On Second-Order Cone Functions

We consider the second-order cone function (SOCF) $f: {\mathbb R}^n \to \mathbb R$ defined by $f(x)= c^T x + d -\|A x + b \|$. Every SOCF is concave. We give necessary and sufficient conditions for strict concavity of $f$. The parameters $A \in {\mathbb R}^{m \times n}$ and $b \in {\mathbb R}^m$ are not uniquely determined. We show that every SOCF can be written in the form $f(x) = c^T x + d -\sqrt{\delta^2 + (x-x_*)^TM(x-x_*)}$. We give necessary and sufficient conditions for the parameters $c$, $d$, $\delta$, $M = A^T A$, and $x_*$ to be uniquely determined. We also give necessary and sufficient conditions for $f$ to be bounded above.


Introduction
Second-order cone programming is an important convex optimization problem [9,14,19,17].A second-order cone constraint has the form ∥Ax + b∥ ≤ c T x + d, where ∥ • ∥ is the Euclidean norm.This second-order cone constraint is equivalent to the inequality f (x) ≥ 0, where f is what we call a second-order cone function.The solution set of the constraint is convex, and the function f is concave [9,15].
In the following definition, we use R to denote the set of real numbers and R m×n to denote the set of m × n matrices with real entries.Of course, m and n are positive integers.
Definition 1.A second-order cone function (SOCF) is a function f : R n → R that can be written as with parameters c ∈ R n , d ∈ R, A ∈ R m×n , and b ∈ R m .
In second-order cone programming a linear function of x is minimized subject to one or more second-order cone constraints, along with the constraint F x = g, where F ∈ R p×n and g ∈ R p .The solution set of F x = g is an affine subspace, and we will show that the restriction of an SOCF to an affine subspace is another SOCF.Thus, from a mathematical point of view the constraint F x = g is not necessary, although in applications it can be convenient.In this paper we do not consider the constraint F x = g but instead focus on understanding the family of SOCFs.
There are interior-point methods for solving second-order cone programming problems.
The current research was started to get a deeper understanding of SOCFs to improve interior-point algorithms for finding the weighted analytic center of a system of secondorder cone constraints [1,3,5].The current work can lead to improved algorithms.
In this paper, we give a thorough description of the family of SOCFs.In the form of Equation (1.1), the parameters A and b are not uniquely determined, since ∥Ax + b∥ = ∥Q(Ax + b)∥ = ∥(QA)x + (Qb)∥ for any orthogonal m × m matrix Q.We show that every SOCF can be written in the form with the parameters δ ≥ 0, x * ∈ R n , and the positive semidefinite replace the parameters A and b.We show that these new parameters are unique if and It is known that every SOCF f is concave [9,15].We show that f is strictly concave if and only if rank(A) = n and b ̸ ∈ col(A), where col(A) denotes the column space of A.
In terms of the new parameters, the SOCF is strictly concave if and only if M is positive definite and δ > 0.
In the case where M is positive definite, we show that f is bounded above if and only Our results have computational implications for convex optimization problems involving second-order constraints such as the problem of minimizing weighted barrier functions presented in [3,1].This is related to the problem of finding a weighted analytic center for second-order cone constraints given in [5].There are also computational implications for the problem of computing the region of weighted analytic centers of a system of several second-order cone constraints.This is under investigation as part of our current research which is an extension of the work given in [5].
In the problems presented in [3,1,5], the boundedness of the feasible region guarantees the existence of a minimizer, and the strict convexity of the barrier function guarantees the uniqueness of the minimizer.Also, the strict convexity of the barrier function affects how quickly we can find the minimizer using these algorithms.The determination of the strict concavity of f is related to the strict convexity of the barrier function.The boundedness of the feasible region of the SOC constraints system is also related to the boundedness of f .If a single f is bounded, then the feasible region of the SOC constraints system is bounded.
Convex optimization algorithms perform well and more efficiently when the problem is known to be bounded and the objective function is strictly convex.If a second-order cone function is strictly concave, its gradient and Hessian matrix are defined, and the Hessian is invertible.The corresponding barrier function is similarly well-behaved, and Newton's method and Newton-based methods work well for the problem.However, many optimization problems are not bounded or have objective functions that are not strictly convex.Our results would allow one to recognize convex optimization problems involving second-order cone constraints (as in [3,1,5]) that can be solved efficiently, or to assist in reformulating those that are hard to solve.

