A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

Liang Fang^I,*; Zengzhe Feng^II

^ICollege of Mathematics and System Science, Taishan University, 271021, Tai'an, P.R. China. E-mail: [e-mail]

^IICollege of Information Engineering, Taishan Medical University, 271016, Tai'an, P.R. China. E-mail: [e-mail]

ABSTRACT

A new smoothing function of the well known Fischer-Burmeister function is given. Based on this new function, a smoothing Newton-type method is proposed for solving second-order cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Numerical results demonstrate the effectiveness of the algorithm.

Mathematical subject classification: 90C25, 90C30, 90C51, 65K05, 65Y20.

Key words: second-order cone programming, smoothing method, interior-point method, Q-quadratic convergence, central path, strong semismoothness.

1 Introduction

The second order cone (SOC) in

ⁿ (n > 2), also called the Lorentz cone orthe ice-cream cone, is defined as

_n = {(x₁; x₂)| x₁∈ , x₂∈ ^n-1 and x₁> ||x₂||)},

here and below, ||·|| refers to the standard Euclidean norm, n is the dimensionof _n, and for convenience, we write x = (x₁; x₂) instead of (x₁, )^T. It is easy to verify that the SOC _n is self-dual, that is

_n = = {s ∈ ⁿ : s^T x > 0, for all x ∈ _n}.

We may often drop the subscripts if the dimension is evident from the context.

For any x = (x₁; x₂), y = (y₁; y₂) ∈ ×^n-1, their Jordan product is defined as [5]

x º y = (x^T y; x₁y₂ + y₁x₂).

Second-order cone programming (SOCP) problems are to minimize a linear function over the intersection of an affine space with the Cartesian product of a finite number of SOCs. The study of SOCP is vast important as it covers linear programming, convex quadratic programming, quadratically constraint convex quadratic optimization as well as other problems from a wide range of applications in many fields, such as engineering, control, optimal control and design, machine learning, robust optimization and combinatorial optimization and so on [13, 24, 4, 23, 29, 22, 18, 10, 11].

Without loss of generality, we consider the SOCP problem with a single SOC

and its dual problem

where c ∈ ⁿ, A ∈ ^m×n and b ∈ ^m, with an inner product ‹·, · ›, are given data. x ∈ is variable and the set is an SOC of dimension n. Note that our analysis can be easily extended to the general case with Cartesian product of SOCs.

We call x ∈ primal feasible if Ax = b. Similarly (y, s) ∈ ^m× is called dual feasible if A^T y + s = c. For a given primal-dual feasible point (x,y,s) ∈ ×^m×, ‹x,s › is called the duality gap due to the well known weak dual theorem, i.e., ‹x,s › > 0, which follows that

‹c,x › - ‹b,y › = ‹A^T y + s,x › - ‹Ax,y › = ‹x,s › > 0.

Let us note that ‹x,s › = 0 is sufficient for optimality of primal and dual feasible (x,y,s) ∈ ×^m×.

Throughout the paper, we make the following Assumption:

Assumption 2.1. Both (PSOCP) and its dual (DSOCP) are strictly feasible.

It is well known that under the Assumption 2.1, the SOCP is equivalent to its optimality conditions:

where ‹x, s › = 0 is usually referred to as the complementary condition.

There are an extensive literature focusing on interior-point methods (IPMs) for (PSOCP) and (DSOCP) (see, e.g., [1, 17, 6, 23, 16, 11] and references therein). IPMs typically deal with the following perturbation of the optimality conditions:

where µ > 0 and e = (1; 0) ∈ ×^n-1 is identity element. This set of conditions are called the central path conditions as they define a trajectory approaching the solution set as µ ↓ 0. Conventional IPMs usually apply a Newton-type method to the equations in (4) with a suitable line search dealing with constraints x ∈ and s ∈ explicitly.

