Since P∞

k=1µ_{k}_{k} < ∞, using Lemma 3.7 with v_{k} := D(ζ^{k}, ζ^{∗}) ≥ 0 and β_{k} := µ_{k}_{k} ≥
0 yields that the sequence {D(ζ^{k}, ζ^{∗})} converges. Thus, by Proposition 3.10(e), the
sequence {ζ^{k}} is bounded and consequently has an accumulation point. Without any
loss of generality, let bζ ∈ F be an accumulation point of {ζ^{k}}. Then, there exists a
subsequence {ζ^{k}^{j}} → bζ for some k_{j} → ∞. Since f is lower semi-continuous, we obtain
f (bζ) = lim inf_{k}_{j}_{→∞}f (ζ^{k}^{j}). On the other hand, f (ζ^{k}^{j}) → f_{∗} by part (a). The two sides
imply that f (bζ) = f∗. Therefore, bζ is a solution of the CSOCP. The proof is thus
complete.

where {λ_{k}} is a sequence of positive parameters, and H : IR^{n} × IR^{n} → (−∞, ∞] is a
proximal distance with respect to int(K) (see Def. 3.1) which plays the same role as the
Euclidean distance kx − yk^{2} in the classical proximal algorithms (see, e.g., [105, 132]),
but possesses certain more desirable properties to force the iterates to stay in K ∩ V, thus
eliminating the constraints automatically. As will be shown, such proximal distances can
be produced with an appropriate closed proper univariate function.

In the rest of this section, we focus on the case where K = K^{n}, and all the analysis can
be carried over to the case where K has the direct product structure. Unless otherwise
stated, we make the following minimal assumption for the CSOCP (3.64):

(A1) domf ∩ (V ∩ int(K^{n})) 6= ∅ and f∗ := inf{f (x) | x ∈ V ∩ K^{n}} > −∞.

Definition 3.2. An extended-valued function H : IR^{n} × IR^{n} → (−∞, ∞] is called a
proximal distance with respect to int(K^{n}) if it satisfies the following properties:

(P1) domH(·, ·) = C_{1}× C_{2} with int(K^{n}) × int(K^{n}) ⊂ C_{1}× C_{2} ⊆ K^{n}× K^{n}.

(P2) For each given y ∈ int(K^{n}), H(·, y) is continuous and strictly convex on C1, and it
is continuously differentiable on int(K^{n}) with dom∇_{1}H(·, y) = int(K^{n}).

(P3) H(x, y) ≥ 0 for all x, y ∈ IR^{n}, and H(y, y) = 0 for all y ∈ int(K^{n}).

(P4) For each fixed y ∈ C_{2}, the sets {x ∈ C_{1} : H(x, y) ≤ γ} are bounded for all γ ∈ IR.

Definition 3.2 has a little difference from Definition 2.1 of [10] for a proximal distance
w.r.t. int(K^{n}), since here H(·, y) is required to be strictly convex over C_{1} for any fixed
y ∈ int(K^{n}). We denote D(int(K^{n})) by the family of functions H satisfying Definition
3.2. With a given H ∈ D(int(K^{n})), we have the following basic iterative algorithm for
(3.64).

Interior Proximal Algorithm (IPA). Given H ∈ D(int(K^{n})) and x^{0} ∈ V ∩ int(K^{n}).

For k = 1, 2, . . . , with λk > 0 and εk ≥ 0, generate a sequence {x^{k}} ⊂ V ∩ int(K^{n}) with
g^{k} ∈ ∂_{ε}_{k}f (x^{k}) via the following iterative scheme:

x^{k} := argminλkf (x) + H(x, x^{k−1}) | x ∈ V

(3.66) such that

λ_{k}g^{k}+ ∇_{1}H(x^{k}, x^{k−1}) = A^{T}u^{k} for some u^{k} ∈ IR^{m}. (3.67)
The following proposition implies that the IPA is well-defined, and moreover, from its
proof we see that the iterative formula (3.66) is equivalent to the iterative scheme (3.65).

When ε_{k} > 0 for any k ∈ N (the set of natural numbers), the IPA can be viewed as an
approximate interior proximal method, and it becomes exact if ε_{k}= 0 for all k ∈ N.

Proposition 3.14. For any given H ∈ D(int(K^{n})) and y ∈ int(K^{n}), consider the problem
f_{∗}(y, τ ) = inf {τ f (x) + H(x, y) | x ∈ V} with τ > 0. (3.68)
Then, for each ε ≥ 0, there exist x(y, τ ) ∈ V ∩ int(K^{n}) and g ∈ ∂_{ε}f (x(y, τ )) such that

τ g + ∇_{1}H(x(y, τ ), y) = A^{T}u (3.69)
for some u ∈ IR^{m}. Moreover, for such x(y, τ ), we have

τ f (x(y, τ )) + H(x(y, τ ), y) ≤ f∗(y, τ ) + ε. (3.70)
Proof. Set F (x, τ ) := τ f (x)+H(x, y)+δ_{V∩K}^{n}(x), where δ_{V∩K}^{n}(x) is the indicator function
defined on the set V ∩ K^{n}. Since domH(·, y) = C_{1} ⊂ K^{n}, it is clear that

f∗(y, τ ) = inf {F (x, τ ) | x ∈ IR^{n}} . (3.71)
Since f∗ > −∞, it is easy to verify that for any γ ∈ IR the following relation holds

{x ∈ IR^{n}| F (x, τ ) ≤ γ} ⊂ {x ∈ V ∩ K^{n}| H(x, y) ≤ γ − τ f_{∗}}

⊂ {x ∈ C_{1}| H(x, y) ≤ γ − τ f∗} ,

which together with (P4) implies that F (·, τ ) has bounded level sets. In addition, by (P1)-(P3), F (·, τ ) is a closed proper and strictly convex function. Hence, the problem (3.71) has a unique solution, to say x(y, τ ). From the optimality conditions of (3.71), we get

0 ∈ ∂F (x(y, τ )) = τ ∂f (x(y, τ )) + ∇1H(x(y, τ ), y) + ∂δV∩K^{n}(x(y, τ ))

where the equality is due to [131, Theorem 23.8] and domf ∩ (V ∩ int(K^{n})) 6= ∅. Notice
that dom ∇_{1}H(·, y) = int(K^{n}) and dom ∂δV∩K^{n}(·) = V ∩ K^{n}. Therefore, the last equation
implies x(y, τ ) ∈ V ∩ int(K^{n}), and there exists g ∈ ∂f (x(y, τ )) such that

−τ g − ∇_{1}H(x(y, τ ), y) ∈ ∂δ_{V∩K}^{n}(x(y, τ )).

On the other hand, by the definition of δV∩K^{n}(·), it is not hard to derive that

∂δV∩K^{n}(x) = Im(A^{T}), ∀x ∈ V ∩ int(K^{n}).

The last two equations imply that (3.69) holds for ε = 0. When ε > 0, (3.69) also holds
for such x(y, τ ) and g since ∂f (x(y, τ )) ⊂ ∂εf (x(y, τ )). Finally, since for each y ∈ int(K^{n})
the function H(·, y) is strictly convex, and since g ∈ ∂_{ε}f (x(y, τ )), we have

τ f (x) + H(x, y) ≥ τ f (x(y, τ )) + H(x(y, τ ), y)

+hτ g + ∇_{1}H(x(y, τ ), y), x − x(y, τ )i − ε

= τ f (x(y, τ )) + H(x(y, τ ), y) + hA^{T}u, x − x(y, τ )i − ε

= τ f (x(y, τ )) + H(x(y, τ ), y) − ε for all x ∈ V,

where the first equality is from (3.69) and the last one is by x, x(y, τ ) ∈ V. Thus,
f_{∗}(y, τ ) = inf{τ f (x) + H(x, y) | x ∈ V} ≥ τ f (x(y, τ )) + H(x(y, τ ), y) − ε.

In the following, we focus on the convergence behaviors of the IPA with H from
several subclasses of D(int(K^{n})), which also satisfy one of the following properties.

(P5) For any x, y ∈ int(K^{n}) and z ∈ C_{1}, H(z, y) − H(z, x) ≥ h∇_{1}H(x, y), z − xi;

(P5’) For any x, y ∈ int(K^{n}) and z ∈ C_{2}, H(y, z) − H(x, z) ≥ h∇_{1}H(x, y), z − xi.

(P6) For each x ∈ C_{1}, the level sets {y ∈ C_{2}| H(x, y) ≤ γ} are bounded for all γ ∈ IR.

Specifically, we denote F_{1}(int(K^{n})) and F_{2}(int(K^{n})) by the family of functions H ∈
D(int(K^{n})) satisfying (P5) and (P5’), respectively. If C_{1} = K^{n}, we denote F_{1}(K^{n}) by
the family of functions H ∈ D(int(K^{n})) satisfying (P5) and (P6). If C_{2} = K^{n}, we write
F_{2}(int(K^{n})) as F (K^{n}). It is easy to see that the class of proximal distance F (int(K^{n}))
(respectively, F (K^{n})) in [10] subsumes the (H, H) with H ∈ F_{1}(int(K^{n})) (respectively,
F_{1}(K^{n})), but it does not include any (H, H) with H ∈ F_{2}(int(K^{n})) (respectively, F_{2}(K^{n})).

Proposition 3.15. Let {x^{k}} be the sequence generated by the IPA with H ∈ F_{1}(int(K^{n}))
or H ∈ F_{2}(int(K^{n})). Set σ_{ν} =Pν

k=1λ_{k}. Then, the following results hold.

