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NONSYMMETRIC CIRCULAR CONE PROGRAMMING

PENGFEI MA, YANQIN BAI, AND JEIN-SHAN CHEN

Abstract. In this paper, we consider a particular conic optimization problem over nonsymmetric circular cone. This class of optimization problem has been found useful in optimal grasping manipulation problems for multi-fingered robots. We first intro- duce a pair of logarithmically homogeneous self-concordant barrier function for circular cone and its dual cone. Then, based on these two logarithmically homogeneous self- concordant barrier functions and their related properties, we present an interior point algorithm for circular cone optimization problem. Furthermore, we derive the itera- tion bound for this interior point algorithm. Finally, we show some numerical tests to demonstrate the performance of the proposed algorithm.

1. Introduction

Nonsymmetric circular cone programming problems are convex programming problems because their objectives are linear functions and their feasible sets are the intersection of an affine space with the Cartesian product of a finite number of circular cones. The circular cone [10] is defined as

Cθn : = {(x1, x2:n)T ∈ R × Rn−1| cos θkxk ≤ x1}

= {(x1, x2:n)T ∈ R × Rn−1|kx2:nk ≤ x1tan θ}.

(1.1)

where θ ∈ (0,π2) is called rotation angle, k · k denotes the Euclidean norm and (Cθn) is the dual cone of Cθn. It is easy to verify that

(Cθn) = Cnπ

2−θ = {(x1, x2:n)T ∈ R × Rn−1 | sin θkxk2 ≤ x1}.

(1.2)

The geometric illustration of a circular cone, its dual cone, and a second order cone are depicted in Figure 1.

A nonsymmetric circular cone programming problem is an optimization problem with the following form:

minx cTx s.t. Ax = b,

x ∈ K, (1.3)

2010 Mathematics Subject Classification. 90C25, 90C51, 90C60.

Key words and phrases. Conic optimization problem, interior point methods (IPMs), self-concordant barriers.

Corresponding author. This research was supported by a grant from the National Natural Science Foundation of China (No.11371242).

1

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−1

−0.5 0

0.5 1

−1

−0.5 0 0.5 1 0 0.5 1 1.5 2

π/3 π/4 π/6

Figure 1. The graph of circular cone in R3 where K ⊂ Rn is the Cartesian product of several circular cones, i.e.,

(1.4) K = Cθn1

1 × Cθn2

2 × · · · × CθnN

N, with n =

N

P

j=1

nj and θj ∈ (0, π2) for j = 1, 2, . . . , N . Furthermore, we partition the vectors x, c and matrix A as x = (x1; x2; . . . ; xN) with xj ∈ Cθnj

j, c = (c1; c2; . . . ; cN) with cj ∈ Rnj, and A = (A1, A2, . . . , AN) with Aj ∈ Rm×nj, and b ∈ Rm. Without loss of generality, we assume that the matrix A has full row rank, i.e., rank(A)=m. Obviously, the problem (1.3) is also expressed as

minx cTx s.t. Ax = b,

fi(x) ≤ 0, i = 1, 2, . . . , N, (1.5)

where fi(x) = kxi2:nk2− (xi1)2· tan2θi for i = 1, 2, . . . , N .

The problem (1.5) is a second order cone programming (SOCO) problem if θj = π4 for j = 1, 2, . . . , N . It is well-known that second order cone programming problems have had widely applications (see, e.g., [1, 15]). Moreover, the circular cone described by (1.1) with θ 6= π4 naturally arises in many real-life engineering problems [6, 7, 12, 14]. One example is to formulate optimal grasping manipulation for multi-fingered robots. The grasping force of the i-th finger can be expressed in the local coordinate frame ni, oi, ti by fi = (fni, foi, fti)T where fni, foi and fti are the components of fi along ni, oi and

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ti, respectively. To ensure no slipping at a contact point, the components of the contact force fi must satisfy the contact constraint

k(foi, fti)k ≤ µfni, (1.6)

where µ is the static friction coefficient of the substrate. In fact, (1.6) geometrically represent a circular cone with rotation angle β = tan−1µ (see Figure 2).

Figure 2. Circular cone at a grasp point

By (1.1) and (1.2), as long as rotation angle θ = π4, the circular cone and its dual cone reduce to the well-known second order cone (also known as the Lorentz cone and the ice-cream cone) given by

Ln:= {(x1, x2:n)T ∈ R × Rn−1|kx2:nk ≤ x1}.