Recently smoothing Newton methods [2, 14, 25, 19, 20, 7, 15, 8, 12] haveattracted a lot of attention partially due to their superior numerical performances. However, some algorithms [2, 19] depend on the assumptions of uniform nonsingularity and strict complementarity. Without the uniform nonsingularity assumption, the algorithm given in [27] usually needs to solve two linear systems of equations and to perform at least two line searches at each iteration. Lastly, Qi, Sun and Zhou [20] proposed a class of new smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities under a nonsingularity assumption. The method in [20] was shown to be locally superlinearly/quadratically convergent without strict complementarity. Moreover, the smoothing methods available are mostly for solving the complementarity problems [2, 3, 19, 20, 7, 8], but there is little work on smoothing methods for the SOCP.

Under certain assumptions, IPMs and smoothing methods are globally convergent in the sense that every limit point of the generated sequence is a solution of optimality conditions (3). However, with the exception of the infeasible IPMs [2, 21, 20], they need a feasible starting point.

Fukushima, Luo and Tseng studied Lipschitzian and differential properties of several typical smoothing functions for second-order cone complementarity problems. They derived computable formulas for their Jacobians, which provide a theoretical foundation for constructing corresponding non-interior point methods. The purpose of this paper is just to present such a non-interior point method for problem (PSOCP), which employs a new smoothing function to characterize the central path conditions. We stress on the demonstration of the global convergence and locally quadratic convergence of the proposed algorithm.

The new smoothing algorithm to be discussed here is based on perturbed optimality conditions (4) and the main difference from IPMs is that we reformulate (4) as a smoothing linear system of equations. It is shown that our algorithm has the following good properties:

(i) The algorithm can start from an arbitrary initial point;

(ii) The algorithm needs to solve only one linear system of equations and perform only one line search at each iteration;

(iii) The algorithm is globally and locally Q-quadratically convergent under mild assumption, without strict complementarity. The result is stronger than the corresponding results for IPMs.

The following notations and terminologies are used throughout the paper.We use "," for adjoining vectors and matrices in a row and ";" for adjoining them in a column.

ⁿ (n > 1) denotes the space of n-dimensional real column vectors, and ⁿ×^m is identified with ^n+m. Denote x² = x º x. For any x,y ∈ ⁿ, we write x y or x y (respectively, x y or x y) if x - y ∈ or y - x ∈ (respectively, x - y ∈ int or y - x ∈ int, where int denotes the interior of ). ₊(₊₊) denotes the set of nonnegative (positive) reals. For x ∈ ⁿ with eigenvalues λ₁ and λ₂, we can define the Frobenius norm

Since both eigenvalues of e are equal to one, ||e||_F = . For any x, y ∈ ⁿ, the Euclidean inner product and norm are denoted by ‹x, y › = x^T y and ||x|| = respectively.

The paper is organized as follows. In Section 2, we give the equivalent formulation of the perturbed optimality conditions and some preliminaries. Asmoothing function associated with the SOC and its properties are given in Section 3. In Section 4, we describe our algorithm. The convergence of the new algorithm is analyzed in Section 5. Numerical results are shown in Section 6.

2 Preliminaries

For any vector x = (x₁; x₂) ∈ ×^n-1, we define its spectral decomposition associated with SOC as

where the spectral values λ_i and the associated spectral vectors u_i of x aregiven by

for i = 1,2, with any ω ∈ ^n-1 such that || ω|| = 1. If x₂≠ 0, then the decomposition (5) is unique. Some interesting properties of λ₁, λ₂ and u₁, u₂ are summarized below.

Property 2.1. For any x = (x₁; x₂) ∈ ×^n-1, the spectral values λ₁, λ₂ and spectral vectors u₁, u₂ as given by (6) and (7), have the following properties:

(i) u₁ + u₂ = e.

(ii) u₁ and u₂ are idempotent under the Jordan product, i.e., = u_i, i = 1,2.

(iii) u₁ and u₂ are orthogonal under the Jordan product, i.e., u₁º u₂ = 0, and have length /2.

(iv) λ₁, λ₂ are nonnegative (respectively, positive) if and only if x ∈ (respectively, x ∈ int).

Given an element x = (x₁; x₂) ∈ ⁿ, we define the arrow-shaped matrix

where I represents the (n - 1) × (n - 1) identity matrix. It is easy to verify that x º s = L_x º s = L_s º x = L_xL_se for any s ∈ ⁿ. Moreover, L_x is symmetric positive definite if and only if x ∈ int, i.e., x 0.