(a) f (x^{ν}) − f (x) ≤ σ^{−1}_{ν} H(x, x^{0}) + σ^{−1}_{ν} Pν

k=1σ_{k}ε_{k} for any x ∈ V ∩ C_{1} if H ∈ F_{1}(int(K^{n}));

f (x^{ν})−f (x) ≤ σ_{ν}^{−1}H(x^{0}, x)+σ^{−1}_{ν} Pν

k=1σ_{k}ε_{k} for any x ∈ V ∩C_{2} if H ∈ F_{2}(int(K^{n})).

(b) If σ_{ν} → +∞ and ε_{k} → 0, then lim inf_{ν→∞}f (x^{ν}) = f∗.
(c) The sequence {f (x^{k})} converges to f∗ whenever P∞

k=1ε_{k} < ∞.

(d) If X∗ 6= ∅, then {x^{k}} is bounded with all limit points in X∗ under (d1) or (d2) below:

(d1) X∗ is bounded and P∞

k=1ε_{k} < ∞;

(d2) P∞

k=1λ_{k}ε_{k}< ∞ and H ∈ F_{1}(K^{n}) (or H ∈ F_{2}(K^{n})).

Proof. The proofs are similar to those of [10, Theorem 4.1]. For completeness, we here
take H ∈ F_{2}(int(K^{n})) for example to prove the results.

(a) Since g^{k} ∈ ∂_{ε}_{k}f (x^{k}), from the definition of the subdifferential, it follows that
f (x) ≥ f (x^{k}) + hg^{k}, x − x^{k}i − ε_{k}, ∀x ∈ IR^{n}.

This together with equation (3.67) implies that

λ_{k}(f (x^{k}) − f (x)) ≤ h∇_{1}H(x^{k}, x^{k−1}), x − x^{k}i + λ_{k}ε_{k}, ∀x ∈ V ∩ C_{2}.
Using (P5’) with x = x^{k}, y = x^{k−1} and z = x ∈ V ∩ C2, it then follows that

λ_{k}(f (x^{k}) − f (x)) ≤ H(x^{k−1}, x) − H(x^{k}, x) + λ_{k}ε_{k}, ∀x ∈ V ∩ C_{2}. (3.72)

Summing over k = 1, 2, . . . , ν in this inequality yields that

−σ_{ν}f (x) +

ν

X

k=1

λ_{k}f (x^{k}) ≤ H(x^{0}, x) − H(x^{ν}, x) +

ν

X

k=1

λ_{k}ε_{k}. (3.73)

On the other hand, setting x = x^{k−1} in (3.72), we obtain

f (x^{k}) − f (x^{k−1}) ≤ λ^{−1}_{k} H(x^{k−1}, x^{k−1}) − H(x^{k}, x^{k−1}) + ε_{k}≤ ε_{k}. (3.74)
Multiplying the inequality by σ_{k−1} (with σ_{0} ≡ 0) and summing over k = 1, . . . , ν, we get

ν

X

k=1

σ_{k−1}f (x^{k}) −

ν

X

k=1

σ_{k−1}f (x^{k−1}) ≤

ν

X

k=1

σ_{k−1}ε_{k}.

Noting that σ_{k} = λ_{k}+ σ_{k−1} with σ_{0} ≡ 0, the above inequality can reduce to
σ_{ν}f (x^{ν}) −

ν

X

k=1

λ_{k}f (x^{k}) ≤

ν

X

k=1

σ_{k−1}ε_{k}. (3.75)

Adding the inequalities (3.73) and (3.75) and recalling that σ_{k} = λ_{k}+ σ_{k−1}, it follows
that

f (x^{ν}) − f (x) ≤ σ_{ν}^{−1}H(x^{0}, x) − H(x^{ν}, x) + σ^{−1}_{ν}

ν

X

k=1

σ_{k}ε_{k}, ∀x ∈ V ∩ C_{2},

which immediately implies the desired result due to the nonnegativity of H(x^{ν}, x).

(b) If σ_{ν} → +∞ and ε_{k} → 0, then applying Lemma 2.2(ii) of [10] with a_{k} = ε_{k} and
b_{ν} := σ^{−1}_{ν} Pν

k=1λ_{k}ε_{k} yields σ_{ν}^{−1}Pν

k=1λ_{k}ε_{k}→ 0. From part(a), it then follows that
lim inf

ν→∞ f (x^{ν}) ≤ inf {f (x) | x ∈ V ∩ int(K^{n})} .
This together with f (x^{ν}) ≥ inf {f (x) | x ∈ V ∩ K^{n}} implies that

lim inf

ν→∞ f (x^{ν}) = inf {f (x) | x ∈ V ∩ int(K^{n})} = f∗.

(c) From (3.74), 0 ≤ f (x^{k}) − f∗ ≤ f (x^{k−1}) − f∗+ ε_{k}. Using Lemma 2.1 of [10] with γ_{k}≡ 0
and vk= f (x^{k}) − f∗, we have that {f (x^{k})} converges to f∗ whenever P∞

k=1εk < ∞.

(d) If the condition (d1) holds, then the sets {x ∈ V ∩ K^{n}| f (x) ≤ γ} are bounded for all
γ ∈ IR, since f is closed proper convex and X∗ = {x ∈ V ∩ K^{n}| f (x) ≤ f∗}. Note that
(3.74) implies {x^{k}} ⊆ {x ∈ V ∩ K^{n}| f (x) ≤ f (x^{0}) +Pk

j=1εj}. Along withP∞

k=1εk < ∞,
clearly, {x^{k}} is bounded. Since {f (x^{k})} converges to f∗ and f is l.s.c., passing to the
limit and recalling that {x^{k}} ⊂ V ∩ K^{n} yields that each accumulation point of {x^{k}} is a
solution of (3.64).

Suppose that the condition (d2) holds. If H ∈ F_{2}(K^{n}), then inequality (3.72) holds for
each x ∈ V ∩ K^{n}, and particularly for x_{∗} ∈ X_{∗}. Consequently,

H(x^{k}, x∗) ≤ H(x^{k−1}, x∗) + λ_{k}ε_{k} ∀x∗ ∈ X∗. (3.76)
Summing over k = 1, 2, . . . , ν for the last inequality, we obtain

H(x^{ν}, x∗) ≤ H(x^{0}, x∗) +

ν

X

k=1

λ_{k}ε_{k}.

This, by (P4) and P∞

k=1λ_{k}ε_{k} < ∞, implies that {x^{k}} is bounded, and hence has an
accumulation point. Without loss of generality, let ˆx ∈ K^{n} be an accumulation point of
{x^{k}}. Then there exists a subsequence {x^{k}^{j}} such that x^{k}^{j} → ˆx as j → ∞. From the
lower semicontinuity of f and part(c), we get f (ˆx) ≤ lim_{j→+∞}f (x^{k}^{j}) = f∗, which means
that ˆx is a solution of (3.64). If H ∈ F_{1}(K^{n}), then the last inequality becomes

H(x∗, x^{ν}) ≤ H(x∗, x^{0}) +

ν

X

k=1

λ_{k}ε_{k}.

By (P6) and P∞

k=1λkεk < ∞, we also have that {x^{k}} is bounded, and hence has an
accumulation point. Using the same arguments as above, we get the desired result.
An immediate byproduct of the above analysis yields the following global rate of
convergence estimate for the IPA with H ∈ F_{1}(K^{n}) or H ∈ F_{2}(K^{n}).

Proposition 3.16. Let {x^{k}} be the sequence given by the IPA with H ∈ F_{1}(K^{n}) or
F_{2}(K^{n}). If X∗ 6= ∅ andP∞

k=1ε_{k}< ∞, then f (x^{ν}) − f∗ = O(σ_{ν}^{−1}).

Proof. The result is direct by setting x = x^{∗} for some x^{∗} ∈ X∗ in the inequalities of
Proposition 3.15(a), and noting that 0 < ^{σ}_{σ}^{k}

ν ≤ 1 for all k = 1, 2, · · · , ν.

To establish the global convergence of {x^{k}} to an optimal solution of (3.64), we need
to make further assumptions on X∗ or the proximal distances in F1(K^{n}) and F2(K^{n}).

We denote bF_{1}(K^{n}) by the family of functions H ∈ F_{1}(K^{n}) satisfying (P7)-(P8) below,
Fb_{2}(K^{n}) by the family of functions H ∈ F_{2}(K^{n}) satisfying (P7’)–(P8’) below, and ¯F (K^{n})
by the family of functions H ∈ F_{2}(K^{n}) satisfying (P7’)-(P9’) below:

(P7) For any {y^{k}} ⊆ int(K^{n}) converging to y^{∗} ∈ K^{n}, we have H(y^{∗}, y^{k}) → 0;

(P8) For any bounded sequence {y^{k}} ⊆ int(K^{n}) and any y^{∗} ∈ K^{n} with H(y^{∗}, y^{k}) → 0,
there holds that λ_{i}(y^{k}) → λ_{i}(y^{∗}) for i = 1, 2;

(P7’) For any {y^{k}} ⊆ int(K^{n}) converging to y^{∗} ∈ K^{n}, we have H(y^{k}, y^{∗}) → 0;

(P8’) For any bounded sequence {y^{k}} ⊆ int(K^{n}) and any y^{∗} ∈ K^{n} with H(y^{k}, y^{∗}) → 0,
there holds that λ_{i}(y^{k}) → λ_{i}(y^{∗}) for i = 1, 2;

(P9’) For any bounded sequence {y^{k}} ⊆ int(K^{n}) and any y^{∗} ∈ K^{n} with H(y^{k}, y^{∗}) → 0,
there holds that y^{k} → y^{∗}.

It is easy to see that all previous subclasses of D(int(K^{n})) have the following relations:

Fb_{1}(K^{n}) ⊆ F_{1}(K^{n}) ⊆ F_{1}(int(K^{n})), F¯_{2}(K^{n}) ⊆ bF_{2}(K^{n}) ⊆ F_{2}(K^{n}) ⊆ F_{2}(int(K^{n})).

