https://doi.org/10.1007/s12190-016-1065-0 O R I G I NA L R E S E A R C H
Numerical comparisons based on four smoothing functions for absolute value equation
B. Saheya1 · Cheng-He Yu2 · Jein-Shan Chen2
Received: 24 March 2016 / Published online: 26 October 2016
© Korean Society for Computational and Applied Mathematics 2016
Abstract The system of absolute value equation, denoted by AVE, is a non- differentiable NP-hard problem. Many approaches have been proposed during the past decade and most of them focus on reformulating it as complementarity problem and then solve it accordingly. Another approach is to recast the AVE as a system of nonsmooth equations and then tackle with the nonsmooth equations. In this paper, we follow this path. In particular, we rewrite it as a system of smooth equations and pro- pose four new smoothing functions along with a smoothing-type algorithm to solve the system of equations. The main contribution of this paper focuses on numeri- cal comparisons which suggest a better choice of smoothing function along with the smoothing-type algorithm.
Keywords Smoothing function· Smoothing algorithm · Singular value · Convergence Mathematics Subject Classification 26B05· 26B35 · 65K05 · 90C33
B
Jein-Shan Chen [email protected] B. Saheya[email protected] Cheng-He Yu
1 College of Mathematical Science, Inner Mongolia Normal University, Hohhot 010022, Inner Mongolia, People’s Republic of China
2 Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan
1 Introduction
The absolute value equation (AVE) is in the form of
Ax + B|x| = b, (1)
where A∈ Rn×n, B ∈ Rn×n, B = 0, and b ∈ Rn. Here|x| means the componentwise absolute value of vector x ∈ Rn. When B = −I , where I is the identity matrix, the AVE (1) reduces to the special form:
Ax − |x| = b. (2)
It is known that the AVE (1) was first introduced by Rohn in [20] and recently has been investigated by many researchers, for example, Hu and Huang [8], Jiang and Zhang [9], Ketabchi and Moosaei [10], Mangasarian [11–15], Mangasarian and Meyer [16], Prokopyev [17], and Rohn [22].
In particular, Mangasarian and Meyer [16] show that the AVE (1) is equivalent to the bilinear program, the generalized LCP (linear complementarity problem), and the standard LCP provided 1 is not an eigenvalue of A. With these equivalent reformu- lations, they also show that the AVE (1) is NP-hard in its general form and provide existence results. Prokopyev [17] further improves the above equivalence which indi- cates that the AVE (1) can be equivalently recast as LCP without any assumption on A and B, and also provides a relationship with mixed integer programming. In general, if solvable, the AVE (1) can have either unique solution or multiple (e.g., exponentially many) solutions. Indeed, various sufficiency conditions on solvability and non-solvability of the AVE (1) with unique and multiple solutions are discussed in [16,17,21]. Some variants of the AVE, like the AVE associated with second-order cone and the absolute value programs (AVP), are investigated in [5] and [23], respectively.
As for its numerical solvers, many numerical methods for solving the AVEs (1)–(2) have been proposed. A parametric successive linearization algorithm for the AVE (1) that terminates at a point satisfying necessary optimality conditions is studied in [12].
The generalized Newton algorithm for the AVE (2) is investigated in [13], in which it was proved that this algorithm converges linearly from any starting point to the unique solution of the AVE (2) under the condition thatA−1 < 14. The generalized Newton algorithm with semismooth and smoothing Newton steps combined into the algorithm is considered in [24]. The smoothing-type algorithms for solving the AVEs (1)–(2) are studied in [1,8,9]. A branch and bound method for the AVP, which is an extension of the AVE, is studied in [23].
