined by

${{\nu}_{0}}$ is typically set as 1.

The solution of the HSD model fully describes the property of the primal problem and dual problem, that is summarized by the following theorem.

Theorem 1. Let $\left(\mathbf{x},\mathbf{y},\mathbf{s},\kappa,\tau\right)$ be an optimal solution of the HSD model Eq. (11). Then, the following relations hold [Terlaky et al., 2002, Wang, 2003]:

a) $\mathbf{x}^{T}\mathbf{s}=0$ and $\kappa\tau=0$;

b) If $\tau>0$, then $\left(\mathbf{x},\mathbf{y},\mathbf{s}\right)/\tau$ is the optimal solution of the primal problem Eq. (2) and the dual problem Eq. (LABEL:SOCP_DUAL);

c) If $\kappa>0$, then at least one of the inequities $\mathbf{b}^{T}\mathbf{y}>0$ and $\mathbf{c}^{T}\mathbf{x}<0$ hold. If the first inequality holds then the primal problem Eq. (2) is infeasible. If the second inequality holds then the dual problem Eq. (LABEL:SOCP_DUAL) is infeasible.

Proof. see corollary 1.3.1 of [Wang, 2003].

The central path is defined by

The central path approaches the optimal solution of the HSD model as $\mu\to 0$, that corresponds to the solution or an infeasibility certificate of the SOCP problem.

Thirdly, we introduce the scaling matrix, that can be applied to scale the searching directions. Monteiro and Tsuchiya [2000] introduces a group of scaling auto-morphism that maps the cone $\mathcal{K}^{i}$ onto itself, as the following group of matrices

where $\theta^{i}$ is a positive scaling value, and

The set of scaling matrice for a vector in $\mathcal{K}$ is defined as

where $i\in\left\{1,\cdots,k\right\}$ and

The solution variables $\mathbf{x}$ and $\mathbf{s}$ are scaled by

For convenience, we denote a scaling matrix by

The following proposition gives several important properties of the scaling matrix.

Proposition 1 [Andersen et al., 2003]. Let $\mathbf{D}\in\mathcal{G}$, then

Proof. See Lemma 3 of [Andersen et al., 2003].

For a vector $\mathbf{v}\in\text{int}\mathbf{(}K)$, Tsuchiya [1999] introduces a special scaling matrix $\mathbf{T}_{v}$ defined as

where

for $\mathbf{u}\in\mathcal{R}^{n}$.

Proposition 2 [Tsuchiya, 1999]. For a vector $\mathbf{v}\in\text{int}\mathbf{(}K)$, $\mathbf{T}_{v}$ is the unique scaling matrix which maps $\mathbf{e}$ to $\mathbf{T}_{v}$. Moreover, $\mathbf{T}_{v}\succ 0$.

Proof. See Proposition 2.1 of [Tsuchiya, 1999].

Finally, it is ready to define the scaling-invariant neighborhoods of the central path. For $(\mathbf{x},\mathbf{s},\kappa,\tau)\in\text{int}(\mathcal{K})\times\text{int}(\mathcal{K})\times R_{+}\times R_{+}$, define the central path coefficient

With a little abuse of notation, we also use $\mu$ instead of $\mu(\mathbf{x},\mathbf{s},\kappa,\tau)$ for simplicity. The central path distances are defined as

where

for $\mathbf{v}\in\mathcal{K}$.

The neighborhoods of the central path determined by the above distance are

where the parameter $\gamma\in(0,1)$. For $(\mathbf{x},\mathbf{y},\mathbf{s},\kappa,\tau)\in\text{int}(\mathcal{C})$, it is easy to show $d_{\infty}(\mathbf{x},\mathbf{s},\kappa,\tau)\leq d_{2}(\mathbf{x},\mathbf{s},\kappa,\tau)$, so that $\mathcal{N}_{2}(\gamma)\subseteq\mathcal{N}_{\infty}(\gamma)$.

The following proposition shows the invariance property of $\mathbf{w}_{xs}$.