Proposition 3.17. Let {x^{k}} be generated by the IPA with H ∈ F_{1}(int(K^{n})) or F_{2}(int(K^{n})).

Suppose that X∗ is nonempty, P∞

k=1λ_{k}ε_{k} < ∞ and P∞

k=1ε_{k} < ∞.

(a) If X_{∗} is a single point set, then {x^{k}} converges to an optimal solution of (3.64).

(b) If X∗ at least includes two elements and for any x^{∗} = (x^{∗}_{1}, x^{∗}_{2}), ¯x^{∗} = (¯x^{∗}_{1}, ¯x^{∗}_{2}) ∈ X∗

with x^{∗} 6= ¯x^{∗}, it holds that x^{∗}_{1} 6= ¯x^{∗}_{1} or kx^{∗}_{2}k 6= k¯x^{∗}_{2}k, then {x^{k}} converges to an
optimal solution of (3.64) whenever H ∈ bF_{1}(K^{n}) (or H ∈ bF_{2}(K^{n})).

(c) If H ∈ ¯F_{2}(K^{n}), then {x^{k}} converges to an optimal solution of (3.64).

Proof. Part (a) is direct by Proposition 3.15(d1). We next consider part (b). Assume
that H ∈ bF2(K^{n}). Since P∞

k=1λkεk < ∞, from (3.76) and Lemma 2.1 of [10], it follows
that the sequence {H(x^{k}, x)} is convergent for any x ∈ X_{∗}. Let ¯x be the limit of
a subsequence {x^{k}^{l}}. By Proposition 3.15(d2), ¯x ∈ X∗. Consequently, {H(x^{k}, ¯x)} is
convergent. By (P7’), H(x^{k}^{l}, ¯x) → 0, and so H(x^{k}, ¯x) → 0. Along with (P8’), λ_{i}(x^{k}) →
λi(¯x) for i = 1, 2, i.e.,

x^{k}_{1} − kx^{k}_{2}k → ¯x_{1}− k¯x_{2}k and x^{k}_{1} + kx^{k}_{2}k → ¯x_{1}+ k¯x_{2}k as k → ∞.

This implies that x^{k}_{1} → ¯x_{1} and kx^{k}_{2}k → k¯x_{2}k. Together with the given assumption for
X∗, we have that x^{k}→ ¯x. Suppose that H ∈ bF_{1}(K^{n}). The inequality (3.76) becomes

H(x∗, x^{k}) ≤ H(x∗, x^{k−1}) + λ_{k}ε_{k}, ∀x∗ ∈ X∗,

and using (P7)-(P8) and the same arguments as above then yields the result. Part(c) is direct by the arguments above and the property (P9’).

When all points in the nonempty X∗ lie on the boundary of K^{n}, we must have x^{∗}_{1} 6= ¯x^{∗}_{1}
or kx^{∗}_{2}k 6= k¯x^{∗}_{2}k for any x^{∗} = (x^{∗}_{1}, x^{∗}_{2}), ¯x^{∗} = (¯x^{∗}_{1}, ¯x^{∗}_{2}) ∈ X∗ with x^{∗} 6= ¯x^{∗}, and the
assump-tion for X∗ in (b) is automatically satisfied. Since the solutions of (3.64) are generally on
the boundary of K^{n}, the assumption for X∗ in Proposition 3.17(b) is much weaker than
the one in Proposition 3.17(a).

Up to now, we have studied two types of convergence results for the IPA by the class
in which the proximal distance H lies. Proposition 3.15 and Proposition 3.16 show that
the largest, and less demanding, classes F_{1}(int(K^{n})) and F_{2}(int(K^{n})) provide reasonable
convergence properties for the IPA under minimal assumptions on the problem’s data.

This coincides with interior proximal methods for convex programming over nonnegative
orthant cones; see [10]. The smallest subclass ¯F_{2}(K^{n}) of F_{2}(int(K^{n})) guarantees that
{x^{k}} converges to an optimal solution provided that X∗ is nonempty. The smaller class
Fb_{2}(K^{n}) may guarantee the global convergence of the sequence {x^{k}} to an optimal solution
under an additional assumption except the nonempty of X∗. Moreover, we will illustrate
that there are indeed examples for the class ¯F_{2}(K^{n}). For the smallest subclass bF_{1}(K^{n})
of F_{1}(int(K^{n})), the analysis shows that it seems hard to find an example, although it
guarantees the convergence of {x^{k}} to an optimal solution by Proposition 3.17(b).

Next, we provide three kinds of ways to construct a proximal distance w.r.t. int(K^{n})
and analyze their own advantages and disadvantages. All of these ways exploit a l.s.c.

(lower semi-continuous) proper univariate function to produce such a proximal distance.

In addition, with such a proximal distance and the Euclidean distance, we obtain the regularized ones.

The first way produces the proximal distances for the class F_{1}(int(K^{n})). This way is
based on the compound of a univariate function φ and the determinant function det(·),
where φ : IR → (−∞, ∞] is a l.s.c. proper function satisfying the following conditions:

(B1) domφ ⊆ [0, ∞), int(domφ) = (0, ∞), and φ is continuous on its domain;

(B2) for any t1, t2 ∈ domφ, there holds that

φ(t^{r}_{1}t^{1−r}_{2} ) ≤ rφ(t_{1}) + (1 − r)φ(t_{2}), ∀r ∈ [0, 1]; (3.77)
(B3) φ is continuously differentiable on int(domφ) with dom(φ^{0}) = (0, ∞);

(B4) φ^{0}(t) < 0 for all t ∈ (0, ∞), lim_{t→0}^{+}φ(t) = ∞, and lim_{t→∞}t^{−1}φ(t^{2}) ≥ 0.

With such a univariate φ, we define the function H : IR^{n}× IR^{n}→ (−∞, ∞] as in (3.15):

H(x, y) := φ(det(x)) − φ(det(y)) − h∇φ(det(y)), x − yi, ∀x, y ∈ int(K^{n}).

∞, otherwise. (3.78)

By the conditions (B1)-(B4), we may prove that H has the following properties.

Proposition 3.18. Let H be defined as in (3.78) with φ satisfying (B1)-(B4). Then, the following hold.

(a) For any fixed y ∈ int(K^{n}), H(·, y) is strictly convex over int(K^{n}).

(b) For any fixed y ∈ int(K^{n}), H(·, y) is continuously differentiable on int(K^{n}) with

∇_{1}H(x, y) = 2φ^{0}(det(x))

x_{1}

−x_{2}

− 2φ^{0}(det(y))

y_{1}

−y_{2}

(3.79)
for all x ∈ int(K^{n}), where x = (x_{1}, x_{2}), y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}.

(c) H(x, y) ≥ 0 for all x, y ∈ IR^{n}, and H(y, y) = 0 for all y ∈ int(K^{n}).

(d) For any y ∈ int(K^{n}), the sets {x ∈ int(K^{n}) | H(x, y) ≤ γ} are bounded for all γ ∈ IR.

(e) For any x, y ∈ int(K^{n}) and z ∈ int(K^{n}), the following three point identity holds
H(z, y) = H(z, x) + H(x, y) + h∇_{1}H(x, y), z − xi.

Proof. (a) It suffices to prove φ(det(x)) is strictly convex on int(K^{n}). By Proposition
1.8(a), there has

det(αx + (1 − α)z) > (det(x))^{α}(det(z))^{1−α}, ∀α ∈ (0, 1),

for all x, z ∈ int(K^{n}) and x 6= z. Since φ^{0}(t) < 0 for all t ∈ (0, +∞), we have that φ is
decreasing on (0, +∞). This, together with the condition (B2), yields that

φ [det(αx + (1 − α)z)] < φ(det(x))^{α}(det(z))^{1−α}

≤ αφ[det(x)] + (1 − α)φ[det(z)], ∀α ∈ (0, 1)

for any x, z ∈ int(K^{n}) and x 6= z. This means that φ(det(x)) is strictly convex on int(K^{n}).

(b) Since det(x) is continuously differentiable on IR^{n} and φ is continuously differentiable
on (0, ∞), we have that φ(det(x)) is continuously differentiable on int(K^{n}). This means
that for any fixed y ∈ int(K^{n}), H(·, y) is continuously differentiable on int(K^{n}). By a
simple computation, we immediately obtain the formula in (3.79).

(c) Since φ(det(x)) is strictly convex and continuously differentiable on int(K^{n}), we have
φ(det(x)) > φ(det(y)) − h∇φ(det(y)), x − yi,

for any x, y ∈ int(K^{n}) with x 6= y. This implies that H(y, y) = 0 for all y ∈ int(K^{n}). In
addition, from the inequality and the continuity of φ on its domain, it follows that

φ(det(x)) ≥ φ(det(y)) − h∇φ(det(y)), x − yi

for any x, y ∈ int(K^{n}). By the definition of H, we have H(x, y) ≥ 0 for all x, y ∈ IR^{n}.
(d) Let {x^{k}} ⊆ int(K^{n}) be a sequence with kx^{k}k → ∞. For any fixed y = (y_{1}, y_{2}) ∈
int(K^{n}), we next prove that the sequence {H(x^{k}, y)} is unbounded by three cases, and
then the desired result follows. For convenience, we write x^{k} = (x^{k}_{1}, x^{k}_{2}) for each k.

Case 1: the sequence {det(x^{k})} has a zero limit point. Without loss of generality, we
assume that det(x^{k}) → 0 as k → ∞. Together with lim_{t→0}^{+}φ(t) = ∞, it readily follows
that lim_{k→∞}φ(det(x^{k})) → ∞. In addition, for each k we have that

h∇φ(det(y)), x^{k}i = 2φ^{0}(det(y))(x^{k}_{1}y1 − (x^{k}_{2})^{T}y2)

≤ 2φ^{0}(det(y))y_{1}(x^{k}_{1} − kx^{k}_{2}k) ≤ 0, (3.80)

where the inequality is true by using φ^{0}(t) < 0 for all t > 0, the Cauchy-Schwartz
Inequality, and y ∈ int(K^{n}). Now from (3.78), it then follows that lim_{k→∞}H(x^{k}, y) = ∞.