Properties of Second-Order Cone Functions
The SOCFs on R (that is, n = 1) are the simplest to understand, and give insight into the general case.One important property of SOCFs is that their restriction to an affine subspace is another SOCF.We will frequently restrict to a 1-dimensional affine subspace.
Remark 3. Let f : R n → R be written in the form of Equation (1.1).The restriction of which is an SOCF on R k with the variable y.

Recall that a function
, and all t ∈ (0, 1).The function is strictly concave if the inequality is strict.A twice differentiable function f : R → R is concave if f ′′ (x) ≤ 0 for all x, and strictly concave if the inequality is strict.
is piecewise linear with a downward bend at x * , and hence concave but not strictly concave.
So far, we have proved that f is concave but not strictly concave if A = 0 or b ∈ col(A).
Assume that A ̸ = 0 and b ̸ ∈ col(A).Then δ > 0, and f strictly concave, since is defined and negative for all x.
Theorem 5. Every second-order cone function f is concave.Furthermore, f is strictly concave if and only if rank(A) = n and b ̸ ∈ col(A), using the parameters in Definition 1.
Proof.Let x 0 ̸ = x 1 ∈ R n , and define v = x 1 − x 0 .Let g : R → R be defined by directly from the definition that f is (strictly) concave if and only if g is (strictly) concave for all x 0 ̸ = x 1 .Note that Av ∈ R m .If Av = 0, then g is linear.If Av ̸ = 0, then are all real numbers.Thus, g is a second-order cone function of one variable.By Lemma 4, g is concave for all choices of x 0 and x 1 , and hence f is concave.
and there exist then there exist x 0 such that Ax 0 + b = 0. Thus t * = 0 and δ = 0, and g is piecewise linear with a downward corner.Thus, if rank(A) < n or b ∈ col(A) (or both), we can find x 0 ̸ = x 1 such that g is concave but not strictly concave, and hence f is not strictly concave.
Now assume rank(A) = n and b ̸ ∈ col(A).It follows that Av ̸ = 0 and δ > 0 for all x 0 ̸ = x 1 .Lemma 4 implies that g is strictly concave for all x 0 ̸ = x 1 , and it follows that f is strictly concave.
Note that A ∈ R n×n cannot satisfy rank(A) = n and b ̸ ∈ col(A).Therefore, any SOCF with A ∈ R n×n is concave but not strictly concave.This theorem uses the Moore-Penrose Inverse of a matrix, also called the pseudoinverse, which has many interesting properties found in [16].For example, x = A + b is the least squares solution to Ax = b, where The next theorem mentions the well-known fact that A T A is a positive semidefinite matrix, which means that it is symmetric with non-negative eigenvalues.A positive definite matrix is a symmetric matrix with all positive eigenvalues.If A ∈ R m×n then A T A is positive definite if and only if the rank of A is n.
The last equality uses the definitions of M and δ.The result follows.
The image of the square in R 2 under A is the light blue parallelogram in R 3 , shown on The proof Theorem 10, to follow, is subtle.While it is obvious that changing one parameter will change the function f , it is difficult to eliminate the possibility that more than one parameter can be changed while leaving the function unchanged.For example, with the form of Equation (1.1), the function f is unchanged when A → QA and b → Qb for an orthogonal matrix Q.The strategy in the proof is to uniquely determine one parameter at a time in a specific order.Theorem 10.Assume an SOCF is written in the form of Equation (2.1), and that the same SOCF is written with possibly different parameters satisfying the same requirements, so for all x.
• If M = 0 (the zero matrix), then c = c, M = 0, d − δ = d − δ, and x * arbitrary, and As a consequence, the parameterization of an SOCF in the form of Equation (2.1) is unique if and only if M is positive definite.
Proof.Recall that M, M are positive semidefinite.It follows that M v = 0 if and only if v T M v = 0. Also, recall that δ, δ are non-negative real numbers.
For nonzero v ∈ R n and t ∈ [0, ∞), we consider the function f (vt) and its asymptotic behavior as t → ∞.
The third equation uses the fact that t ≥ 0, and the fourth equation uses the Taylor series The fourth equation describes the slant asymptote of the graph of f (vt), and is crucial for the remainder of the proof.
For all v ̸ = 0, Equation (2.2) implies that A similar expression where c is replaced by c holds.
, then the slope of the slant asymptote is the same for both sets of parameters, so again cT v = c T v.This holds for all v, so c = c.
For all v ̸ = 0, Equation (2.2) implies that Assume M ̸ = 0. Then there exists v ∈ R n that satisfies M v ̸ = 0. Using Equation (2.4)The last row of A ∈ R (n+1)×n is all 0s, and the last component of b ∈ R n+1 is δ.
Proof.Note that M 1/2 is symmetric, and Remark 13.It follows from this theorem that any SOCF can be defined in the form of Equation (1.1) with A ∈ R (n+1)×n .While A is an m × n matrix with any m, using m > n + 1 is never needed.
Case I: δ = 0.In this case f (x) = c T x − √ x T M x.Let v be any nonzero vector in is a critical point of f at which f is not differentiable, and f has a critical point at x * .To determine if f has a global maximum at 0, define g Note that g v is a linear function giving the value of f along a ray starting at 0 ∈ R n with the direction vector v.The function f is bounded above if and only if the slope of g v is non-positive for all directions v.
Note that E is an ellipsoid centered at 0, since M is positive definite.Furthermore, g v (0) = 0, so f is bounded above if and only if the maximum value of f , restricted to E, is non-positive.We compute this maximum value using the method of Lagrange multipliers.The extreme values of f restricted to E occur at places where , which occurs at x + .Thus, the maximum slope of g v occurs when v is a positive scalar multiple of M −1 c, and that maximum slope has the same sign as All functions have the same positive definite matrix M .The parameters c and δ are chosen to illustrate Theorem 14, which says that f is bounded above if and only if c T M −1 c ≤ 1.The six parts of Theorem 14 correspond to the six contour plots.The contour with f (x) = −1 is a thick red curve, and the spacing between contours is ∆f = 0.5.description leads to new results about SOCFs.We characterize the critical points and global maxima of f , depending on the parameters.We give necessary and sufficient conditions for f to be bounded above, and for the set {x ∈ R n | f (x) ≥ 0} to be bounded.
Our results can lead to improved algorithms for optimization problems involving secondorder cone constraints.