Proposition 3 Tsuchiya [1999]. Suppose that $(\mathbf{x},\mathbf{s})\in\text{int}(\mathcal{K})\times\text{int}(\mathcal{K})$. Let $\mathbf{w}\triangleq\mathbf{w}_{xs}$ and $\mathbf{\bar{w}}\triangleq\mathbf{w}_{\bar{x}\bar{s}}$, $i\in{1,\cdots,k}$. Then:

a) $\mathbf{\bar{w}}_{0}^{(i)}=\mathbf{w}_{0}^{(i)}$ and $\|\mathbf{\bar{w}}_{1}^{(i)}\|=\|\mathbf{w}_{1}^{(i)}\|$;

b) $\lambda_{\min}(\mathbf{\bar{w}}_{0}^{(i)})=\lambda_{\min}(\mathbf{w}_{0}^{(i)})$ and $\lambda_{\max}(\mathbf{\bar{w}}_{0}^{(i)})=\lambda_{\max}(\mathbf{w}_{0}^{(i)})$;

c) $d_{2}(\mathbf{\bar{x}},\mathbf{\bar{s}})=d_{2}(\mathbf{x},\mathbf{s})$ and $d_{\infty}(\mathbf{\bar{x}},\mathbf{\bar{s}})=d_{\infty}(\mathbf{x},\mathbf{s})$.

Proof. See Proposition 2.4 of Tsuchiya [1999].

Using Proposition 3, it is easy to verify that for a point $(\mathbf{x},\mathbf{y},\mathbf{s},\kappa,\tau)$ on the central path defined by Eq. (13), the distances $d_{2}(\mathbf{x},\mathbf{s},\kappa,\tau)=d_{\infty}(\mathbf{x},\mathbf{s},\kappa,\tau)=0$.

In this subsection, we briefly introduce the infeasible IPM for SOCP studied in the paper.

The SOCP problem can be solved by loosely track the central path towards the optimal solution of the HSD model. In such a way, the updating direction in each iteration is computed by solving

where $\mathbf{X}=\text{mat}(\mathbf{x})$, $\mathbf{S}=\text{mat}(\mathbf{s})$. $\mu$ is $\mu(\mathbf{x},\mathbf{s},\kappa,\tau)$ defined in Eq. (23), which is proportional to the complementarity gap $\mathbf{x}^{T}\mathbf{s}+\kappa\tau$.

Instead of using the searching direction obtained by solving Eq (27) directly, the MZ family of drections are applied to speed up the convergence. The directions are calculated by solving a scaled linear system, as

where

The scaled linear system Eq (28) is obtained by using the scale variable $\bar{\mathbf{x}},\bar{\mathbf{s}}$ instead of $\mathbf{x}$ and $\mathbf{s}$ in the HSD model Eq. (11).

The MZ family of directions include the HRVW/KSH/M directions and NT directions as special cases. For example, in the NT scaling, the scaling matrix $\mathbf{D}$ is symmetric and satisfies $\mathbf{x}=\mathbf{D}^{2}\mathbf{s}$, and the multiplication with $\mathbf{D}$ or $\mathbf{D}^{2}$ can be calculated in $O(n)$ complexity.

Then, the infeasible IPM for SOCP is presented as Algorithm 1.

In Algorithm 1, the stop criteria is as follows:

where the primal residual $\mathbf{r}_{p}$ and the dual residual $\mathbf{r}_{d}$ are defined as

For convience, we also define

The criteria indicates that the primal residual, dual residual, and complementarity gap are reduce by a certain ratio $\epsilon$. When the criteria is satisfied, the final status is decided using an approximate version of Theorem 1: if $\tau\geq\epsilon\max(1,\kappa)$, then the problem is solved; if $\tau<\epsilon\max(1,\kappa)$, the primal problem is infeasible if $\mathbf{b}^{T}\mathbf{y}>0$, and the dual problem is infeasible if $\mathbf{c}^{T}\mathbf{x}<0$. Some important issues in implementation such as the maximum iteration number limit and rounding errors are not considered in the theoretical analysis of this paper.

Algorithm 1 obtains the solution for the HSD model. If the problem is solved, the solution for the SOCP problem is $(\mathbf{x}/\tau,\mathbf{y}/\tau,\mathbf{s}/\tau)$. In oder to guarantee the convergence of Algorithm 1, the parameters $\gamma,\delta$ should be chosen to satisfy certain conditions, which will be presented in Proposition 4 in the next section.