Case 2: the sequence {det(x^{k})} is unbounded. Noting that det(x^{k}) > 0 for each k, we
must have det(x^{k}) → +∞ as k → ∞. Since φ is decreasing on its domain, we have that

φ(det(x^{k}))
kx^{k}k =

√2φ(λ_{1}(x^{k})λ_{2}(x^{k}))

p(λ1(x^{k}))^{2}+ (λ_{2}(x^{k}))^{2} ≥ φ[(λ_{2}(x^{k}))^{2}]
λ_{2}(x^{k}) .

Note that λ_{2}(x^{k}) → ∞ in this case, and from the last equation and (B4) it follows that
lim

k→∞

φ(det(x^{k}))

kx^{k}k ≥ lim

k→∞

φ[(λ_{2}(x^{k}))^{2}]
λ2(x^{k}) ≥ 0.

In addition, since {_{kx}^{x}^{k}kk} is bounded, we without loss of generality assume that
x^{k}

kx^{k}k → ˆx = (ˆx_{1}, ˆx_{2}) ∈ IR × IR^{n−1}.
Then, ˆx ∈ K^{n}, kˆxk = 1, and ˆx_{1} > 0 (if not, ˆx = 0), and hence

k→∞lim

∇φ(det(y)), x^{k}
kx^{k}k

= h∇φ(det(y)), ˆxi

= 2φ^{0}(det(y))(ˆx_{1}y_{1}− ˆx^{T}_{2}y_{2})

≤ 2φ^{0}(det(y))ˆx_{1}(y_{1}− ky_{2}k)

< 0.

The two sides show that lim_{k→∞} ^{H(x}_{kx}k^{k}k^{,y)} > 0, and consequently lim_{k→∞}H(x^{k}, y) = ∞.

Case 3: the sequence {det(x^{k})} has some limit point ω with 0 < ω < ∞. Without loss
of generality, we assume that det(x^{k}) → ω as k → ∞. Since {x^{k}} is unbounded and
{x^{k}} ⊂ int(K^{n}), we must have x^{k}_{1} → ∞. In addition, by (3.80) and φ^{0}(t) < 0 for t > 0,

−h∇φ(det(y)), x^{k}i ≥ −2φ^{0}(det(y))(x^{k}_{1}y_{1}− kx^{k}_{2}kky_{2}k) ≥ −2φ^{0}(det(y))x^{k}_{1}(y_{1}− ky_{2}k).

This along with y ∈ int(K^{n}) implies that −h∇φ(det(y)), x^{k}i → +∞ as k → ∞. Noting
that φ(det(x^{k})) is bounded, from (3.78) it follows that lim_{k→∞}H(x^{k}, y) → ∞.

(e) For any x, y ∈ int(K^{n}) and z ∈ int(K^{n}), from the definition of H it follows that
H(z, y) − H(z, x) − H(x, y) = h∇φ(det(x)) − ∇φ(det(y)), z − xi

= h∇_{1}H(x, y), z − xi,

where the last equality is by part (b). The proof is thus complete.

Proposition 3.18 shows that the function H defined by (3.15) with φ satisfying (B1)–

(B4) is a proximal distance w.r.t. int(K^{n}) and dom H = int(K^{n}) × int(K^{n}). Also,
H ∈ F_{1}(int(K^{n})). The conditions (B1) and (B3)-(B4) are easy to check, whereas by
Lemma 2.2 of [123] we have the following important characterizations for the condition
(B2).

Lemma 3.8. A function φ : (0, ∞) → IR satisfies (B2) if and only if one of the following conditions holds:

(a) the function φ(exp(·)) is convex on IR;

(b) φ(t_{1}t_{2}) ≤ 1

2 φ(t^{2}_{1}) + φ(t^{2}_{2}) for any t_{1}, t_{2} > 0;

(c) φ^{0}(t) + tφ^{00}(t) ≥ 0 if φ is twice differentiable.

Example 3.8. Let φ : (0, ∞) → IR be φ(t) = − ln t, if t > 0.

∞, otherwise.

Solution. It is easy to verify that φ satisfies (B1)-(B4). By formula (3.78), the induced proximal distance is

H(x, y) :=

− lndet(x)

det(y) +2x^{T}Jny

det(y) − 2, ∀x, y ∈ int(K^{n}),

∞, otherwise,

where J_{n} is a diagonal matrix with the first entry being 1 and the rest (n − 1) entries
being −1. This is exactly the proximal distance given by [10]. Since H ∈ F_{1}(int(K^{n})),
we have the results of Proposition 3.15(a)-(d1) if the proximal distance is used for the
IPA.

Example 3.9. Take φ(t) = t^{1−q}/(q −1) (q > 1) if t > 0, and otherwise φ(t) = ∞.

Solution. It is not hard to check that φ satisfies (B1)-(B4). In light of (3.78), we compute that

H(x, y) :=

(det(x))^{1−q}− (det(y))^{1−q}

q − 1 + 2x^{T}J_{n}y

(det(y))^{q} − (det(y))^{1−q}, ∀x, y ∈ int(K^{n}),

∞, otherwise,

where J_{n}is the diagonal matrix same as Example 3.8. Since H ∈ F (int(K^{n})), when using
the proximal distance for the IPA, the results of Proposition 3.15(a)-(d1) hold.

We should emphasize that using the first way can not produce the proximal distances
of the class F1(K^{n}), and so bF1(K^{n}), since the condition lim_{t→0}^{+}φ(t) = ∞ is necessary
to guarantee that H has the property (P4), but it implies that the domain of H(·, y)
for any y ∈ int(K^{n}) can not be continuously extended to K^{n}. Thus, when choosing such
proximal distances for the IPA, we can not apply Proposition 3.15(d2) and Proposition
3.17.

The other two ways are both based on the compound of the trace function tr(·) and a vector-valued function induced by a univariate φ via (1.9). For convenience, in the

sequel, for any l.s.c. proper function φ : IR → (−∞, ∞], we write d : IR × IR → (−∞, ∞]

as

d(s, t) := φ(s) − φ(t) − φ^{0}(t)(s − t), if s ∈ domφ, t ∈ dom(φ^{0}).

∞, otherwise. (3.81)

The second way also produces the proximal distances for the class F1(int(K^{n})), which
requires φ : IR → (−∞, ∞] to be a l.s.c. proper function satisfying the conditions:

(C1) domφ ⊆ [0, +∞) and int(domφ) = (0, ∞);

(C2) φ is continuous and strictly convex on its domain;

(C3) φ is continuously differentiable on int(domφ) with dom(φ^{0}) = (0, ∞);

(C4) for any fixed t > 0, the sets {s ∈ domφ | d(s, t) ≤ γ} are bounded with all γ ∈ IR;

for any fixed s ∈ domφ, the sets {t > 0 | d(s, t) ≤ γ} are bounded with all γ ∈ IR.

Let φ^{soc} be the vector-valued function induced by φ via (1.9) and write dom(φ^{soc}) = C_{1}.
Clearly, C_{1} ⊆ K^{n} and intC_{1}= int(K^{n}). Define the function H : IR^{n}× IR^{n} → (−∞, ∞] by
H(x, y) := tr(φ^{soc}(x)) − tr(φ^{soc}(y)) − h∇tr(φ^{soc}(y)), x − yi, ∀x ∈ C_{1}, y ∈ int(K^{n}).

∞, otherwise. (3.82)

Using (1.6), Proposition 1.3, Lemma 3.3, the conditions (C1)-(C4), and similar arguments to [116, Proposition 3.1 and Proposition 3.2] (also see Section 3.1), it is not difficult to argue that H has the following favorable properties.

Proposition 3.19. Let H be defined by (3.82) with φ satisfying (C1)-(C4). Then, the following hold.

(a) For any fixed y ∈ int(K^{n}), H(·, y) is continuous and strictly convex on C1.
(b) For any fixed y ∈ int(K^{n}), H(·, y) is continuously differentiable on int(K^{n}) with

∇_{1}H(x, y) = ∇tr(φ^{soc}(x)) − ∇tr(φ^{soc}(y)) = 2 [(φ^{0})^{soc}(x) − (φ^{0})^{soc}(y)] .
(c) H(x, y) ≥ 0 for all x, y ∈ IR^{n}, and H(y, y) = 0 for any y ∈ int(K^{n}).

(d) H(x, y) ≥P2

i=1d(λ_{i}(x), λ_{i}(y)) ≥ 0 for any x ∈ C_{1} and y ∈ int(K^{n}).

(e) For any fixed y ∈ int(K^{n}), the sets {x ∈ C_{1}| H(x, y) ≤ γ} are bounded for all γ ∈ IR;

for any fixed x ∈ C_{1}, the sets {y ∈ int(K^{n}) | H(x, y) ≤ γ} are bounded for all γ ∈ IR.

(f ) For any x, y ∈ int(K^{n}) and z ∈ C_{1}, the following three point identity holds:

H(z, y) = H(z, x) + H(x, y) + h∇_{1}H(x, y), z − xi.

Proposition 3.19 shows that the function H defined by (3.82) with φ satisfying
(C1)-(C4) is a proximal distance w.r.t. int(K^{n}) with dom H = C_{1}× int(K^{n}), and furthermore,
such proximal distances belong to the class F_{1}(int(K^{n})). In particular, when domφ =
[0, ∞), they also belong to the class F_{1}(K^{n}). We next present some specific examples.