3 A smoothing function associated with the SOC and its properties

First, let us introduce a smoothing function. In [7], it has been shown that the vector valued Fischer-Burmeister function ϕ_FB(x, s) : ⁿ × ⁿ → ⁿ defined by

satisfies the following important property

The Fischer-Burmeister function has many interesting properties. However,it is typically nonsmooth, because it is not derivable at (0; 0) ∈ ×^n-1, which limits its practical applications. Recently, some smoothing methods are presented, such as the method using Chen-Harker-Kanzow-Smale smoothing function (see [9, 28] and its references therein).

We now smoothing the function ϕ_FB, so that we get a characterization ofthe central path conditions (4). By smoothing the symmetrically perturbed ϕ_FB, we obtain the new vector-valued function Φ : → ⁿ, defined by

where = {( µ, x, s) ∈ ₊₊ × ⁿ × ⁿ : µ > ||||_F}.

Proposition 3.1. For any ( µ₁, x, s), ( µ₂,x, s) ∈ ,

|| Φ( µ₁, x, s) - Φ( µ₂, x, s)||_F < | µ₁ - µ₂|.

Proof. For any ( µ₁, x, s), ( µ₂, x, s) ∈ , without loss of generality, we assume µ₁> µ₂ > . Thus, we have

which completes the proof.

□

As we will show, the function Φ( µ, x, s) have many good properties that can be used to characterize the central path conditions (4). Φ( µ, x, s) is smooth for any ( µ, x, s) ∈ . This property plays an important role in the analysis of the quadratic convergence of our smoothing Newton method. Semismoothness is a generalization concept of smoothness, which was originally introduced in [15] and then extended by L. Qi in 1993. Semismooth functions includesmooth functions, piecewise smooth functions, convex and concave functions, etc. The composition of (strongly) semismooth functions is still a (strongly) semismooth function.

Definition 3.1. Suppose that H : ⁿ → ^m is a locally Lipschitz continuous function. H is said to be semismooth at x ∈ ⁿ if H is directionally differentiable at x and for any V ∈ ∂H(x + Δx)

H(x + Δx) - H(x) - V( Δx) = o(|| Δx||).

H is said to be p-order (0 < p < ∞) semismooth at x if H is semismoothat x and

H(x + Δx) - H(x) - V( Δx) = O(|| Δx||^1+p).

In particular, H is called strongly semismooth at x if H is 1-order semismooth at x.

The following concept of a smoothing function of a nondifferentiable function was introduced by Hayashi, Yamashita and Fukushima [8].

Definition 3.2. A function H : ⁿ → ^m is said to be a semismooth (respectively, p-order semismooth) function if it is semismooth (respectively, p-order semismooth) everywhere in ⁿ.

In fact, the function Φ( µ, x, s) given by (10) is a smoothing function of ϕ_FB(x, s). Thus, we can solve a family of smoothing subproblems Φ( µ, x, s) = 0 for µ > 0 and obtain a solution of Φ_FB(x, s) = 0 by letting µ ↓ 0.

Definition 3.3 [8]. For a nondifferentiable function g : ⁿ → ^m, we consider a function g _µ : ⁿ → ^m with a parameter µ > 0 that has the following properties:

(i) g _µ is differentiable for any µ > 0;

(ii) (x) = g(x) for any x ∈ⁿ.

Such a function g _µ is called a smoothing function of g.

Theorem 3.1. For any x, s ∈ ⁿ, let w := whose spectral decomposition associated with SOC is w = λ₁u₁ + λ₂u₂. If µ > ||||_F , then the following results hold:

(i) w 0; (ii) µe w; (iii) µ²e

w².

Proof. Assume the spectral decomposition of w associated with SOC is w = λ₁u₁ + λ₂u₂. From µ > ||||_F we have

Thus, we have 0 < λ_i < µ, i = 1,2, which means that (i) holds. By

0 < λ_i < µ, i = 1, 2, µe - w = ( µ - λ₁)u₁ + ( µ - λ₂)u₂ 0,

which yields (ii), and µ²e - w² = ( µ² - )u₁ + ( µ² - )u₂ 0, which gives (iii).

