An alternative approach for a distance inequality associated with the second-order cone and the circular cone

The circular cone [1, 2] is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation. Let \({\mathcal {L}}_{\theta }\) denote the circular cone in \(\mathbb {R}^{n}\), which is defined by with \(\|\cdot\|\) denoting the Euclidean norm and \(\theta\in(0,\frac{\pi}{2})\). When \(\theta= \frac{\pi}{4}\), the circular cone \({\mathcal {L}}_{\theta }\) reduces to the well-known second-order cone (SOC) \(\mathcal {K}^{n}\) [3, 4] (also called the Lorentz cone), i.e., In particular, \({\mathcal {K}}^{1}\) is the set of nonnegative reals \(\mathbb {R}_{+}\). It is well known that the second-order cone \(\mathcal{K}^{n}\) is a special kind of symmetric cones [5]. But when \(\theta\neq\frac{\pi}{4}\), the circular cone \({\mathcal {L}}_{\theta }\) is a non-symmetric cone [1, 6, 7].

In [2], Zhou and Chen showed that there is a special relationship between the SOC and the circular cone as follows: where \(I_{n-1}\) is the \((n-1) \times(n-1)\) identity matrix. Based on the relationship (2) between the SOC and circular cone, Miao et al. [8] showed that circular cone complementarity problems can be transformed into the second-order cone complementarity problems. Furthermore from the relationship (2), we have Besides the relationship between second-order cone and circular cone, some topological structures play important roles in theoretical analysis for optimization problems. For example, the projection formula onto a cone facilitates designing algorithms for solving conic programming problems [9‐13]; the distance formula from an element to a cone is an important factor in the approximation theory; the tangent cone and normal cone are crucial in analyzing the structure of the solution set for optimization problems [14‐16]. From the above illustrations, an interesting question arises: What is the relationship between second-order cone and circular cone regarding the projection formula, the distance formula, the tangent cone and normal cone, and so on? The issue of the tangent cone and normal cone has been studied in [2], Theorem 2.3. In this paper, we focus on the other two issues.

More specifically, we provide an alternative approach to achieve an inequality which was obtained in [2], Theorem 2.2. Although the proof is a bit longer than the existing one, the new approach offers a way to clarify when the equality holds, which is helpful for further studying in the relationship between the second-order cone programming problems and the circular cone programming problems.

In order to study the relationship between second-order cone and circular cone, we need to recall some background materials. For any vector \(x=(x_{1},x_{2})\in \mathbb {R}\times \mathbb {R}^{n-1}\), the spectral decomposition of x with respect to second-order cone is given by where \(\lambda_{1}(x)\), \(\lambda_{2}(x)\), \(u_{x}^{(1)}\), and \(u_{x}^{(2)}\) are expressed as with \(w=\frac{x_{2}}{\|x_{2}\|}\) if \(x_{2}\neq0\), or any vector in \(\mathbb {R}^{n-1}\) satisfying \(\|w\|=1\) if \(x_{2}=0\). In the setting of the circular cone \({\mathcal {L}}_{\theta }\), Zhou and Chen [2] gave the following spectral decomposition of \(x\in \mathbb {R}^{n}\) with respect to \({\mathcal {L}}_{\theta }\): where \(\mu_{1}(x)\), \(\mu_{2}(x)\), \(v_{x}^{(1)}\), and \(v_{x}^{(2)}\) are expressed as with \(w=\frac{x_{2}}{\|x_{2}\|}\) if \(x_{2}\neq0\), or any vector in \(\mathbb {R}^{n-1}\) satisfying \(\|w\|=1\) if \(x_{2}=0\). Moreover, \(\lambda_{1}(x)\), \(\lambda_{2}(x)\) and \(u_{x}^{(1)}\), \(u_{x}^{(2)}\) are called the spectral values and the spectral vectors of x associated with \({\mathcal {K}}^{n}\), whereas \(\mu_{1}(x)\), \(\mu_{2}(x)\) and \(v_{x}^{(1)}\), \(v_{x}^{(2)}\) are called the spectral values and the spectral vectors of x associated with \({\mathcal {L}}_{\theta }\), respectively.

To proceed, we denote \(x_{+}\) (resp. \(x^{\theta }_{+}\)) the projection of x onto \({\mathcal {K}}^{n}\) (resp. \({\mathcal {L}}_{\theta }\)); also we set \(a_{+} = \max\{ a, 0 \}\) for any real number \(a \in \mathbb {R}\). According to the spectral decompositions (3) and (5) of x, the expressions of \(x_{+}\) and \(x^{\theta }_{+}\) can be obtained explicitly, as stated in the following lemma.

Based on the expression of the projection \(x_{+}^{\theta }\) onto \({\mathcal {L}}_{\theta }\) in Lemma 1.1, it is easy to obtain, for any \(x=(x_{1},x_{2})\in \mathbb {R}\times \mathbb {R}^{n-1}\), the explicit formula of projection of \(x\in \mathbb {R}^{n}\) onto the dual cone \({\mathcal {L}}_{\theta }^{\ast}\) (denoted by \((x^{\theta })_{+}^{\ast}\)):

In this section, we give the main results of this paper.

Now, applying Theorem 2.2, we can obtain the relation on the distance formulas associated with the second-order cone and the circular cone. Note that when \(\theta =\frac{\pi}{4}\), we know \({\mathcal {L}}_{\theta }={\mathcal {K}}^{n}\) and \(Ax=x\). Thus, it is obvious that \(\operatorname{dist}(Ax, {\mathcal {K}}^{n}) = \operatorname{dist}(x,{\mathcal {L}}_{\theta })\). In the following theorem, we only consider the case \(\theta \neq\frac{\pi}{4}\).