Among the aforementioned approaches, many of them focus on reformulating it as complementarity problem and then solve it accordingly. An alternative approach is to recast the AVE as a system of nonsmooth equations and then tackle with the nonsmooth equations by applying nonsmooth Newton algorithm [18] or smoothing Newton algorithm [19]. In this paper, we follow the latter pathway. More specifically, we rewrite it as a system of smooth equations and propose four new smoothing func- tions along with a smoothing-type algorithm to solve the system of equations. To see
this, motivated by the approach in [1,9], we define Hi : Rn+1→ Rn+1as
Hi(μ, x) =
μ
Ax+ Bi(μ, x) − b
forμ ∈ R and x ∈ Rn (3)
wherei : Rn+1→ Rnis given by
i(μ, x) :=
⎡
⎢⎢
⎢⎣
φi(μ, x1) φi(μ, x2) φi(μ, x... n)
⎤
⎥⎥
⎥⎦ for μ ∈ R and x ∈ Rn (4)
with four various smoothing functionsφi : R2→ R that will be introduced later. The role ofφilooks similar to the functionφpused in [9]. However, they are substantially different. More specifically, the functionφpemployed in [9] is strongly semismooth onR2, whereas eachφi proposed in this paper is continuously differentiable onR2. Now, we present the exact form for each functionφi, which is defined as below:
φ1(μ, t) = μ ln
1+ e−μt + ln
1+ eμt
(5)
φ2(μ, t) =
⎧⎪
⎪⎨
⎪⎪
⎩
t if t ≥ μ2, t2
μ +μ
4 if −μ2 < t < μ2,
−t if t ≤ −μ2.
(6)
φ3(μ, t) =
4μ2+ t2 (7)
φ4(μ, t) =
⎧⎨
⎩ t2
2μ if |t| ≤ μ,
|t| −μ2 if |t| > μ.
(8)
Some of the smoothing functions have appeared in other contexts for other optimization problems, but they are all novel ones for dealing with the AVE (1). The main idea in this paper is showing that the AVE (1) has a solution if and only if Hi(μ, x) = 0, φi
is continuously differentiable at any (μ, t) ∈ R++ × R, and limμ↓0φi(μ, x) =
|x|. Then, with these four new smoothing functions, we consider the smoothing-type algorithm studied in [7,25] to solve the AVE (1). In other words, we reformulate the AVE (1) as parameterized smooth equations and then employ a smoothing-type algorithm to solve it. In addition, we show that the algorithm is well-defined under the assumption that the minimal singular value of the matrix A is strictly greater than the maximal singular value of the matrix B. We also show that the proposed algorithm is globally and locally quadratically convergent no matter which smoothing function φiis used. Numerical implementations and comparisons based on these four different φi are reported as well. From the numerical results, we conclude thatφ2is the best choice of smoothing function when we apply the proposed smoothing-type algorithm.
More detailed reports will be seen in Sect.4.
Fig. 1 Graphs of|t| and all four φi(μ, t) with μ = 0.1
2 Smoothing Reformulation
In this section, we depict the graphs of φi for i = 1, 2, 3, 4 and investigate their properties. Then, we show the equivalent reformulation that Hi(μ, x) = 0 if and only if x solves the AVE (1), and talk about the condition to guarantee the unique solvability of the AVE (1). We begin with showing the pictures ofφifor i = 1, 2, 3, 4, see Fig.1.
From Fig.1, we see thatφ2is the one which best approximates the function|t| under the sense that it is closest to|t| among all φi for i = 1, 2, 3, 4. To see this, we adopt the max norm to measure the distance of two real-valued functions. In other words, for given two real-valued functions f and g, the distance between them is defined as
f − g∞= max
t∈R{ f (t) − g(t)} . Now, for any fixedμ > 0, we know that
|t|→∞lim φi(μ, t) − |t| =0, for i = 1, 2, 3.
This implies that
maxt∈Rφi(μ, t) − |t| = |φi(μ, 0)| , for i = 1, 2, 3.
Since,φ1(μ, 0) = (2 ln 2)μ ≈ 1.4μ, φ2(μ, 0) = μ4, andφ3(μ, 0) = 2μ, we obtain
φ1(μ, t) − |t|
∞= (2 ln 2)μ ≈ 1.4μ
φ2(μ, t) − |t|
∞= μ
φ3(μ, t) − |t| 4
∞= 2μ
On the other hand, we see that
tlim→∞φ4(μ, t) − |t| = μ
2 and φ4(μ, 0) = 0, which says
maxt∈Rφ4(μ, t) − |t| = μ 2. Hence, we obtain
φ4(μ, t) − |t|
∞= μ
2. From all the above, we conclude that
φ3(μ, t)−|t|
∞>φ1(μ, t)−|t|
∞>φ4(μ, t) − |t|
∞>φ2(μ, t)−|t|
∞. (9) This shows thatφ2is the function amongφi, i = 1, 2, 3, 4 which best approximates the function|t|. In fact, for fixed μ > 0, there has the local behavior that
φ3(μ, t) > φ1(μ, t) > φ2(μ, t) > |t| > φ4(μ, t). (10)
A natural question arises here, does the smoothing algorithm based onφ2 perform best among allφ1, φ2, φ3, φ4? This will be answered in Sect.4.