In the HSD model, $\kappa$ and $\tau$ are variables in $\mathcal{K}_{L}^{1}$, but they act differently from other variables in LCs. Although we find it is theoretically not difficult to consider them as additional terms, the analysis in such a way would be long and tedious. In order to simplify the analysis, we propose a equivalent form of the HSD model in this subsection.

Consider the following change of variables

where $\hat{\mathbf{x}},\hat{\mathbf{s}}\in\mathcal{K}\times\mathcal{K}_{L}^{1}$ and $\mathbf{y}\in\mathcal{R}^{p}$.

It is easy to verify that the linear system Eq. (27) can be written as

where $\hat{\mathbf{X}}=\text{mat}(\hat{\mathbf{x}})$, $\hat{\mathbf{S}}=\text{mat}(\hat{\mathbf{s}})$. It is worth noting that the second and third equations in Eq. (27) are combined into the second equation in Eq. (34).

The scaled linear system Eq. (28) is written as

where the scaling matrix

The central path coefficient is also simplified as

Because the two linear systems are identical, the IPM solving the HSD model by Eq. (28) is identical to solving the model by Eq. (35). Without loss of generality, the last LC $\mathcal{K}_{L}^{1}$ can be viewed as a SOC $\mathcal{K}_{S}^{1}$. Then, with a little abuse of notation, Eq. (35) is a special case of the following generalized linear system

where the variables are redefined for simplicity, as

The central path coefficient for the generalized linear system Eq. (38) is defined as

where $\mathbf{x}^{T}\mathbf{s}$ is the complementarity gap . The residuals are defined as

For the special case Eq. (35), the generalized linear system Eq. (38) has 1 additional dimensional in both $\mathbf{x}$ and $\mathbf{s}$ compared with the original form Eq. (28), to substitute $\kappa$ and $\tau$. Meanwhile, its cone constraint $\mathcal{K}$ is also 1-dimensional higher. The scaling matrix $\mathbf{D}\in\mathcal{G}$ generates the MZ family of searching directions. The residual $\hat{\mathbf{r}}_{p}$ is identical to the primal residual $\mathbf{r}_{p}$ of the HSD model defined in Eq. (12), $\hat{\mathbf{r}}_{d}$ is a combination of the dual residual and gap as $(\mathbf{r}_{d},r_{g})$, and the central path coefficient $\mu(\mathbf{x},\mathbf{s})$ is identical to $\mu(\mathbf{x},\mathbf{s},\kappa,\tau)$ defined in Eq. (23).

In the following, it sufficient to study the convergence properties to reduce the residuals $\hat{\mathbf{r}}_{p}$, $\hat{\mathbf{r}}_{d}$, and the complementarity gap which is proportional to $\mu$ by solving the linear system Eq. (38), and the results apply to the process by solving Eq. (28).

For the generalized linear system Eq. (38), the central path distances are

The central path neighborhoods are

where the parameter $\gamma\in(0,1)$. It is easy to verify the central path distances and neighborhoods are identical to the definition Eq. (24) and Eq. (26) for the special case Eq. (35). We also have $d_{\infty}(\mathbf{x},\mathbf{s})\leq d_{2}(\mathbf{x},\mathbf{s})$, and $\mathcal{N}_{2}(\gamma)\subseteq\mathcal{N}_{\infty}(\gamma)$.

Tsuchiya [1999] and Monteiro and Tsuchiya [2000] provide valuable tools for our study. For the convenience of analysis, we introduce Lemma 1-4 in [Monteiro and Tsuchiya, 2000] as Lemma 1-4 (see the paper for the proofs).

Define the following notations as

Lemma 1 [Monteiro and Tsuchiya, 2000]. For any $\mathbf{x}\in\text{int}(\mathcal{K})$, the matrice $\mathbf{X}$ and $\mathbf{T}_{x}$ satisfy:

a)

where

for $\mathbf{v}\in\mathcal{R}^{n}$, and $\mathbf{P}_{v}$ is the orthogonal projection matrix onto the subspace orthogonal to $\mathbf{v}_{2:n}$, as

We also use the notation

b)

c) $\mathbf{X}$ and $\mathbf{T}_{x}$ commute and $\mathbf{X}\succeq\mathbf{T}_{x}$.