□

Now, we give the main properties of Φ( µ, x, s):

Theorem 3.2. (i) Φ( µ, x, s) is globally Lipschitz continuous for any ( µ, x, s) ∈ . Moreover, Φ( µ, x, s) is continuously differentiable at any ( µ, x, s) ∈ with its Jacobian

(ii) ( µ, x, s) = ϕ_FB(x, s) for any (x, s) ∈ ⁿ × ⁿ. Thus, Φ( µ, x, s) is a smoothing function of Φ_FB(x, s).

Proof. (i) It is not difficult to show that Φ( µ, x, s) is globally Lipschitz continuous, and continuously differentiable at any ( µ, x, s) ∈ . Now we prove (12). For any ( µ, x, s) ∈ , from (10) and by simply calculation, we have

which yield (12).

Next, we prove (ii). For any x = (i₁; x₂) ∈ × ^n-1 and s = (s₁; s₂) ∈ × ^n-1, denote w = whose spectral decomposition associated with SOC is w = λ₁u₁ + λ₂u₂. From Theorem 3.1, we have w 0, and 0 < λ_i< µ, i = 1,2. Therefore, we have

Thus, we have ( µ, x, s) = ϕ_FB(x, s). Therefore, it follows from (i) and Definition 3.3 that Φ( µ, x, s) is a smoothing function of ϕ_FB(x, s).

□

4 Description of the algorithm

Based on the smoothing function (10) introduced in the previous section, the aim of this section is to propose the smoothing Newton-type algorithm for theSOCP and show the well-definedness of it under suitable assumptions.

Let z := ( µ, x, c - A^T y) ∈ . By using the smoothing function (10), we define the function G( µ, x, y) : × ⁿ × ^m → × ^m × ⁿ by

In view of (9) and (13), z*: = ( µ*, x*, y*) is a solution of the system G(z) = 0 if and only if (x*, y*, c - A^T y*) solves the optimality conditions (3).

It is well-known that problems (PSOCP) and (DSOCP) are equivalent to (13) [1, 20]. Therefore, z* is a solution of G(z) = 0 if and only if (x*, y*, c -A^T y*) is the optimal solution of (PSOCP) and (DSOCP). Then we can apply Newton's method to the nonlinear system of equations G(z) = 0.

Let γ ∈ (0, 1) and define the function β:

^n+m+1 → ₊ by

Now we are in the position to give a formal description of our algorithm.

Algorithm 4.1. (A smoothing Newton-type method for SOCP).

Step 0. Choose constants δ ∈ (0, 1), σ ∈ (0, 1), and µ₀∈ ₊₊, and let := ( µ₀, 0, 0) ∈ . Let (x₀, y₀) ∈ ⁿ × ^m be arbitrary initial point and z₀ := ( µ₀, x₀, y₀). Choose γ ∈ (0, 1) such that γ µ₀ < 1/2. Set k := 0.

Step 1. If G(z_k) = 0, then stop. Else, let β_k: = β(z_k).

Step 2. Compute Δz_k: = ( Δ µ_k, Δx_k, Δy_k) by solving the following system of linear equations

Step 3. Let ν_k be the smallest nonnegative integer ν such that

Let λ_k := .

Step 4. Set z_k+1 := z_k + λ_k Δz_k and k := k + 1. Go to step 1.

In order to analyze our algorithm, we study the Lipschitzian, smoothness and differential properties of the function G(z) given by (13). Moreover, we derive the computable formula for the Jacobian of the function G(z) and give the condition for the Jacobian to be nonsingular.

Throughout the rest of this paper, we make the following assumption:

Assumption 4.1. The matrix A has full row rank, i.e., all the row vectors of A are linearly independent.

Lemma 4.1 [7]. For any x, s ∈ ⁿ and any ω 0, we have

Moreover, (17) remains true when "" is replaced by "" everywhere.

Theorem 4.1. Let z := ( µ, x, y) ∈ and G : → be defined by (13). Then the following results hold.