Proposition 2.1 Letφi : R2→ R for i = 1, 2, 3, 4 be defined as in (5), (6), (7) and (8), respectively. Then, we have
(a) φi is continuously differentiable at(μ, t) ∈ R++× R;
(b) limμ↓0φi(μ, t) = |t| for any t ∈ R.
Proof (a) In order to prove the continuous differentiability ofφi, we need to write out the expressions of ∂φi∂t(μ,t) and∂φi∂μ(μ,t); and then show the continuity of ∂φi∂t(μ,t) and
∂φi(μ,t)
∂μ .
(i) For i= 1, we compute that
∂φ1(μ, t)
∂t = 1
1+ e−μt − 1 1+ eμt ,
∂φ1(μ, t)
∂μ =
ln
1+ e−μt + ln
1+ eμt + t
μ
−1
1+ e−μt + 1 1+ eμt
.
Then, it is clear to see that ∂φ1∂t(μ,t) and ∂φ1∂μ(μ,t) are continuous. Hence, φ1 is continuously differentiable.
(ii) For i= 2, we compute that
∂φ2(μ, t)
∂t =
⎧⎨
⎩
1 if t≥ μ2,
2tμ if −μ2 < t < μ2,
−1 if t ≤ −μ2.
∂φ2(μ, t)
∂μ =
⎧⎪
⎨
⎪⎩
0 if t≥ μ2,
−
μt
2
+14 if −μ2 < t < μ2, 0 if t≤ −μ2.
Then, it can be verified that∂φ2∂t(μ,t) and∂φ2∂μ(μ,t) are continuous because
lim
t→μ2
∂φ2(μ, t)
∂t = lim
t→μ2
2t μ = 1, lim
t→−μ2
∂φ2(μ, t)
∂t = lim
t→−μ2
2t μ = −1.
and
lim
t→μ2
∂φ2(μ, t)
∂μ = lim
t→μ2
−
t μ
2
+1 4
= 0,
lim
t→−μ2
∂φ2(μ, t)
∂μ = lim
t→−μ2
−
t μ
2
+1 4
= 0.
Hence,φ2is continuously differentiable.
(iii) For i= 3, we compute that
∂φ3(μ, t)
∂t = t
4μ2+ t2,
∂φ3(μ, t)
∂μ = 4μ
4μ2+ t2.
Again it is clear to see that ∂φ3∂t(μ,t) and ∂φ3∂μ(μ,t) are continuous. Hence, φ3 is continuously differentiable.
(iv) For i= 4, we compute that
∂φ4(μ, t)
∂t =
⎧⎨
⎩
1 if t > μ,
μt if −μ ≤ t ≤ μ,
−1 if t < −μ.
∂φ4(μ, t)
∂μ =
⎧⎪
⎨
⎪⎩
−12 if t > μ,
−12
μt
2
if −μ ≤ t ≤ μ,
−12 if t < −μ.
Then, we conclude that∂φ4∂t(μ,t) and∂φ4∂μ(μ,t) are continuous by checking
tlim→μ
∂φ4(μ, t)
∂t = lim
t→μ
t μ = 1,
t→−μlim
∂φ4(μ, t)
∂t = lim
t→−μ
t μ = −1.
and
tlim→μ
∂φ4(μ, t)
∂μ = lim
t→μ
−1 2 ×
t μ
2
= −1 2,
t→−μlim
∂φ4(μ, t)
∂μ = lim
t→−μ
−1 2 ×
t μ
2
= −1 2.
Hence,φ4is continuously differentiable.
From all the above, we prove thatφiis continuously differentiable at(μ, t) ∈ R++×R.
(b) For i = 1, 2, 3, 4, we always have the following:
μ→0lim
∂φi(μ, t)
∂t =
1 if t> 0,
−1 if t < 0,
which verifies part (b).
For subsequent needs in convergence analysis and numerical implementations, we summarize the gradient of eachφi as below.
∇φ1(μ, t) =
⎡
⎢⎣
ln(1 + e−μt ) + ln(1 + eμt) +μt
−1
1+e− tμ + 1
1+eμt
1
1+e− tμ − 1
1+eμt
⎤
⎥⎦ .