(1.7)

It is clear that the second order cone is a symmetric cone. However, the circular cone is a nonsymmetric cone because it is not self-dual, i.e., K 6= K unless θ = π4. A main difference between circular cone constraints and most of the other cone constraints [2, 5, 11, 13, 24, 25] is that the circular cone is nonsymmetric, which makes the problem (1.5) more challenging.

As mentioned in [28], there is a close relation between Cθn and Ln as below Ln= AθCθn where Aθ :=tan θ 0

0 En−1

 .

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We point out that, with the above transformation, it is possible to construct a new inner product which ensures the circular cone Cθn to be self-dual. More precisely, we define an inner product associated with Aθ as hx, yiAθ := hAθx, Aθyi. Then, we have

(Cθn) = {x | hx, yiAθ ≥ 0, ∀y ∈ Cθn} = {x | hAθx, Aθyi ≥ 0, ∀y ∈ A−1θ Ln}

= {x | hAθx, yi ≥ 0, ∀y ∈ Ln} = {x | Aθx ∈ Ln}

= A−1θ Ln= Cθn.

However, under this new inner product the second-order cone is not self-dual, because (Ln) = {x | hx, yiAθ ≥ 0, ∀y ∈ Ln} = {x | hAθx, Ayi ≥ 0, ∀y ∈ Ln}

= {x | hA2θx, yi ≥ 0, ∀y ∈ Ln} = {x | A2θx ∈ Ln} = A−2θ Ln.

Since we cannot find an inner product such that the circular cone and second-order cone are both self-dual simultaneously, we must choose an inner product from the standard inner product or the new inner product associated with Aθ. In view of the well-known properties regarding second-order cone and second-order cone programming (in which many results are based on the Jordan algebra and second-order cones are considered as self-dual cones), we adopt the standard inner product in this paper.

Some researchers have investigated circular cones and nonsymmetric circular cone pro- gramming problems. In [9, 27–29], Chen et al. paid a lot attentions to study some properties of circular cone and vector-valued functions associated with circular cones. In [30], Zhou et al. established complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular cone programming problems. Moreover, Bai et al. considered kernel function-based interior point algorithm for solving the problem (1.3) or (1.5) in [3]. They conclude that the problem (1.5) is polynomial-time solvable.

At the same time, Bai et al. also investigated kernel function-based interior point algo- rithm for convex quadratic circular cone programming problems in [4].

Recently, Nesterov [23] proposed a new interior point algorithm that is based on an extension of the ideas of self-scaled optimization to the nonsymmetric conic optimization.

The author developed a 4n-self-concordant barrier for an n-dimensional p-cone, which is a special case of nonsymmetric cone. Matsukawa and Yoshise [16] proposed a primal barrier function phase I algorithm for solving conic optimization problems over doubly nonnegative cone. Skajaa and Ye [26] designed a homogeneous interior point algorithm for nonsymmetric convex conic optimization. All these IPMs are designed based on self- concordant barrier functions for its corresponding cone.

Self-concordant barrier functions are presented by Nesterov and Nemirovski [17]. They play an important role in the powerful polynomial-time IPMs for convex programming.

Several classes of interior point algorithms for linear programming are extended to non- linear setting in terms of self-concordant barrier functions for convex region. Following the work of Nesterov and Nemirovski, many articles have issued using this type of func- tion to construct barrier functions for IPMs [18–20]. In [22] and [8], the authors also

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presented 3-self-concordant barriers for the nonsymmetric power cone and the exponen- tial cone, respectively. Therefore, cone programming problems with the power cone or the exponential cone constraints have effective interior point algorithm [26].

Inspirited by the nice properties of self-concordant barrier functions, in this paper, we consider a particular conic optimization problem over nonsymmetric circular cone, which has been found useful application in optimal grasping manipulation problems for multi- fingered robots. We first introduce a pair of logarithmically homogeneous self-concordant barrier function for circular cone and its dual cone. Then, based on these two logarith- mically homogeneous self-concordant barrier functions and their related properties, we present an interior point algorithm for circular cone optimization problem. Furthermore, we derive the iteration bound for this interior point algorithm. Finally, we show some numerical tests to demonstrate the performance of the proposed algorithm.