Example 3.10. Let φ(t) = t ln t − t if t ≥ 0, and otherwise φ(t) = ∞, where we stipulate 0 ln 0 = 0.

Solution. It is easy to verify that φ satisfies (C1)-(C4) with domφ = [0, ∞). By formulas (1.9) and (3.82), we compute that H has the following expression:

H(x, y) = tr(x ◦ ln x − x ◦ ln y + y − x), ∀x ∈ K^{n}, y ∈ int(K^{n}).

∞, otherwise.

Example 3.11. Let φ(t) = t^{p} − t^{q} if t ≥ 0, and otherwise φ(t) = ∞, where p ≥ 1 and
0 < q < 1.

Solution. We can show that φ satisfies the conditions (C1)-(C4) with dom(φ) = [0, ∞).

When p = 1 and q = 1/2, from formulas (1.9) and (3.82), we derive that

H(x, y) =

tr

"

y^{1}^{2} − x^{1}^{2} + (tr(y^{1}^{2})e − y^{1}^{2}) ◦ (x − y)
2pdet(y)

#

, ∀x ∈ K^{n}, y ∈ int(K^{n}).

∞, otherwise.

Example 3.12. Let φ(t) = −t^{q} if t ≥ 0, and otherwise φ(t) = ∞, where 0 < q < 1.

Solution. We can show that φ satisfies the conditions (C1)-(C4) with domφ = [0, ∞).

Now

H(x, y) = (1 − q)tr(y^{q}) − tr(x^{q}) + tr(qy^{q−1}◦ x), ∀x ∈ K^{n}, y ∈ int(K^{n}).

∞, otherwise.

Example 3.13. Let φ(t) = − ln t + t − 1 if t > 0, and otherwise φ(t) = ∞.

Solution. It is easy to check that φ satisfies (C1)-(C4) with domφ = (0, ∞). The induced proximal distance is

H(x, y) = tr(ln y) − tr(ln x) + 2hy^{−1}, xi − 2, ∀x, y ∈ int(K^{n}).

∞, otherwise.

By a simple computation, we obtain that the proximal distance is same as the one given by Example 3.8, and the one induced by φ(t) = − ln t (t > 0) via formula (3.82).

Clearly, the proximal distances in Examples 3.10–3.12 belong to the class F_{1}(K^{n}).

Also, by Proposition 3.20 below, the proximal distances in Examples 3.10–3.11 also satisfy (P8) since the corresponding φ also satisfies the following condition (C5):

(C5) For any bounded sequence {a^{k}} ⊆ int(domφ) and a ∈ domφ such that lim

k→∞d(a, a^{k})

= 0, there holds that a = limk→∞a^{k}, where d is defined as in (3.81).

Proposition 3.20. Let H be defined as in (3.82) with φ satisfying (C1)-(C5) and
domφ = [0, ∞). Then, for any bounded sequence {y^{k}} ⊆ int(K^{n}) and y^{∗} ∈ K^{n} such
that H(y^{∗}, y^{k}) → 0, we have λi(y^{k}) → λi(y^{∗}) for i = 1, 2.

Proof. From Proposition 3.19(d) and the nonnegativity of d, for each k we have
H(y^{∗}, y^{k}) ≥ d(λ_{i}(y^{∗}), λ_{i}(y^{k})) ≥ 0, i = 1, 2.

This, together with the given assumption H(y^{∗}, y^{k}) → 0, implies that
d(λi(y^{∗}), λi(y^{k})) → 0, i = 1, 2.

Notice that {λ_{i}(y^{k})} ⊂ int(domφ) and λ_{i}(y^{∗}) ∈ K^{n} for i = 1, 2 by Property 1.1(c). From
the condition (C5), we immediately obtain λ_{i}(y^{k}) → λ_{i}(y^{∗}) for i = 1, 2.

Nevertheless, we should point out that the proximal distance H given by (3.82) with φ satisfying (C1)-(C4) and domφ = [0, ∞) generally does not have the property (P7), even if φ satisfies the condition (C6) below. This fact will be illustrated by Example 3.14.

(C6) For any {a^{k}} ⊆ (0, ∞) converging to a ∈ [0, ∞), lim_{k→∞}d(a^{∗}, a^{k}) → 0.

Example 3.14. Let H be the proximal distance induced by the entropy function φ in Example 3.10.

Solution. It is easy to verify that φ satisfies the conditions (C1)-(C6). Here we shall
present a sequence {y^{k}} ⊂ int(K^{3}) which converges to y^{∗} ∈ K^{3}, but H(y^{∗}, y^{k}) → ∞. Let

y^{k} =

p2(1 + e^{−k}^{3})

√

1 + k^{−1}− e^{−k}^{3}

√

1 − k^{−1}+ e^{−k}^{3}

∈ int(K^{3}) and y^{∗} =

√2 1 1

∈ K^{3}.

By the expression of H(y^{∗}, y^{k}), i.e., H(y^{∗}, y^{k}) = tr(y^{∗}◦ ln y^{∗}) − tr(y^{∗}◦ ln y^{k}) + tr(y^{k}− y^{∗}),
it suffices to prove that lim_{k→∞}−tr(y^{∗} ◦ ln y^{k}) = ∞ since lim_{k→∞}tr(y^{k}− y^{∗}) = 0 and
tr(y^{∗}◦ ln y^{∗}) = λ_{2}(y^{∗}) ln(λ_{2}(y^{∗})) < ∞. By the definition of ln y^{k}, we have

tr(y^{∗}◦ ln y^{k}) = ln(λ_{1}(y^{k})) y_{1}^{∗}− (y_{2}^{∗})^{T}y¯_{2}^{k} + ln(λ_{2}(y^{k})) y_{1}^{∗}+ (y_{2}^{∗})^{T}y¯_{2}^{k}

(3.83)

for y^{∗} = (y_{1}^{∗}, y_{2}^{∗}), y^{k}= (y_{1}^{k}, y_{2}^{k}) ∈ IR × IR^{2} with ¯y_{2}^{k} = y^{k}_{2}/ky_{2}^{k}k. By computing,
ln(λ_{1}(y^{k})) = ln√

2 − ln 1 +p

1 + e^{−k}^{3}

− k^{3},
y_{1}^{∗}− (y_{2}^{∗})^{T}y¯_{2}^{k} = 1

ky^{k}_{2}k

−k^{−1}+ e^{−k}^{3}
1 +√

1 + k^{−1}− e^{−k}^{3} + k^{−1}− e^{−k}^{3}
1 +√

1 − k^{−1}+ e^{−k}^{3}

! .

The last two equalities imply that lim_{k→∞}ln(λ_{1}(y^{k})) y_{1}^{∗}− (y_{2}^{∗})^{T}y¯_{2}^{k} = −∞. In addition,
by noting that y^{k}_{2} 6= 0 for each k, we compute that

lim

k→∞ln(λ_{2}(y^{k})) y_{1}^{∗}− (y_{2}^{∗})^{T}y¯_{2}^{k} = ln(λ2(y^{k}))

y_{1}^{∗}+ (y_{2}^{∗})^{T} y_{2}^{∗}
ky_{2}^{∗}k

= λ_{2}(y^{∗}) ln(λ_{2}(y^{∗})).

From the last two equations, we immediately have lim_{k→∞}−tr(y^{∗}◦ ln y^{k}) = ∞.
Thus, when the proximal distance in the IPA is chosen as the one given by (3.82)
with φ satisfying (C1)-(C6) and domφ = [0, ∞), Proposition 3.17(b) may not apply, i.e.

the global convergence to an optimal solution may not be guaranteed. This is different
from interior proximal methods for convex programming over nonnegative orthant cones
by noting that φ is now a univariate Bregman function. Similarly, it seems hard to find
examples for the class F+(K^{n}) in [10] so that Theorem 2.2 therein can apply for since it
also requires (P7).

The third way will produce the proximal distances for the class F2(int(K^{n})), which
needs a l.s.c. proper function φ : IR → (−∞, ∞] satisfying the following conditions:

(D1) φ is strictly convex and continuous on domφ, and φ is continuously differentiable
on a subset of domφ, where dom(φ^{0}) ⊆ domφ ⊆ [0, ∞) and int(domφ^{0}) = (0, ∞);

(D2) φ is twice continuously differentiable on int(domφ) and lim_{t→0}^{+}φ^{00}(t) = ∞;

(D3) φ^{0}(t)t − φ(t) is convex on dom(φ^{0}), and φ^{0} is strictly concave on dom(φ^{0});

(D4) φ^{0} is SOC-concave on dom(φ^{0}).

With such a univariate φ, we define the proximal distance H : IR^{n}× IR^{n}→ (−∞, ∞] by
H(x, y) := tr(φ^{soc}(y)) − tr(φ^{soc}(x)) − h∇tr(φ^{soc}(x)), y − xi, ∀x ∈ C_{1}, y ∈ C_{2},

∞, otherwise. (3.84)

where C_{1} and C_{2} are the domain of φ^{soc} and (φ^{0})^{soc}, respectively. By the relation between
dom(φ) and dom(φ^{0}), obviously, C_{2} ⊆ C_{1} ⊆ K^{n} and intC_{1} = intC_{2} = int(K^{n}).

Lemma 3.9. Let φ : IR → (−∞, ∞] be a l.s.c. proper function satisfying (D1)-(D4).

Then, the following hold.

(a) tr [(φ^{0})^{soc}(x) ◦ x − φ^{soc}(x)] is convex in C_{1} and continuously differentiable on intC_{1}.

(b) For any fixed y ∈ IR^{n}, h(φ^{0})^{soc}(x), yi is continuously differentiable on intC_{1}, and
moreover, it is strictly concave over C_{1} whenever y ∈ int(K^{n}).