∇φ2(μ, t) =
ξ1
ξ2
, where ξ1=
⎧⎪
⎨
⎪⎩
0 if t ≥ μ2,
−
μt
2
+14 if −μ2 < t < μ2, 0 if t ≤ −μ2.
ξ2=
⎧⎨
⎩
1 if t ≥ μ2,
2tμ if −μ2 < t < μ2,
−1 if t ≤ −μ2.
∇φ3(μ, t) =
⎡
⎣
4μ
√4μ2+t2
√ t 4μ2+t2
⎤
⎦ .
∇φ4(μ, t) =
v1
v2
, where v1=
⎧⎪
⎪⎨
⎪⎪
⎩
−12 if t> μ,
−12
μt
2
if −μ ≤ t ≤ μ,
−12 if t< −μ.
v2=
⎧⎨
⎩
1 if t > μ,
μt if −μ ≤ t ≤ μ,
−1 if t < −μ.
In fact, Proposition2.1can be also depicted by geometric views. In particular, from Figs.2,3,4and5, we see that whenμ ↓ 0, φi is getting closer to|t|, which verifies Proposition2.1(b).
Now, in light of Proposition2.1, we obtain the equivalent reformulation Hi(μ, x)
= 0 for the AVE (1).
Fig. 2 Graphs ofφ1(μ, t) with μ = 0.01, 0.1, 0.3, 0.5
Fig. 3 Graphs ofφ2(μ, t) with μ = 0.01, 0.1, 0.3, 0.5
Fig. 4 Graphs ofφ3(μ, t) with μ = 0.01, 0.1, 0.3, 0.5
Fig. 5 Graphs ofφ4(μ, t) with μ = 0.01, 0.1, 0.3, 0.5
Proposition 2.2 Leti(μ, x) for i = 1, 2, 3, 4 be defined as in (4). Then, we have (a) Hi(μ, x) = 0 if and only if x solves the AVE (1);
(b) Hiis continuously differentiable onRn+1\ {0} with the Jacobian matrix given by
∇ Hi(μ, x) :=
1 0
B∇1i(μ, x) A + B ∇2i(μ, x)
(11)
where
∇1i(μ, x) :=
⎡
⎢⎢
⎢⎢
⎣
∂φi(μ,x1)
∂φi∂μ(μ,x2)
∂μ...
∂φi(μ,xn)
∂μ
⎤
⎥⎥
⎥⎥
⎦,
∇2i(μ, x) :=
⎡
⎢⎢
⎢⎢
⎣
∂φi(μ,x1)
∂x1 0 · · · 0
0 ∂φi∂x(μ,x2)
2 · · · 0 ... ... ... ...
0 · · · 0 ∂φi∂x(μ,xn)
n
⎤
⎥⎥
⎥⎥
⎦.
Proof This result follows from Proposition2.1immediately and the computation of
the Jacobian matrix is straightforward.
For completeness, we also talk about the unique solvability of the AVE (1), which is presumed in our numerical implementations. The following assumption and propo- sition are both employed from [9]. Assumption2.3will be also used to guarantee that
∇ Hi(μ, x) is invertible at any (μ, x) ∈ R++× Rn, see Proposition3.2in Sect.3.
Assumption 2.3 The minimal singular value of the matrix A is strictly greater than the maximal singular value of the matrix B.
Proposition 2.4 ([9, Proposition 2.3]) The AVE (1) is uniquely solvable for any b∈ Rn if Assumption2.3is satisfied.
3 A smoothing-type algorithm
From Proposition 2.2, we know that the AVE (1) is equivalent to Hi(μ, x) = 0.
Accordingly, in this section, we consider the smoothing-type algorithm as in [1,9] to solve Hi(μ, x) = 0. In fact, this type of algorithm has been also proposed for solving other kinds of problems, see [2,7,25] and references therein.
Algorithm 3.1 (A smoothing-type algorithm)
Step 0 Choose δ, σ ∈ (0, 1), μ0 > 0, x0 ∈ Rn. Set z0 := (μ, x0). Denote e0 := (1, 0) ∈ R × Rn. Chooseβ > 1 such that
min
1, Hi(z0) 2
≤ βμ0. Set k:= 0.
Step 1 IfHi(zk) = 0, stop.
Step 2 Setτk := min
1, Hi(zk)
, and computezk := (μk, xk) ∈ R × Rn by using
∇ Hi(zk)zk = −Hi(zk) + (1/β)τk2e0, (12) where∇ Hi(·) is defined as in (11).