The paper is organized as follows. In Section 2, we recall basic concepts and properties on self-concordant barrier functions. In Section 3, we introduce a pair of logarithmically homogeneous self-concordant barrier functions for circular cone and its dual cone, respec- tively. In Section 4, we discuss optimality conditions and central paths of nonsymmetric circular cone programming problems. In Section 5, based on logarithmically homoge- neous self-concordant barrier function for circular cone and its dual cone, we present an interior point algorithm for nonsymmetric circular cone programming. In Section 6, we implement our algorithm by several random examples to show the performance of the algorithm. Finally, we conclude and give further research in Section 7.

2. Preliminaries

As mentioned in the Introduction, self-concordant barrier functions are crucial to IPMs.

In order to proceed with our discussion, we recall some basic concepts and properties of self-concordant barrier functions which will be used in this paper. The materials can be found in [17] and [21], we here omit their proofs.

Given a closed convex function f (x)(dom f ) with open domain and fix a point x ∈ domf and a direction u ∈ Rn, we consider the function

φ(t) = f (x + tu),

depending on the variable t ∈ domφ(x; ·) ⊆ R. Then, we denote Df (x)[u] = φ0(t) = h∇f (x), ui, D2f (x)[u, u] = φ00(t) = h∇2f (x)u, ui, D3f (x)[u, u, u] = φ000(t).

With these notations, self-concordant function and self-concordant barrier function are defined as follows.

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Definition 2.1. A closed convex function F ∈ C3 (three times continuously differen- tiable) with open domain C is called self-concordant if

|D3F (x)[h, h, h]| ≤ 2(D2F (x)[h, h])3/2, (2.1)

for all x ∈ domF and for all h ∈ Rn.

Definition 2.2. A self-concordant function F is a ν-self-concordant barrier for a closed convex set K if

∇F (x)T(∇2F (x))−1∇F (x) ≤ ν, ∀x ∈ K.

The value ν is called the parameter of the barrier F and K is interior of the set K.

In order to prove our Theorem 3.3, we need a property regarding ν-self-concordant barrier under an affine transformation.

Lemma 2.3. Let F : C ⊆ Rn → R be a ν-self-concordant barrier, A : Rp → Rn such that A(y) = By + b for B ∈ Rn×p and b ∈ Rn. Assume A(Rp) ∩ C 6= ∅. Define C+:= A−1(C) = {y ∈ Rp : A(y) ∈ C} ⊆ Rp. Then ˜F : C+→ R defined as

F (y) = F (A(y))˜ is a ν-self-concordant barrier for C+.

For proper cones, Nesterov and Nemirovski have presented a special class of barriers in [17] and [21]. The definition is stated as follows.

Definition 2.4. Let K be a proper cone, F : K → R a twice continuously differentiable, convex barrier function. F is called ν-logarithmically homogeneous for K if

(2.2) F (tx) = F (x) − ν ln t

for any x ∈ K and any t > 0.

In order to introduce logarithmically homogeneous barrier of dual cone of the circular cone, we employ the following Definition 2.5 and Lemma 2.6.

Definition 2.5. Let F be a ν-logarithmically homogeneous barrier for K. Its conjugate is defined as

F(s) = sup

x∈K

{−sTx − F (x)}.

Lemma 2.6. Let F : K → R be a ν-self-concordant barrier for K. Then F(s) is a ν-self-concordant barrier for K.

For the positive orthant, the second order cone, and the cone of positive semidefinite matrices, we list their self-concordant barriers in Table 1. Note that the three cones lead to linear programming problems, second order cone programming problems, and semidefinite programming problems, respectively. In all these examples, the cones are symmetric and the barriers are self-scaled [19]. However, in general, this cannot be true.

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Table 1. self-concordant barriers over some convex cones

Cones Self-concordant barriers Parameters Conjugate functions Rn+= {x ∈ Rn: x ≥ 0} F (x) = −

n

P

i=1

ln(xi) ν = n F(s) = F (s) − n {(x1, x2:n)T ∈ R × Rn−1|kx2:nk ≤ x1} F (x1, x2:n) = − ln(x21− kx2:nk2) ν = 2 F(s) = F (s) + 2 ln 2 − 2

Sn+= {X ∈ Sn: X  0} F (X) = − ln det(X) ν = n F(s) = F (s) − n

3. The barrier function for circular cone

In this section we introduce logarithmically homogeneous self-concordant barrier func- tions for the circular cone and its dual cone.