Proof. (a) Let ψ(t) := φ^{0}(t)t−φ(t). Then, by (D2) and (D3), ψ(t) is convex on domφ^{0} and
continuously differentiable on int(domφ^{0}) = (0, +∞). Since tr [(φ^{0})^{soc}(x) ◦ x − φ^{soc}(x)] =
tr[ψ^{soc}(x)], using Lemma 3.3(b) and (c) immediately yields part(a).

(b) From (D2) and Lemma 3.3(a), (φ^{0})^{soc}(·) is continuously differentiable on int C_{1}. This
implies that hy, (φ^{0})^{soc}(x)i for any fixed y is continuously differentiable on intC_{1}. We next
show that it is also strictly concave in C_{1} whenever y ∈ int(K^{n}). Note that tr[(φ^{0})^{soc}(·)]

is strictly concave on C_{1} since φ^{0} is strictly concave on dom(φ^{0}). Consequently,
tr[(φ^{0})^{soc}(βx + (1 − β)z)] > βtr[(φ^{0})^{soc}(x)] + (1 − β)tr[(φ^{0})^{soc}(z)], ∀0 < β < 1
for any x, z ∈ C_{1} and x 6= z. This implies that

(φ^{0})^{soc}(βx + (1 − β)z) − β(φ^{0})^{soc}(x) − (1 − β)(φ^{0})^{soc}(z) 6= 0.

In addition, since φ^{0} is SOC-concave on dom(φ^{0}), it follows that

(φ^{0})^{soc}[βx + (1 − β)z] − β(φ^{0})^{soc}(x) − (1 − β)(φ^{0})^{soc}(z) K^{n} 0.

Thus, for any fixed y ∈ int(K^{n}), the last two equations imply that

hy, (φ^{0})^{soc}[βx + (1 − β)z] − β(φ^{0})^{soc}(x) − (1 − β)(φ^{0})^{soc}(z)i > 0.

This shows that hy, (φ^{0})^{soc}(x)i for any fixed y ∈ int(K^{n}) is strictly convex on C_{1}.
Using the conditions (D1)-(D4) and Lemma 3.9, and following the same arguments as
[116, Propositions 4.1 and 4.2] (also see Section 3.2, Propositions 3.8-3.9), we may prove
the following proposition.

Proposition 3.21. Let H be defined as in (3.84) with φ satisfying (D1)-(D4). Then, the following hold.

(a) H(x, y) ≥ 0 for any x, y ∈ IR^{n}, and H(y, y) = 0 for any y ∈ int(K^{n}).

(b) For any fixed y ∈ C_{2}, H(·, y) is continuous in C_{1}, and it is strictly convex on C_{1}
whenever y ∈ int(K^{n}).

(c) For any fixed y ∈ C_{2}, H(·, y) is continuously differentiable on int(K^{n}) with

∇_{1}H(x, y) = 2∇(φ^{0})^{soc}(x)(x − y).

Moreover, dom∇_{1}H(·, y) = int(K^{n}) whenever y ∈ int(K^{n}).

(d) H(x, y) ≥P2

i=1d(λ_{i}(y), λ_{i}(x)) ≥ 0 for any x ∈ C_{1} and y ∈ C_{2}.

(e) For any fixed y ∈ C_{2}, the sets {x ∈ C_{1}| H(x, y) ≤ γ} are bounded for all γ ∈ IR.

(f ) For all x, y ∈ int(K^{n}) and z ∈ C_{2}, H(x, z) − H(y, z) ≥ 2h∇_{1}H(y, x), z − yi.

Proposition 3.21 demonstrates that the function H defined by (3.84) with φ satisfying
(D1)-(D4) is a proximal distance w.r.t. the cone int(K^{n}) and possesses the property (P5’),
and therefore belongs to the class F_{2}(int(K^{n})). If, in addition, domφ = [0, ∞), then H
belongs to the class F_{2}(K^{n}). The conditions (D1)–(D3) are easy to check, and for the
condition (D4), we can employ the characterizations in [41, 44] to verify whether φ^{0} is
SOC-concave or not. Some examples are presented as follows.

Example 3.15. Let φ(t) = t ln t − t + 1 if t ≥ 0, and otherwise φ(t) = ∞.

Solution. It is easy to verify that φ satisfies (D1)–(D3) with domφ = [0, ∞) and
dom(φ^{0}) = (0, ∞). By Example 2.12(c), φ^{0} is SOC-concave on (0, ∞). Using
formu-las (1.9) and (3.84), we have

H(x, y) = tr(y ◦ ln y − y ◦ ln x + x − y), ∀x ∈ int(K^{n}), y ∈ K^{n}.

∞, otherwise.

Example 3.16. Let φ(t) = t^{q+1}

q +1 if t ≥ 0, and otherwise φ(t) = ∞, where 0 < q < 1.

Solution. It is easy to show that φ satisfies (D1)-(D3) with domφ = [0, ∞) and dom(φ^{0}) =
[0, ∞). By Example 2.12, φ^{0} is also SOC-concave on [0, ∞). By (1.9) and (3.84), we
com-pute that

H(x, y) =

_{1}

q+1tr(y^{q+1}) + _{q+1}^{q} tr(x^{q+1}) − tr(x^{q}◦ y), ∀ x ∈ int(K^{n}), y ∈ K^{n}.

∞, otherwise.

Example 3.17. Let φ(t) = (1 + t) ln(1 + t) + t^{q+1}

q +1 if t ≥ 0, and otherwise φ(t) = ∞, where 0 < q < 1.

Solution. We can verify that φ satisfies (D1)-(D3) with domφ = dom(φ^{0}) = [0, ∞).

From Example 2.12, φ^{0} is also SOC-concave on [0, ∞). Using (1.9) and (3.84), it is not
hard to compute that for any x, y ∈ K^{n},

H(x, y) = tr [(e + y) ◦ (ln(e + y) − ln(e + x))] − tr(y − x) + 1

q + 1tr(y^{q+1}) + q

q + 1tr(x^{q+1}) − tr(x^{q}◦ y).

Note that the proximal distances in Example 3.16 and Example 3.17 belong to the
class F_{2}(K^{n}). By Proposition 3.22 below, the ones in Example 3.16 and Example 3.17
also belong to the class bF_{2}(K^{n}).

Proposition 3.22. Let H be defined as in (3.84) with φ satisfying (D1)-(D4). Suppose
that domφ = dom(φ^{0}) = [0, ∞). Then, H possesses the properties (P7’) and (P8’).

Proof. By the given assumption, C_{1} = C_{2} = K^{n}. From Proposition 3.21(b), the function
H(·, y^{∗}) is continuous on K^{n}. Consequently, lim_{k→∞}H(y^{k}, y^{∗}) = H(y^{∗}, y^{∗}) = 0.

From Proposition 3.21(d), H(y^{k}, y^{∗}) ≥ d(λi(y^{∗}), λi(y^{k})) ≥ 0 for i = 1, 2. This together
with the assumption H(y^{k}, y^{∗}) → 0 implies d(λ_{i}(y^{∗}), λ_{i}(y^{k})) → 0 for i = 1, 2. From this,
we necessarily have λ_{i}(y^{k}) → λ_{i}(y^{∗}) for i = 1, 2. Suppose not, then the bounded sequence
{λ_{i}(y^{k})} must have another limit point ν_{i}^{∗} ≥ 0 such that ν_{i}^{∗} 6= λ_{i}(y^{∗}). Without loss of
generality, we assume that limk∈K,k→∞λi(y^{k}) = ν_{i}^{∗}. Then, we have

d(ν_{i}^{∗}, λ_{i}(y^{∗})) = lim

k→∞d(ν_{i}^{∗}, λ_{i}(y^{k})) = lim

k∈K,k→∞d(ν_{i}^{∗}, λ_{i}(y^{k})) = d(ν_{i}^{∗}, ν_{i}^{∗}) = 0,

where the first equality is due to the continuity of d(s, ·) for any fixed s ∈ [0, ∞), and
the second one is by the convergence of {d(ν_{i}^{∗}, λi(y^{k}))} implied by the first equality. This
contradicts the fact that d(ν_{i}^{∗}, λ_{i}(y^{∗})) > 0 since ν_{i}^{∗} 6= λ_{i}(y^{∗}).

As illustrated by the following example, the proximal distance generated by (3.84)
with φ satisfying (D1)-(D4) generally does not belong to the class ¯F_{2}(K^{n}).

Example 3.18. Let H be the proximal distance as in Example 3.15.

Solution. Let

y^{k} =

√2
(−1)^{k}_{k+1}^{k}
(−1)^{k}_{k+1}^{k}

for each k and y^{∗} =

√2 1 1

.

It is not hard to check that the sequence {y^{k}} ⊆ int(K^{3}) satisfies H(y^{k}, y^{∗}) → 0. Clearly,
the sequence y^{k} 9 y^{∗} as k → ∞, but λ_{1}(y^{k}) → λ_{1}(y^{∗}) = 0 and λ_{2}(y^{k}) → λ_{2}(y^{∗}) = 2√

2.

Finally, let H_{1} be a proximal distance produced via one of the ways above, and define
H_{α}(x, y) := H_{1}(x, y) + α

2kx − yk^{2}, (3.85)

where α > 0 is a fixed parameter. Then, by Propositions 3.18, 3.19 and 3.21 and the identity

kz − xk^{2} = kz − yk^{2}+ ky − xk^{2} + 2hz − y, y − xi, ∀x, y, z ∈ IR^{n},

it is easily shown that H_{α} is also a proximal distance w.r.t. int(K^{n}). Particularly, when
H_{1} is given by (3.84) with φ satisfying (D1)-(D4) and domφ = dom(φ^{0}) = [0, ∞) (for
example the distances in Examples 3.16 and and Example 3.17), the regularized proximal
distance H_{α} satisfies (P7’) and (P9’), and hence H_{α} ∈ ¯F_{2}(K^{n}). With such a regularized
proximal distance, the sequence generated by the IPA converges to an optimal solution
of (3.64) if X_{∗} 6= ∅.