Step 3 Letαkbe the maximum of the values 1, δ, δ2, · · · such that
Hi(zk+ αkzk) ≤ [1 − σ(1 − 1/β)αk]Hi(zk) (13)
Step 4 Set zk+1:= zk+ αkzk and k:= k + 1. Back to Step 1.
Following the same arguments as in [6,7], the line search (13) in the above scheme is well-defined. In other words, the Algorithm3.1is well-defined and possesses some nice properties.
Proposition 3.2 (a) Suppose that Assumption2.3holds. Then, the Algorithm3.1is well-defined.
(b) Let the sequence zk
be generated by Algorithm3.1. Then, (i) both
Hi(zk)
and{τk} are monotonically decreasing;
(ii) τk2≤ βμkholds for all k;
(iii) the sequence{μk} is monotonically decreasing, and μk> 0 for all k.
Proof Please refer to [7, Remark 2.1] or [9, Proposition 3.1].
The key point in the above scheme is the solvability of Newton equations (12) in Step 2. The following result is regarding this issue. Since theifunction plays almost the same role as the functionpused in [9], the below Proposition3.3can be obtained by mimicking the same arguments as in [9, Theorem 3.2]. We omit its proof and only state it.
Proposition 3.3 Let Hi and∇ Hi be given as in (3) and (11), respectively. Suppose that Assumption2.3holds. Then,∇ Hi(μ, x) is invertible at any (μ, x) ∈ R++× Rn. Next, we discuss the global and local convergence. Again, although the function
i here is continuously differentiable and the functionpused in [9] is only semi- smooth, their roles in the proof are almost the same. Consequently, the arguments for convergence analysis are almost the same. Hence, we also omit the detailed proof and only present the convergence result.
Proposition 3.4 Suppose that Assumption2.3holds and that the sequence zk
is generated by Algorithm3.1. Then,
(a) zk
is bounded;
(b) any accumulation point of zk
is a solution of the AVE (1).
(c) The whole sequence zk
convergence to z∗withzk+1− zk = o
zk− z∗ andμk+1= μ2k.
4 Numerical implementations
In this section, we report the numerical results of Algorithm3.1for solving the AVE (1) and (2). All numerical experiments are carried out in Mathematica 10.0 running on a PC with Intel i5 of 3.00 GHz CPU processor, 4.00 GB Memory and 32-bit Windows 7 operating system.
In our numerical experiments, the stoping criteria for Algorithm3.1isHi(zk) ≤ 1.0e−6. We also stop programs when the total iteration is more than 100. Throughout the computational experiments, the following parameters are used:
δ = 0.5, σ = 0.0001, μ0= 0.1, β = max
1, 1.01 ∗ τ02/μ .
4.1 Experiments on the AVE Ax− |x| = b
In this subsection we consider the simplified form of AVE (2). Consider the ordinary differential equation [4, Example 4.2]:
d2x
dt2 − |x| = (1 − t2), x(0) = −1, x(1) = 0, t ∈ [0, 1]. (14) As explained in [4, Example 4.2], after descretization (by using finite difference method), the above ODE can be recast an AVE in form of
Ax− |x| = b, (15)
where the matrix A is given by
ai, j =
⎧⎪
⎨
⎪⎩
−242, i= j, 121, |i − j| = 1, 0, otherwise.
(16)
We implement the above problems by usingφi, i = 1, 2, 3, 4 and n = 2, 5, 10, 20, . . . , 100, respectively. Every starting point x is randomly generated 10 times from a uni- form distribution on x∈ [−2, 2]. The results are put together in Table1, where Dim denotes the size of problem, N_φi denotes the average number of iterations, T_φi
denotes the average value of the CPU time in seconds, Ar_φi denotes the average value ofH(zk) when Algorithm3.1stop.
From Table1, in terms of the average number of iterations, the efficiency ofφ2(μ, t) is best, followed byφ4(μ, t), φ3(μ, t) and φ1(μ, t). This is especially true for the prob- lem of high dimension ordinary differential equation (14). In terms of time efficiency, φ1(μ, t) is still better than other functions too. In other words, for the AVE (2) arising from the ODE (15), we have
φ2(μ, t) > φ4(μ, t) > φ3(μ, t) > φ1(μ, t)
where “>” means “better performance”.