First, we recall an important Lemma in [3, Theorem 2.3]. The Lemma is critical to our subsequent analysis. Given a rotation angle θ, let

Aθ := tan θ 0

0 En−1

 (3.1)

where En−1 is an n − 1 dimensional unit matrix. It is straightforward to verify that A−1θ = cot θ 0

0 En−1

 . (3.2)

Lemma 3.1. For any x ∈ Cθn and s ∈ (Cθn), there exist ˜x ∈ Ln and ˜s ∈ Ln such that (1) ˜x = Aθx and ˜s = A−1θ s.

(2) ˜xT˜s = 0 if and only if xTs = 0.

Then, we prove the following Lemma 3.2 to obtain a self-concordant barrier of dual cone.

Lemma 3.2. Let h be a convex function on Rn, and let f (x) = h(Ax + b) + aTx + α,

where A is a one-to-one linear transformation from Rn to Rn, a and b are vectors in Rn, and α ∈ R. Then

f(s) = h (A−1)T(a + s) + (A−1b)Ts + α, where α = −α + (A−1b)Ta.

Proof. The substitution y = Ax + b enables us to calculate f as follows f(s) = sup

x

{−sTx − h(Ax + b) − aTx − α}

= sup

y

{−sT A−1(y − b) − h(y) − A−1(y − b)T

a − α}

= sup

y

{−(A−1y)T(s + a) − h(y)} + (A−1b)T(s + a) − α

= sup

y

{−yT(A−1)T(s + a) − h(y)} + (A−1b)Ts + (A−1b)Ta − α

= h (A−1)T(s + a) + (A−1b)Ts + α.



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2 Based on the self-concordant barrier for the second order cone, Lemma 2.3, Lemma 2.6, Lemma 3.1 and Lemma 3.2, we introduce self-concordant barriers for the circular cone and its dual cone.

Theorem 3.3. The function

(3.3) Fθ(x) = − ln(x21· tan2θ − kx2:nk2) is a 2-self-concordant barrier for x ∈ Cθn and the function

(3.4) (Fθ)(s) = − ln(s21· cot2θ − ks2:nk2) + 2 ln 2 − 2 is a 2-self-concordant barrier for s ∈ (Cθn). Furthermore,

(3.5) FK(x) = −

N

X

j=1

ln (xj1)2· tan2θj − kxj2:nk2 is 2N -self-concordant barrier for K.

Proof. By Lemma 4.3.3 in [21], the function

F (x) = − ln(x21− kx2:nk2)

is 2-self-concordant barrier for the second order cone and its conjugate function is F(s) = F (s) + 2 ln 2 − 2.

Using Lemma 2.3 and Lemma 3.2, one has Fθ(x) = F tan θ 0

0 En−1

  x1 x2:n



= − ln(x21· tan2θ − kx2:nk2)

is a 2-self-concordant barrier for Cθn. According to Lemma 3.1, Lemma 2.6 and Lemma 3.2, we obtain

(Fθ)(s) = F

 cot θ 0

0 En−1

  s1 s2:n



= − ln(s21· cot2θ − ks2:nk2) + 2 ln 2 − 2 is a 2-self-concordant barrier for (Cθn). By using [21, Theorem 4.2.2], we complete the

proof. 2 

Suppose that Fθ is a 2-self-concordant barrier of the circular cone. It is clear that the Hessian of Fθ is a positive definite matrices. Using the Hessian of Fθ for any x ∈ (Cθn), we can define the following local norms on Cθn and (Cθn):

khkx =p

hT(∇2Fθ(x))h, f or h ∈ Cθn, (3.6)

kskx = q

sT(∇2Fθ−1(x))s, f or s ∈ (Cθn). (3.7)

Other properties of Fθ is refer to [19, 20].

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4. Self-concordant barrier and central path

In this section, we define central path in terms of FK(x). We assume that the problem (1.5) is strictly feasible, i.e., there exists x0 such that Ax0 = b and fi(x0) < 0 for i = 1, . . . , N . This means that Slater’s constraint qualification holds, so there exits dual optimal λ = (λ1, λ1, . . . , λN) ∈ RN, v ∈ Rm, which together with optimal solution x satisfy the KKT conditions

Ax = b, fi(x) ≤ 0, i = 1, 2, . . . , N λi ≥ 0, i = 1, 2, . . . , N c +

N

X

i=1

λi∇fi(x) + ATv = 0,

λifi(x) = 0, i = 1, 2, . . . , N.