(i) G is globally Lipschitz continuous, and continuously differentiable atany z := ( µ, x, y) ∈ with its Jacobian

(ii) Under Assumption 4.1, G '(z) is nonsingular for any µ > 0.

Proof. By Theorem 3.1, we can easily show that (i) holds. Now we prove (ii). For any fixed µ > 0, let

Δz := ( Δ µ, Δx, Δy) ∈ × ⁿ × ^m.

It is sufficient to prove that the linear system of equations

has only zero solution, i.e., Δ µ = 0, Δx = 0 and Δy = 0. By (18) and (19), we have

Since

It follows from Lemma 4.1 that

Premultiplying (23) by ΔxT and taking into account A Δx = 0, we have

Denote

From (24), we obtain

By (23), is positive definite. Therefore, (26) yields Δ = 0, and it follows from (25) that Δx = 0. Since A has full row rank, (21) implies Δy = 0. Thus the linear system of equations (18) has only zero solution, and hence G'(z) is nonsingular. Thus, the proof is completed.

□

By Theorem 4.1, we can show that Algorithm 4.1 is well-defined.

Theorem 4.2. Suppose that Assumption 4.1 holds. If µ_k > 0, then Algorithm 4.1 is well-defined for any k > 0.

Proof. Since A has full row rank, it follows from Theorem 4.1 that G'(z_k) is nonsingular for any µ_k > 0. Therefore, Step 2 is well-defined at the kth iteration. Then by following the proof of Lemma 5 in [20], we can show the well-definedness of Step 3. The proof is completed.

□

5 Convergence analysis

In this section, we analyze the global and local convergence properties of Algorithm 4.1. It is shown that any accumulation point of the iteration sequence is a solution of the system G(z) = 0. If the accumulation point z* satisfies a nonsingularity assumption, then the iteration sequence converges to z* locally Q-quadratically without any strict complementarity assumption. To show the global convergence of Algorithm 4.1, we need the following Lemma (see [20], Proposition 6).

Lemma 5.1. Suppose that Assumption 4.1 holds. For any ∈ ₊₊ × ⁿ × ^m, and G '() is nonsingular, then there exists a closed neighborhood () of and a positive number ∈ (0, 1] such that for any z = (u, x, y) ∈() and all α ∈ [0, ], we have u ∈ ₊₊, G'(z) is invertible and

Theorem 5.1. Suppose that Assumption 4.1 holds and that {z_k} is the iteration sequence generated by Algorithm 4.1. Then the following results hold.

(i) µ_k ∈ ₊₊ and z_k ∈ Θ for any k > 0, where

(ii) Any accumulation point z* := ( µ*, x*, y*) of {z_k} is a solution of G(z) = 0.

Proof. Suppose that µ_k > 0. It follows from (15) and Step 4 that

Substituting (29) into (30), we have

which, together with µ₀ > 0 and λ_k = ∈ (0, 1) implies that µ_k∈ ₊₊ for any k > 0.

Now we prove z_k ∈ Θ for any k > 0 by induction. Since

β₀ = β(z₀) = γ min {1, ||G(z₀)||²} < γ ∈ (0, 1),

it is easy to see that z₀∈ Θ. Suppose that z_k ∈ Θ, then

We consider the following two cases:

Case (I): If ||G(z_k)|| > 1, then

Since β_k+1 = γ min{1, ||G(z_k+1)||²} < γ, it follows from (16), (31), (32) and (33) that

Case (II): If ||G(z_k)|| < 1, then

By (16), we have ||G(z_k+1)|| < ||G(z_k)|| < 1. From (31), (35), and taking into account β_k+1 = γ||G(z_k+1)||², we have

Combining (34) and (36) yields that z_k ∈ Θ for any k > 0.

Now, we prove (ii). Without loss of generality, we assume that {z_k} converges to z* as k → + ∞. Since {||G(z_k)||} is monotonically decreasing and bounded from below, it follows from the continuity of G(·) that {||G(z_k)||} converges to a nonnegative number G(z*). Then by the definition of β(·), we obtain that { β_k} converges to

β* = γ min {1, ||G(z*)||²}.