To compare the performance of smoothing function φi(μ, t), i = 1, 2, 3, 4, we adopt the performance profile which is introduced in [3] as a means. In other words, we regard Algorithm3.1corresponding to a smoothing functionφi(μ, t), i = 1, 2, 3, 4 as a solver, and assume that there are ns solvers and nptest problems from the test setP which is generated randomly. We are interested in using the iteration number as performance measure for Algorithm3.1with differentφi(μ, t). For each problem p and solver s, let
fp,s = iteration number required to solve problem p by solver s.
Table1Thenumericalresultsofordinarydifferentialequation(14) DimN_φ1T_φ1Ar_φ1N_φ2T_φ2Ar_φ2N_φ3T_φ3Ar_φ3N_φ4T_φ4Ar_φ4 25.10.09673.30E−073.90.00156.92E−085.10.00165.93E−0840.00625.99E−08 55.90.36972.23E−074.10.00317.47E−085.60.00622.21E−084.20.00166.54E−08 106.40.48512.98E−074.30.00942.10E−075.90.00311.05E−074.50.00314.67E−08 205.20.42902.41E−074.90.00781.10E−086.30.00782.13E−0950.00942.46E−09 408.84.41174.66E−076.10.52105.28E−087.30.01726.59E−086.30.01561.88E−07 609.12.42892.31E−076.80.02814.49E−0890.03121.20E−087.70.03121.31E−07 809.82.05143.61E−077.40.03743.21E−109.30.04523.21E−089.20.05933.15E−08 1009.88.23064.44E−077.80.05778.78E−08100.06712.26E−079.50.08272.83E−08
We employ the performance ratio
rp,s := fp,s
min
fp,s : s ∈ S ,
whereS is the four solvers set. We assume that a parameter rp,s≤ rM for all p, s are chosen, and rp,s = rM if and only if solver s does not solve problem p. In order to obtain an overall assessment for each solver, we define
ρs(τ) := 1 np
size
p∈ P : rp,s ≤ τ ,
which is called the performance profile of the number of iteration for solver s. Then, ρs(τ) is the probability for solver s ∈ S that a performance ratio fp,sis within a factor τ ∈ R of the best possible ratio.
We then need to test the four functions for ODE (14) at random starting points. In particular, starting points for each dimension are randomly chosen 20 times from a uniform distribution on x ∈ [−2, 2]. In order to obtain an overall assessment for the four functions, we are interested in using the number of iterations as a performance measure for Algorithm 3.1withφ1(μ, t), φ2(μ, t), φ3(μ, t), and φ4(μ, t), respec- tively. The performance plot based on iteration number is presented in Fig.6. From this figure, we see thatφ2(μ, t) working with Algorithm3.1has the best numerical performance, followed byφ4(μ, t). In other words, in view of “iteration numbers”, there has
φ2(μ, t) > φ4(μ, t) > φ3(μ, t) > φ1(μ, t) where “>” means “better performance”.
We are also interested in using the computing time as performance measure for Algorithm3.1with differentφi(μ, t), i = 1, 2, 3, 4. The performance plot based on
Fig. 6 Performance profile of iteration numbers of Algorithm3.1for the ODE (14)
Fig. 7 Performance profile of computing time of Algorithm3.1for the ODE (14)
computing time is presented in Fig.7. From this figure, we can also see the function φ2(μ, t) has best performance, then followed by φ3(μ, t). Note that the time efficiency ofφ1(μ, t) is very bad. In other words, in view of “computing tim”, there has
φ2(μ, t) > φ3(μ, t) > φ4(μ, t) > φ1(μ, t) where “>” means “better performance”.
In summary, for the special AVE (2) arising from the ODE (14), no matter the number of iterations or the computing time is taken into account, the functionφ2(μ, t) is the best choice for the Algorithm3.1.
4.2 Experiments on the general AVE Ax+ B|x| = b
In this subsection we consider the general AVE (1): Ax+ B|x| = b. Here matrix A (or B) is equal to a normal distribution random matrix minus another one so that we can randomly generate the testing problems.
In order to ensure that Assumption2.3holds, we further modify the matrix A in light of the below conditions.
• If min{wii : i = 1, . . . , n} = 0 with {u, w, v} = SingularValueDecomposition [A], then we set A = u(w + 0.01 × IdentityMatrix[n])v.
• Set A = λmaxλ(BTB)+0.01
min(ATA) A.