(4.1)

We refer to a pair (λ, v) with λi ≥ 0, i = 1, 2, . . . , N and c +

N

P

i=1

λi∇fi(x) + ATv = 0 as dual feasible.

Based on FK(x) and the basic idea of barrier function, a suitable equality constrained problem is given by t > 0:

minx {ft(x) = thc, xi + FK(x) : Ax = b}, (4.2)

which is a penalty problem as nonsymmetric circular cone programming (1.5). The problem (4.2) is an equality constrained problem to which Newton’s method can be applied. By using strongly convex of thc, xi + FK(x), the problem (4.2) has a unique solution for each t > 0.

For any t > 0, we define x(t) as the solution of (4.2). The central path associated with problem (1.3) is defined as the set of points x(t) for t > 0. The points on the central path are characterized by the following necessary and sufficient conditions:

(4.3)  tc + ∇FK(x(t)) + ATv(t) = 0 Ax(t) = b, x(t) ∈ K.

By (4.3), we have

0 = tc + ∇FK(x(t)) + ATv

= tc +

N

X

i=1

1

−fi(x(t))∇fi(x(t)) + ATv.

(4.4)

From (4.4), we can yield a dual feasible point

(4.5) λi(t) = − 1

−tfi(x(t)), i = 1, 2, . . . , N, v(t) = v t.

It is clear that λi(t) > 0 because fi(x(t)) < 0, i = 1, 2, . . . , N . Moreover, by (4.4), we have

c +

N

X

i=1

λi(t)∇fi(x(t)) + ATv(t) = 0.

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We see that x(t) minimizes the Lagrangian L(x, λ, v) = cTx +

N

X

i=1

λifi(x) + vT(b − Ax),

for λ = λ(t) = (λ1(t), λ2(t), . . . , λN(t)), v = v(t). Therefore the dual function g (λ(t), v(t)) = inf

z L (z, λ(t), v(t)) is finite, and

g (λ(t), v(t)) = cTx(t) +

N

X

i=1

λi(t)fi(x(t)) + v(t)T(b − Ax(t))

= cTx(t) −N t .

In particular, the duality gap associated with x(t) and the dual feasible point λ(t), v(t) is N

t . As an important consequence, we have

(4.6) cTx(t) − cTx ≤ N

t .

By the above inequality, it implies that x(t) converges to optimal point x as t → ∞.

5. Interior point algorithm for nonsymmetric circular cone programming

In this section, we discuss the search direction from Newton-type system of (4.3).

Then, based on the search direction, we describe the scheme of our algorithm. Moreover, the iteration bound of the algorithm is computed.

5.1. The search direction. We will go along Newton direction towards the minimizer of (4.2). To compute the Newton direction, we use the gradient and the Hessian of the objective function ft(x), which is given by

∇ft(x) = tc + ∇FK(x), ∇2ft(x) = ∇2FK(x).

The Newton direction 4x(t) is then defined as the direction of linear system (5.1)  ∇2FK(x) AT

A 0

  ∆x(t)

∆v(t)



= − (tc + ∇FK(x)) 0



By (5.1), we have ∆x(t) = −(∇2ft(x))−1(∇ft(x) + AT∆v(t)). We denote the Newton decrement for (4.2) at the point x by

δx,t = k∆x(t)kx =p

∆x(t)T2ft(x)∆x(t).

Obviously, we have δx,t = k∇ft(x) + AT∆v(t)kx.

To complete the following Theorem 5.2, we need the following technical Lemma.

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Lemma 5.1. [8, Theorem 2.3.4] Let F (x) be a self-concordant function, A ∈ Rm×n, b ∈ Rm and x ∈ domF such that Ax = b and δx< 1. Then

ω(δx) ≤ F (x) − F (x) ≤ ωx), (5.2)

ω0x) ≤ kx − xkx ≤ ω0

x), (5.3)

where x denotes an optimal solution for the following problem minx {F (x) : Ax = b},

(5.4)

δx denotes Newton decrement for (5.4) at the point x, ω(t) = t − ln(1 + t), t > −1 and ω(t) = −t − ln(1 − t), t < 1.

Theorem 5.2. For any t > 0 and δx,t ≤ β < 1, then cT(x − x) ≤ 1

tκ(β, N ).

where κ(β, N ) = (β+

2N )β 1−β + N .