To sum up, we may construct a proximal distance w.r.t. the cone int(K^{n}) via three
ways with an appropriate univariate function. The first way in (3.78) can only produce a
proximal distance belonging to F_{1}(int(K^{n})), the second way in (3.82) produces a proximal
distance of F_{1}(K^{n}) if domφ = [0, ∞), whereas the third way in (3.84) produces a proximal
distance of the class bF_{2}(K^{n}) if domφ = dom(φ^{0}) = [0, ∞). Particularly, the regularized
proximal distances Hα in (3.85) with H1 given by (3.84) with domφ = dom(φ^{0}) = [0, ∞)
belong to the smallest class ¯F_{2}(K^{n}). With such regularized proximal distances, we have
the convergence result of Proposition 3.17(c) for the general convex SOCP with X∗ 6= ∅.

For the linear SOCP, we will obtain some improved convergence results for the IPA by exploring the relations between the sequence generated by the IPA and the central path associated to the corresponding proximal distances.

Given a l.s.c. proper strictly convex function Φ with dom(Φ) ⊆ K^{n} and int(domΦ) =
int(K^{n}), the central path of (3.64) associated to Φ is the set {x(τ ) | τ > 0} defined by

x(τ ) := argminn

τ f (x) + Φ(x) | x ∈ V ∩ K^{n}o

for τ > 0. (3.86)
In what follows, we will focus on the central path of (3.64) w.r.t. a distance-like function
H ∈ D(int(K^{n})). From Proposition 3.14, we immediately have the following result.

Proposition 3.23. For any given H ∈ D(int(K^{n})) and ¯x ∈ int(K^{n}), the central path
{x(τ ) | τ > 0} associated to H(·, ¯x) is well defined and is in V ∩ int(K^{n}). For each τ > 0,
there exists g_{τ} ∈ ∂f (x(τ )) such that τ g_{τ}+∇_{1}H(x(τ ), ¯x) = A^{T}y(τ ) for some y(τ ) ∈ IR^{m}.

We next study the favorable properties of the central path associated to H ∈ D(int(K^{n})).

Proposition 3.24. For any given H ∈ D(int(K^{n})) and ¯x ∈ int(K^{n}), let {x(τ ) | τ > 0}

be the central path associated to H(·, ¯x). Then, the following results hold.

(a) The function H(x(τ ), ¯x) is nondecreasing in τ .

(b) The set {x(τ ) | ˆτ ≤ τ ≤ ˜τ } is bounded for any given 0 < ˆτ < ˜τ . (c) x(τ ) is continuous at any τ > 0.

(d) The set {x(τ ) | τ ≥ ¯τ } is bounded for any ¯τ > 0 if X∗ 6= ∅ and domH(·, ¯x) = K^{n}.

(e) All cluster points of {x(τ ) | τ > 0} are solutions of (3.64) if X∗ 6= ∅.

Proof. The proofs are similar to those of Propositions 3–5 of [82].

(a) Take τ_{1}, τ_{2} > 0 and let x^{i} = x(τ_{i}) for i = 1, 2. Then, from Proposition 3.23, we know
x^{1}, x^{2} ∈ V ∩ int(K^{n}) and there exist g^{1} ∈ ∂f (x^{1}) and g^{2} ∈ ∂f (x^{2}) such that

∇1H(x^{1}, ¯x) = −τ1g^{1}+ A^{T}y^{1} and ∇1H(x^{2}, ¯x) = −τ2g^{2}+ A^{T}y^{2} (3.87)
for some y^{1}, y^{2} ∈ IR^{m}. This together with the convexity of H(·, ¯x) yields that

τ_{1}^{−1} H(x^{1}, ¯x) − H(x^{2}, ¯x)

≤ τ_{1}^{−1}h∇_{1}H(x^{1}, ¯x), x^{1}− x^{2}i = hg^{1}, x^{2}− x^{1}i,
τ_{2}^{−1} H(x^{2}, ¯x) − H(x^{1}, ¯x)

≤ τ_{2}^{−1}h∇_{1}H(x^{2}, ¯x), x^{2}− x^{1}i = hg^{2}, x^{1}− x^{2}i. (3.88)
Adding the two inequalities and using the convexity of f , we obtain

τ_{1}^{−1}− τ_{2}^{−1}

H(x^{1}, ¯x) − H(x^{2}, ¯x) ≤ hg^{1}− g^{2}, x^{2}− x^{1}i ≤ 0.

Thus, H(x^{1}, ¯x) ≤ H(x^{2}, ¯x) whenever τ_{1} ≤ τ_{2}. Particularly, from the last two equations,
0 ≤ τ_{1}^{−1}H(x^{1}, ¯x) − H(x^{2}, ¯x)

≤ τ_{1}^{−1}h∇_{1}H(x^{1}, ¯x), x^{1}− x^{2}i (3.89)

≤ hg^{2}, x^{2}− x^{1}i

≤ τ_{2}^{−1}H(x^{1}, ¯x) − H(x^{2}, ¯x) , ∀τ_{1} ≥ τ_{2} > 0.

(b) By part(a), H(x(τ ), ¯x) ≤ H(x(˜τ ), ¯x) for any τ ≤ ˜τ , which implies that
{x(τ ) : τ ≤ ˜τ } ⊆ L_{1} = {x ∈ int(K^{n}) | H(x, ¯x) ≤ H(x(˜τ ), ¯x)} .

Noting that {x(τ ) : ˆτ ≤ τ ≤ ˜τ } ⊆ {x(τ ) : τ ≤ ˜τ } ⊆ L_{1}, the desired result follows by (P4).

(c) Fix ¯τ > 0. To prove that x(τ ) is continuous at ¯τ , it suffices to prove that lim_{k→∞}x(τ_{k})

= x(¯τ ) for any sequence {τk} such that limk→∞τk = ¯τ . Given such a sequence {τk}, and
take ˆτ , ˜τ such that ˆτ > ¯τ > ˜τ . Then, {x(τ ) : ˆτ ≤ τ ≤ ˜τ } is bounded by part (b), and
τ_{k}∈ (ˆτ , ˜τ ) for sufficiently large k. Consequently, the sequence {x(τ_{k})} is bounded. Let ¯y
be a cluster point of {x(τ_{k})}, and without loss of generality assume that lim_{k→∞}x(τ_{k}) = ¯y.

Let K1 := {k : τk ≤ ¯τ } and take k ∈ K1. Then, from (3.89) with τ1 = ¯τ and τ2 = τk,
0 ≤ ¯τ^{−1}[H(x(¯τ ), ¯x) − H(x(τ_{k}), ¯x)]

≤ ¯τ^{−1}h∇_{1}H(x(¯τ ), ¯x), x(¯τ ) − x(τ_{k})i

≤ τ_{k}^{−1}[H(x(¯τ ), ¯x) − H(x(τ_{k}), ¯x)] .

If K_{1} is infinite, taking the limit k → ∞ with k ∈ K_{1} in the last inequality and using the
continuity of H(·, ¯x) on int(K^{n}) yields that

H(x(¯τ ), ¯x) − H(¯y, ¯x) = h∇_{1}H(x(¯τ ), ¯x), x(¯τ ) − ¯yi.

This together with the strict convexity of H(·, ¯x) implies x(¯τ ) = ¯y. If K_{1} is finite, then
K_{2} := {k : τ_{k}≥ ¯τ } must be infinite. Using the same arguments, we also have x(¯τ ) = ¯y.

(d) By (P3) and Proposition 3.23, there exists g_{τ} ∈ ∂f (x(τ )) such that for any z ∈ V ∩K^{n},
H(x(τ ), ¯x) − H(z, ¯x) ≤ τ^{−1}h∇_{1}H(x(τ ), ¯x), x(τ ) − zi = hg_{τ}, z − x(τ )i. (3.90)
In particular, taking z = x^{∗} ∈ X∗ in the last equality and using the fact

0 ≥ f (x^{∗}) − f (x(τ )) ≥ hg_{τ}, x^{∗}− x(τ )i,

we have H(x(τ ), ¯x) − H(x^{∗}, ¯x) ≤ 0. Hence, {x(τ ) | τ > ¯τ } ⊂ {x ∈ int(K^{n}) | H(x, ¯x) ≤
H(x^{∗}, ¯x)}. By (P4), the latter is bounded, and the desired result then follows.

(e) Let ˆx be a cluster point of {x(τ )} and {τ_{k}} be a sequence such that lim_{k→∞}τ_{k} = ∞
and limk→∞x(τk) = ˆx. Write x^{k} := x(τk) and take x^{∗} ∈ X∗ and z ∈ V ∩ int(K^{n}). Then,
for any ε > 0, we have x(ε) := (1 − ε)x^{∗}+ εz ∈ V ∩ int(K^{n}). From the property (P3),

h∇_{1}H(x(ε), ¯x) − ∇_{1}H(x^{k}, ¯x), x^{k}− x(ε)i ≤ 0.

On the other hand, taking z = x(ε) in (3.90), we readily have
τ_{k}^{−1}h∇_{1}H(x^{k}, ¯x), x^{k}− x(ε)i = hg^{k}, x(ε) − x^{k}i
with g^{k} ∈ ∂f (x^{k}). Combining the last two equations, we obtain

τ_{k}^{−1}h∇_{1}H(x(ε), ¯x), x^{k}− x(ε)i ≤ hg^{k}, x(ε) − x^{k}i.