Funding Statement, Data Availability Statement, Conflicts of Interest, and Acknowledgements
This research was performed as part of the employment of the authors at Northern Arizona University.The authors received no specific funding for this research.The data used to support the findings of this study are included within the article.The authors declare that there is no conflict of interest regarding the publication of this article.The authors thank the Cornell University arXiv for posting a preprint of this article [8].

c 5 Figure 1 :
Figure 1: Graphs of second-order cone functions f (x) = cx + d − δ 2 + (x − x * ) 2 , as described in Example 2. In each of the three plots, the parameters c, d, and x * are indicated.The dashed curve has δ = 0, and the solid curve has δ = 0.2.

Lemma 4 .Figure 2 :
Figure 2: The geometry of an SOCF on R. In this case A ∈ R 2 = R 2×1 .Note that Ax * is the point in col(A) that is closest to −b.See Lemma 4.

Figure 3 :
Figure 3: Graphs of the four SOCFs on R 2 defined in Example 6.Note that the graph of the SOCF (b) is indeed a cone.The top row shows functions with rank(A) = 2 and the bottom row shows rank(A) = 1.The left column shows b ̸ ∈ col(A) and the right column shows b ∈ col(A).All the functions graphed are concave, but only the upper left function is strictly concave, in agreement with Theorem 5.