Proof. By using tc = ∇ft(x) − ∇FK(x), we have tcT(x − x(t)) = (∇ft(x) − ∇FK(x))T(x − x(t))

= (∇ft(x) + ATv(t) − ∇FK(x) − ATv(t))T(x − x(t))

= (∇ft(x) + ATv(t) − ∇FK(x))T(x − x(t)) − v(t)TA(x − x(t))

≤ (k∇ft(x) + ATv(t)kx+ k∇FK(x)kx) · kx − x(t)kx

≤ (δx,t+

2N ) · ω0x,t)

≤ (δx,t+√

2N ) δx,t 1 − δx,t

≤ (β +√ 2N )β 1 − β ,

We use Lemma 5.1 on the above second inequality. From (4.6) and the above inequality,

we immediately yield the desired result. 2 

5.2. Interior point algorithm. In this subsection, we describe our algorithm. First, we denote by ∆xit

k the Newton direction at the point x(i) towards the target point x(tk) on the central path. Furthermore, δtk(x(i)) = k∆x(i)tkkx(i) is the Newton decrement of

∆x(i)tk with respect to the current iterate x(i). The algorithm is as follows.

5.3. The iteration bound. In this subsection, we analyze the iteration bound for al- gorithm 5.2. To proceed, two technical Lemmas are needed.

Lemma 5.3. [8, Theorem 2.3.6] Let F (x) be a self-concordant function, A ∈ Rm×n, b ∈ Rm and x ∈ domF such that Ax = b and we define the new iterate

x+= x + 1

1 + δx · ∆x.

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Interior point algorithm for nonsymmetric circular cone programming Input: A ∈ Rm×n with full row rank, b ∈ Rm, c ∈ Rn, θi for

i = 1, 2, . . . , N and FK(x).

Parameter: Choose  > 0, 0 < β ≤ 12, µ > 1 and define κ(β, N ) = (β+

2N )β 1−β + N .

Initialize: k = 0, i = 0, t0 > 0 and x0 satisfies Ax0 = b, x0 ∈ K and δt0(x(0)) ≤ β.

while  · tk < κ(β, N ) do

1) compute Newton direction ∆x(i)tk from (5.1) 2) compute Newton decrement δtk(x(i)) = k∆x(i)tkkx(i)

while δtk(x(i)) > β do a) x(i+1) = x(i)+ 1+δ 1

tk(x(i))· ∆x(i)t

k

b) i = i + 1

c) compute Newton direction ∆x(i)tk from (5.1) d) compute Newton decrement δtk(x(i)) = k∆x(i)tkkx(i) end while

3) update tk+1 = µ · tk

4) k = k + 1 end while

Figure 3. Interior point algorithm for nonsymmetric circular cone programming.

Then x+∈ domF and Ax+ = b. Moreover, we have F (x+) ≤ F (x) − ω(δx),

where ∆x denote the Newton direction for (5.4) at the point x and δx denotes Newton decrement.

By using the above Lemma 5.3, an upper bound on the functional difference is given in the process of inner iterations. From the upper bound, we can yeild an upper bound on the number of iterations from x to x(µt)

Lemma 5.4. Let x ∈ K, Ax = b and δx,t ≤ β. If we update t to µt and impose additionally that β ≤ 12, then we have the following bound on the functional difference:

(5.5) fµt(x) − fµt(x(µt)) ≤ µ(N + 2(√

2N + 1)).

Proof. If we denote ρ = fµt(x) − fµt(x(µt)), ρ1 = fµt(x) − fµt(x(t)) and ρ2 = fµt(x(t)) − fµt(x(µt)), then

ρ = ρ1 + ρ2.

The upper bound on ρ is obtained by adding upper bounds for ρ1 and ρ2.

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First, we drive an upper bound on ρ1. By convexity of FK(x) on x, one has ρ1 = fµt(x) − fµt(x(t))

= µtcTx + FK(x) − µtcTx(t) − FK(x(t))

= µtcT(x − x(t)) + (FK(x) − FK(x(t)))

≤ µtcT(x − x(t)) + h∇FK(x), x − x(t)i

≤ µtcT(x − x(t)) + k∇FK(x)kx· kx − x(t)kx

≤ µ(β +√ 2N )β 1 − β +√

2N · δx,t 1 − δx,t

≤ µ(β +√ 2N )β 1 − β +√

2N · β 1 − β

≤ (µ + 1)(√

2N + 1)

≤ 2µ(√

2N + 1).