It follows from (15) and Theorem 5.1 (i) that

0 < µ_k+1 = (1 - λ_k) µ_k + λ_k β_k µ₀< µ_k,

which implies that { µ_k} converges to µ*. If ||G(z*)|| = 0, then we obtain the desired result. In the following, we suppose ||G(z*)|| > 0. By Lemma 4.1, 0 < β* µ₀< µ*. It follows from Theorem 4.1 that G'(z*) exists and it is invertible. Hence, by Lemma 5.1, there exists a closed neighborhood () of and a positive number ∈ (0, 1] such that for any z = ( µ, x, y) ∈ () and all α ∈ [0, ], we have µ ∈ ₊₊, G'(z) is invertible and

Therefore, for a nonnegative integer such that ∈ (0, ], for all sufficiently large k, we have

||G(z_k + Δz_k)||²< [1 - σ(1 - 2 γ µ₀)]||G(z_k)||².

For all sufficiently large k, since λ_k = > , it follows from (16) that

This contradicts the fact that the sequence {||G(z_k)||} converges to ||G(z*)|| > 0. This completes the proof.

□

To establish the locally Q-quadratic convergence of our smoothing Newton method, we need the following assumption:

Assumption 5.1. Assume that z* satisfies the nonsingularity condition, i.e., all V ∈ ∂G(z*) are nonsingular.

Now we are in the position to give the rate of convergence for Algorithm 4.1.

Theorem 5.2. Suppose that Assumption 4.1 holds and that z* is an accumulation point of the iteration sequence {z_k} generated by Algorithm 4.1. If Assumption 5.1 holds, then:

(i) λ_k ≡ 1 for all z_k sufficiently close to z*;

(ii) {z_k} converges to z* Q-quadratically, i.e., ||z_k+1 - z*|| = O(||z_k - z*||²); moreover, µ_k+1 = O().

Proof. By using Lemma 3.1 and Theorem 4.1, we can prove the theorem similarly as in Theorem 8 of [20]. For brevity, we omit the details here.

□

6 Numerical results

In this section, we conducted some numerical experiments to evaluate the efficiency of Algorithm 4.1. All these experiments were performed on an IBM notebook computer R40e with Intel(R) Pentium(R) 4 CPU 2.00 GHz and 512 MB memory. The operating system was Windows XP SP2 and the implementations were done in MATLAB 7.0.1. For comparison purpose, we also useSDPT3 solver [12] which is an IPM for the SOCP.

For simplicity, we randomly generate six test problems with size m = 50 and n = 100. To be specific, we generate a random matrix A ∈ ^m×n with full row rank and random vectors

x ∈ int, s ∈ int, y ∈ ^m, and then let b := Ax and c := A^T y + s.

Thus the generated problems (PSOCP) and (DSOCP) have optimal solutions and their optimal values coincide, because the set of strictly feasible solutions of (PSOCP) and (DSOCP) are nonempty. Let x₀ = e ∈ ⁿ and y₀ = 0 ∈ ^m be initial points. The parameters used in Algorithm 4.1 were as follows:

µ₀ = 0.01, σ = 0.35, Δ = 0.65 and γ = 0.90.

We used ||H(z)|| < 10^-5 as the stopping criterion.

The results in Table 1 indicate that Algorithm 4.1 performs very well. We also obtained similar results for other random examples.

Acknowledgments. The work is supported by National Natural Science Foundation of China (10571109, 10971122), Natural Science Foundation of Shandong (Y2008A01), Scientific and Technological Project of Shandong Province (2009GG10001012), the Excellent Young Scientist Foundation of Shandong Province (BS2011SF024), and Project of Shandong Province Higher Educational Science and Technology Program (J10LA51). The authors would like to thank the anonymous referees for their valuable comments and suggestions on the paper, which have considerably improved the paper.

Received: 20/VI/10.

Accepted: 03/VIII/10.

#CAM-227/10.

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Corresponding author.

Publication Dates

• Publication in this collection
  06 Jan 2012
• Date of issue
  2011

History

• Received
  20 June 2010
• Accepted
  03 Aug 2010