. 1 )
where M = A T A is positive semidefinite, x * = −A + b, and δ = ∥Ax * + b∥.Proof.It is well-known that the least squares solution to Ax = −b is x * = −A + b, and that Ax * = −AA + b is the orthogonal projection of −b onto col(A).That is, Ax * is the point in col(A) that is closest to −b.Thus, the distance squared from Ax to −b is the distance squared from Ax to Ax * plus the distance squared from Ax * to −b.That is,

Remark 8 .Example 9 .
For A ∈ R m×n , note that rank(A) = n if and only if A T A ∈ R n×n is positive definite.The definition of δ in Theorem 7 makes it clear that b ∈ col(A) if and only if δ = 0. Therefore Theorem 5 implies that an SOCF written in the form of Equation (2.1) is strictly concave if and only if M is positive definite and δ > 0. The left half of Figure 4 shows the critical point and one contour of the SOCF f (x) = −∥Ax + b∥, with

.
The right part of the same Figure shows the geometry behind Theorem 7, which describes how to write the function in the form f

Figure 4 :
Figure 4: The geometry of the second-order cone function f (x) = −∥Ax + b∥, with A ∈ R 3×2 and b ∈ R 3 defined in Example 9.The function can also be written as f (x) = − δ 2 + (x − x * ) T M (x − x * ), were M = A T A. The maximum of f is at x * = −A + b, and the maximum value is f (x * ) = −δ.The orthogonal projection of −b onto the column space of A is Ax * = −AA + b.The distance from Ax * to −b is δ.One contour of f is shown.The image of this contour is a circle of points in col(B) that are equidistant from −b.
along with a similar expression where d is replaced by d, etc.If M ̸ = M , then there issome vector v such that v T M v ̸ = v T M v.This leads to a contradiction since the slope of the slant asymptote in Equation (2.4) would be different.Thus, M = M .Assume M = 0. Then f (x) = c T x + d − δ = c T x + d − δ, since c = c, and M = M = 0. Thus, d − δ = d − δ.

Example 11 .
we find that d = d.At this point we conclude, from the equality of the two expressions for f , that δ2 + (x − x * ) T M (x − x * ) = δ2 + (x − x * ) T M (x − x * ) T for all x.Expanding the quadratic term and canceling like terms, we find that δ 2 −2x T M x * = δ2 − 2x T M x * for all x.Thus δ = δ and M x * = M x * .Now we show that the parameterization of f is unique if and only if M is positive definite.If M is not positive definite there exist x * ̸ = x * such that M x * = M x * .If M is positive definite then M ̸ = 0 and M is invertible, so x * = x * and all of the parameters are unique.Let f (x 1 , x 2 ) = − 4 + (x 1 − 1) 2 be the SOCF on R 2 defined by c = 0, d = 0, δ = 2, M = [ 1 0 0 0 ], and x * = (1, 0).Note that M is not positive definite.The null space of M is span{(0, 1)}.The parameterization is not unique since any x * ∈ {(1, a) | a ∈ R} yields the same SOCF.While many choices of A and b in the form of Equation (1.1) yield the same function, there is a canonical choice for A and b starting with the function in the form of Equation (2.1).Recall that a positive semidefinite matrix M has a unique positive semidefinite square root, denoted M 1/2 .Theorem 12. Let M ∈ R n×n be positive semidefinite, x * ∈ R n , and δ ∈ R. Then δ 2 + (x − x * ) T M (x − x * ) = ∥Ax + b∥ 2 for

Figure 5 :
Figure 5: Contour plots of six different second-order cone functions defined in Example 15.All functions have the same positive definite matrix M .The parameters c and δ are chosen to illustrate Theorem 14, which says that f is bounded above if and only if c T M −1 c ≤ 1.The six parts of Theorem 14 correspond to the six contour plots.The contour with f (x) = −1 is a thick red curve, and the spacing between contours is ∆f = 0.5.