Then, we drive an upper bound on ρ2 as follows:

ρ2 = fµt(x(t)) − fµt(x(µt))

= µtcTx(t) + FK(x(t)) − µtcTx(µt) − FK(x(µt))

= µtcT (x(t) − x(µt)) +

N

X

i=1

 ln − µtλi(t)fi(x(µt))

− ln − µtλi(t)fi(x(t))

= µtcT(x(t) − x(µt)) +

N

X

i=1

ln − µtλi(t)fi(x(µt)) − N ln µ

≤ µtcT(x(t) − x(µt)) − µt

N

X

i=1

λi(t)fi(x(µt)) − N − N ln µ

= µtcTx(t) − µtcTx(µt) +

N

X

i=1

λi(t)fi(x(µt)) + v(t)T(b − Ax(µt)) − N − N ln µ

≤ µtcTx(t) − µtg(λ(t), v(t)) − N − N ln µ

= N (µ − 1 − ln µ)

≤ N µ.

To obtain the fourth equality from the third, we use λi(t) = − 1

tfi(t). In the first inequality we use the fact that ln y ≤ y−1 for y > 0. To obtain fifth equality from the first inequality,

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we use Ax(µt) = b. The second inequality follows from the definition of dual function:

g(λ(t), v(t)) = inf

z cTz +

N

X

i=1

λi(t)fi(z) + v(t)T(b − Az)

!

≤ cTx(µt) +

N

X

i=1

λi(t)fi(x(µt)) + v(t)T(b − Ax(µt)).

The last equality follows from g(λ(t), v(t)) = cTx(t) − N/t.

In a word, we have ρ ≤ µ(N + 2(√

2N + 1)). 2 

Next, we state our main result as follows.

Theorem 5.5. Let β ≤ 12. Then, the algorithm 5.2 terminates after at most k ≤ O(N ln N

t0) iterations with a point xk such that hc, xk− xi ≤ .

Proof. According to Theorem 5.2, we have cT(x − x) ≤ 1

tκ(β, N ).

If we desire cT(x−x) ≤ , then this is guaranteed by finding a point x such that δt(x) ≤ β and 1tκ(β, N ) ≤ . The latter condition is satisfied if µk11t

0κ(β, N ) ≤ , where k1 is the number of outer iterations. Then, k1 is no more than

ln κ(β, N ) − ln(t0)

ln µ = O(ln N

t0).

By Lemma 5.3, we have

ω(δx) ≤ F (x) − F (x+).

As long as δx > β, we can reduce the function fµt(x) at least ω(β) by ω(δx) > ω(β).

Using Lemma 5.4, that means the optimality gap fµt(x) − fµt(x(µt)) will be reduced at most

fµt(x) − fµt(x(µt))

ω(β) ≤ µ(N + 2(√

2N + 1))

ω(β) = O(N )

times before δx ≤ β. We complete the proof. 2 

6. Numerical results

In this section, we give some numerical examples to illustrate the performance of the proposed algorithm for solving nonsymmetric circular cone programming problems de- scribed in Section 5.

Numerical examples are generated randomly and the number of constraints is set as half of dimension for its corresponding problem. For these circular cones, we choose the rotation angles as θ = 12π,π6,π4,π3,12, respectively. The test problems are divided into

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three categories for every fixed rotation angle. In the first group, the problems’ dimen- sion is between 10 and 90, and the problems’ dimension is between 100 and 900 in the second group, whereas it is between 1000 and 2000 in the third group.

The numerical experiments are implemented by using MATLAB R2008b, and on a PC with Intel 2.20 GHz CPU, 2 GB RAM. The parameters were selected as: β = 0.2925 and

 = 1 × 10−5.

The numerical results are listed in Table 2, Table 3 and Table 4. The following nota- tions are used: iter, number of iterations; CPU(s), CPU time (in seconds).