Since the subdifferential set ∂f (x^{k}) for each k is compact and g^{k} ∈ ∂f (x^{k}), the sequence
{g^{k}} is bounded. Taking the limit in the last inequality yields 0 ≤ hˆg, x(ε) − ˆxi, where ˆg
is a limit point of {g^{k}}, and by [131, Theorem 24.4], ˆg ∈ ∂f (ˆx). Taking the limit ε → 0
in the inequality, we get 0 ≤ hˆg, x^{∗}− ˆxi. This implies that f (ˆx) ≤ f (x^{∗}) since x^{∗} ∈ X∗

and ˆg ∈ ∂f (ˆx). Consequently, ˆx is a solution of the CSOCP (3.64).

Particularly, from the following proposition, we also have that the central path is
convergent if H ∈ D(int(K^{n})) satisfies domH(·, ¯x) = K^{n}, where ¯x ∈ int(K^{n}) is a given
point. Notice that H(·, ¯x) is continuous on domH(·, ¯x) by (P2), and hence the assumption
for H is equivalent to saying that H(·, ¯x) is continuous at the boundary of the cone K^{n}.
Proposition 3.25. For any given ¯x ∈ int(K^{n}) and H ∈ D(int(K^{n})) with domH(·, ¯x) =
K^{n}, let {x(τ ) : τ > 0} be the central path associated to H(·, ¯x). If X∗ is nonempty, then
lim_{τ →∞}x(τ ) exists and is the unique solution of min{H(x, ¯x) | x ∈ X_{∗}}.

Proof. Let ˆx be a cluster point of {x(τ )} and {τk} be such that limk→∞τk = ∞ and
lim_{k→∞}x(τ_{k}) = ˆx. Then, for any x ∈ X∗, using (3.89) with x^{1} = x(τ_{k}) and x^{2} = x, we
obtain

[H(x(τ_{k}), ¯x) − H(x, ¯x)] ≤ τ_{k}hg^{k}, x − x(τ_{k})i ≤ τ_{k}[f (x) − f (x(τ_{k}))] ≤ 0,

where the second inequality is since g^{k} ∈ ∂f (x(τ_{k})), and the last one is due to x ∈ X∗.
Taking the limit k → ∞ in the last inequality and using the continuity of H(·, ¯x), we
have H(ˆx, ¯x) ≤ H(x, ¯x) for all x ∈ X∗. Since ˆx ∈ X∗ by Proposition 3.24(e), this shows
that any cluster point of {x(τ ) | τ > 0} is a solution of min{H(x, ¯x) | x ∈ X∗}. By the
uniqueness of the solution of min{H(x, ¯x) | x ∈ X∗}, we have lim_{τ →∞}x(τ ) = x^{∗}.

For the linear SOCP, we may establish the relations between the sequence generated by the IPA and the central path associated to the corresponding distance-like functions.

Proposition 3.26. For the linear SOCP, let {x^{k}} be the sequence generated by the IPA
with H ∈ D(int(K^{n})), x^{0} ∈ V ∩ int(K^{n}) and ε_{k} ≡ 0, and {x(τ ) | τ > 0} be the central path
associated to H(·, x^{0}). Then, x^{k} = x(τ_{k}) for k = 1, 2, . . . under either of the conditions:

(a) H is constructed via (3.78) or (3.82), and {τ_{k}} is given by τ_{k} = Pk

j=0λ_{j} for k =
1, 2, . . .;

(b) H is constructed via (3.84), the mapping ∇(φ^{0})^{soc}(·) defined on int(K^{n}) maps any
vector IR^{n} into ImA^{T}, and the sequence {τ_{k}} is given by τ_{k}= λ_{k} for k = 1, 2, · · · .
Moreover, for any positive increasing sequence {τ_{k}}, there exists a positive sequence {λ_{k}}
with P∞

k=1λk = ∞ such that the proximal sequence {x^{k}} satisfies xk= x(τk).

Proof. (a) Suppose that H is constructed via (3.78). From (3.67) and Proposition 3.18(b), we have

λ_{j}c + ∇φ(det(x^{j})) − ∇φ(det(x^{j−1})) = A^{T}u^{j} for j = 0, 1, 2, . . . . (3.91)
Summing the equality from j = 0 to k and taking τk =Pk

j=0λj, y^{k}=Pk

j=0u^{j}, we get
τ_{k}c + ∇φ(det(x^{k})) − ∇φ(det(x^{0})) = A^{T}y^{k}.

This means that x^{k} satisfies the optimal conditions of the problem

minτkf (x) + H(x, x^{0}) | x ∈ V ∩ int(K^{n}) , (3.92)
and so x^{k} = x(τ_{k}). Now let {x(τ ) : τ > 0} be the central path. Take a positive increasing
sequence {τ_{k}} and let x^{k} ≡ x(τ_{k}). Then from Proposition 3.23 and Proposition 3.18(b),
it follows that

τ_{k}c + ∇φ(det(x^{k})) − ∇φ(det(x^{0})) = A^{T}y^{k} for some y^{k} ∈ IR^{m}.
Setting λ_{k}= τ_{k}− τ_{k−1} and u^{k}= y^{k}− y^{k−1}, from the last equality it follows that

λ_{k}c + ∇φ(det(x^{k})) − ∇φ(det(x^{k−1})) = A^{T}u^{k}.

This shows that {x^{k}} is the sequence generated by the IPA with ε_{k} ≡ 0. If H is given by
(3.82), using Proposition 3.19(b) and the same arguments, we also have the result holds.

(b) Under this case, by Proposition 3.21(c), the above (3.91) becomes
λ_{j}c + ∇(φ^{0})^{soc}(x^{j}) · (x^{j} − x^{j−1}) = A^{T}u^{j} for j = 0, 1, 2, . . . .

Since φ^{00}(t) > 0 for all t ∈ (0, ∞) by (D1) and (D2), from [63, Proposition 5.2] it follows
that ∇(φ^{0})^{soc}(x) is positive definite on int(K^{n}). Thus, the last equality is equivalent to

∇(φ^{0})^{soc}(x^{j})−1

λ_{j}c + (x^{j} − x^{j−1}) =∇(φ^{0})^{soc}(x^{j})−1

A^{T}u^{j} for j = 0, 1, 2, . . . . (3.93)
Summing the equality (3.93) from j = 0 to k and making suitable arrangement, we get
λ_{k}c + ∇(φ^{0})^{soc}(x^{k})(x^{k}− x^{0}) = A^{T}u^{k}+ ∇(φ^{0})^{soc}(x^{k})

k−1

X

j=0

∇(φ^{0})^{soc}(x^{j})−1

(A^{T}u^{j}− λ_{j}c),
which, using the given assumptions and setting τ_{k} = λ_{k}, reduces to

τ_{k}c + ∇(φ^{0})^{soc}(x^{k})(x^{k}− x^{0}) = A^{T}y¯^{k} for some ¯y^{k} ∈ IR^{m}.

This means that x^{k} is the unique solution of (3.92), and hence x^{k} = x(τ_{k}) for any k. Let
{x(τ ) : τ > 0} be the central path. Take a positive increasing sequence {τ_{k}} and define
the sequence x^{k} = x(τ_{k}). Then, from Proposition 3.23 and Proposition 3.21(c),

τ_{k}c + ∇(φ^{0})^{soc}(x^{k})(x^{k}− x^{0}) = A^{T}y^{k} for some y^{k} ∈ IR^{m},
which, by the positive definiteness of ∇(φ^{0})^{soc}(·) on int(K^{n}), implies that

[∇(φ^{0})^{soc}(x^{k})]^{−1}(τ_{k}c − A^{T}y^{k}) + [∇(φ^{0})^{soc}(x^{k−1})]^{−1}(τ_{k−1}c − A^{T}y^{k−1}) + (x^{k}− x^{k−1}) = 0.

Consequently,

τ_{k}c + ∇(φ^{0})^{soc}(x^{k})(x^{k}− x^{k−1}) = ∇(φ^{0})^{soc}(x^{k})[∇(φ^{0})^{soc}(x^{k−1})]^{−1}(A^{T}y^{k−1}− τ_{k−1}c).

Using the given assumptions and setting λ_{k} = τ_{k}, we have

λ_{k}c + ∇(φ^{0})^{soc}(x^{k})(x^{k}− x^{k−1}) = A^{T}u^{k} for some u^{k} ∈ IR^{m}.

for some u^{k} ∈ IR^{m}. This implies that {x^{k}} is the sequence generated by the IPA and the
sequence {λ_{k}} satisfiesP∞

k=1λ_{k} = ∞ since {τ_{k}} is a positive increasing sequence.
From Proposition 3.25 and Proposition 3.26, we readily have the following improved
convergence results of the sequence generated by the IPA for the linear SOCP.

Proposition 3.27. For the linear SOCP, let {x^{k}} be the sequence generated by the IPA
with H ∈ D(int(K^{n})), x^{0} ∈ V ∩ int(K^{n}) and ε_{k}≡ 0. If one of the conditions is satisfied:

(a) H is constructed via (3.82) with domH(·, x^{0}) = K^{n} and P∞

k=0λ_{k} = ∞;

(b) H is constructed via (3.84) with domH(·, x^{0}) = K^{n}, the mapping ∇(φ^{0})^{soc}(·) defined
on int(K^{n}) maps any vector in IR^{n} into ImA^{T}, and lim_{k→∞}λ_{k}= ∞;

and X∗ 6= ∅, then {x^{k}} converges to the unique solution of min{H(x, x^{0}) | x ∈ X∗}.

## Chapter 4

## SOC means and SOC inequalities

In this chapter, we present some other types of applications of the aforementioned SOC-functions, SOC-convexity, and SOC-monotonicity. These include so-called SOC means, SOC weighted means, and a few SOC trace versions of Young, H¨older, Minkowski in-equalities, and Powers-Størmer’s inequality. We believe that these results will be helpful in convergence analysis of optimizations involved with SOC. Many materials of this chap-ter are extracted from [36, 77, 78], the readers can look into them for more details.