Table 2. The numerical results on circular cone programming (small size)

θ 12π π6 π4 π3 12

(n, m) iter CPU(s) iter CPU(s) iter CPU(s) iter CPU(s) iter CPU(s) (10,5) 28 0.085950 23 0.007586 24 0.064620 23 0.009108 23 0.008762 (20,10) 28 0.014009 25 0.027221 25 0.029289 24 0.116081 23 0.127255 (30,15) 29 0.018051 26 0.036856 25 0.103724 24 0.015807 24 0.084869 (40,20) 29 0.089729 27 0.145478 26 0.048549 25 0.074739 24 0.021951 (50,25) 30 0.117831 27 0.133267 26 0.076434 26 0.072427 26 0.034333 (60,30) 30 0.071203 27 0.102295 26 0.101482 26 0.045275 25 0.099405 (70,35) 30 0.139458 28 0.064778 27 0.122284 26 0.117338 26 0.114639 (80,40) 31 0.277781 29 0.157181 27 0.101396 26 0.120880 26 0.151339 (90,45) 30 0.110020 29 0.264767 27 0.123550 26 0.097459 26 0.185536

Table 3. The numerical results on circular cone programming (medium size)

θ 12π π6 π4 π3 12

(n, m) iter CPU(s) iter CPU(s) iter CPU(s) iter CPU(s) iter CPU(s) (100,50) 31 0.191587 29 0.122548 27 0.150536 26 0.112776 26 0.119702 (200,100) 32 0.718956 30 0.699293 28 0.661747 28 0.876931 27 0.556061 (300,150) 33 2.237525 30 1.822652 29 1.553632 29 3.204657 28 1.749023 (400,200) 33 4.153084 31 3.754879 30 3.025065 29 3.204657 29 3.734951 (500,250) 33 6.700060 31 6.510338 30 5.288580 29 6.166231 29 5.865587 (600,300) 34 10.678081 32 10.461737 30 8.000719 29 9.698626 29 8.716132 (700,350) 34 15.733862 32 14.290476 30 11.815231 29 13.848642 29 13.128566 (800,400) 35 20.898460 32 20.421453 30 16.041597 30 19.409672 29 17.971985 (900,450) 35 29.560119 32 27.392799 31 23.314514 30 25.578036 30 24.746241

From Table 2, Table 3 and Table 4, we see that the iterative number of our algorithm ranges from 25 to 40. In particular, when the rotation angle is getting smaller, the iter- ations become larger. The computing time for θ = π4 is always less, no matter what size the problem is. Another phenomenon is that, with the increase of the dimension, the iterations of our algorithm become more and the computing time get longer.

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Table 4. The numerical results on circular cone programming (large size)

θ 12π π6 π4 π3 12

(n, m) iter CPU(s) iter CPU(s) iter CPU(s) iter CPU(s) iter CPU(s) (1000,500) 35 37.090044 32 35.099251 31 27.448155 30 32.902582 30 33.633529 (1100,550) 35 44.537521 33 45.148683 31 35.335342 30 43.566852 30 42.799847 (1200,600) 34 54.978862 33 56.331029 31 43.534430 31 52.749817 30 51.741102 (1300,650) 35 62.850570 33 70.953161 32 53.073101 30 64.660031 31 61.763158 (1400,700) 36 82.304524 33 81.420084 32 64.888901 31 77.799324 31 75.506688 (1500,750) 35 94.100085 33 91.887821 32 77.277800 31 92.013727 31 88.459140 (1600,800) 36 111.545756 34 112.266897 31 86.991472 31 107.168125 31 104.693379 (1700,850) 36 128.554900 33 136.690869 32 102.955529 31 122.958628 31 123.637996 (1800,900) 35 141.416163 34 156.262517 31 116.271943 31 138.461658 31 136.811174 (1900,950) 36 166.054804 33 178.260190 32 139.689364 31 159.19593 31 154.884698 (2000,1000) 36 189.208820 33 198.598245 32 163.310439 32 202.917734 31 182.404055

7. Conclusions

In this paper, based on the algebraic relationship between the second cone and the circular cone, we introduce a logarithmically homogeneous self-concordant barrier func- tions for circular cone and its dual cone. By using logarithmically homogeneous self- concordant barrier function of circular cone, we investigate an interior point algorithm to solve nonsymmetric circular cone programming and derive the iteration bound. Finally, The numerical results show the effectiveness of the proposed algorithm.

At last, we point out that the proposed algorithm may be extended as infeasible initial point for nonsymmetric circular cone programming. Moreover, we can explore how to solve large scale problems. We leave them as our future research work.

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(P.F. Ma) Department of Mathematics, Shanghai University, Shanghai, 200444, China E-mail address: mathpengfeima@126.com

(Y.Q Bai) Department of Mathematics, Shanghai University, Shanghai, 200444, China E-mail address: yqbai@shu.edu.cn

(Jein-Shan Chen) Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

E-mail address: jschen@math.ntnu.edu.tw

參考文獻

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