Then, it is clear to verify that Assumption2.3is satisfied for such A. Moreover, we set p =2RandomVariate [NormalDistribution[ ],{n, 1}] and b = Ap + B|p| so that the testing problems are solvable.
We implement the above problems forφi, i = 1, 2, 3, 4 and n = 2, 5, 10, 20, . . . , 100, respectively. Every case is randomly generated 10 times for testing. The numerical results are listed in Table2. From Table2, in terms of the number of iterations and computation time, the efficiency ofφ2(μ, t) is best, followed by φ4(μ, t). The iteration
Table2Thenumericalresultsofexperiments DimN_φ1T_φ1Ar_φ1N_φ2T_φ2Ar_φ2N_φ3T_φ3Ar_φ3N_φ4T_φ4Ar_φ4 26.20.45965.00E−73.60.00318.56E−87.10.00161.79E−73.908.04E−8 57.40.22466.05E−74.10.00318.39E−89.60.00944.73E−74.30.00167.53E−8 1010.21.07332.23E−74.30.00628.26E−817.20.01874.79E−74.70.00317.53E−8 2019.83.78305.00E−74.80.00629.95E−826.30.04991.86E−75.90.00941.06E−7 3028.75.05754.46E−75.60.01401.00E−743.20.12955.22E−89.30.02651.82E−7 4038.63.09356.52E−77.10.02345.60E−854.10.21371.65E−711.90.03749.14E−8 5042.71.90165.37E-75.30.02187.73E−861.50.31201.93E−810.40.04375.88E−8 6052.12.52725.61E−76.60.03595.90E−878.70.49761.05E−813.90.07181.15E−7 7060.23.70506.10E−79.90.06241.12E−794.40.73321.80E−718.70.12641.26E−7 8058.04.12464.31E−78.90.06406.03E−898.50.88453.88E−817.50.14205.35E−8 9078.211.1706.28E−710.00.09052.23E−7114.31.27451.46E−720.90.20281.46E−7 10072.212.2114.77E−77.50.07091.62E−7110.81.64771.31E−716.90.18811.34E−7
Fig. 8 Performance profile of iteration numbers of Algorithm3.1for general AVE
number ofφ1(μ, t) is less than φ3(μ, t), but the computing time of φ1(μ, t) is more thanφ3(μ, t).
Figure8shows the performance profile of iteration number in Algorithm3.1in the range ofτ ∈ [1, 15] for four solvers on 100 test problem which are generated randomly.
The four solvers correspond to Algorithm3.1withφ1(μ, t), φ2(μ, t), φ3(μ, t), and φ4(μ, t), respectively. From this figure, we see that φ2(μ, t) working with Algorithm 3.1has the best numerical performance, followed byφ4(μ, t). In summary, from the viewpoint of “iteration numbers”, we conclude that
φ2(μ, t) > φ4(μ, t) > φ1(μ, t) > φ3(μ, t) where “>” means “better performance”.
Finally, we are also interested in using the computing time as performance measure for Algorithm3.1with differentφi(μ, t), i = 1, 2, 3, 4. The performance plot based on computing time is presented in Fig.9. From this figure, we can also see the function
Fig. 9 Performance profile of computing time of Algorithm3.1for general AVE
φ2(μ, t) has best performance, then followed by φ4(μ, t). Note that the time efficiency ofφ1(μ, t) is very bad. Again, from the viewpoint of “computing time”, we conclude that
φ2(μ, t) > φ4(μ, t) > φ3(μ, t) > φ1(μ, t)
where “>” means “better performance”.
5 Conclusion
In this paper, we recast the AVE (1) as a system of smooth equations. Accordingly, we have proposed four smoothing functions along with a smoothing-type algorithm studied in [1,9] to solve it. As mentioned in Sect.2, there holds the local behavior shown as in (10):
φ3(μ, t) > φ1(μ, t) > φ2(μ, t) > |t| > φ4(μ, t).
andφ2(μ, t) is the one which best approximates the function |t| shown as in (9), i.e.,
φ3(μ, t) − |t|
∞>φ1(μ, t) − |t|
∞>φ4(μ, t) − |t|
∞>φ2(μ, t) − |t|
∞. Surprisingly,φ2(μ, t) is also the best choice of smoothing function no matter when the iteration number or the computing time is taken into account. For the “iteration”
aspect, the order of numerical performance from good to bad is
φ2(μ, t) > φ4(μ, t) > φ1(μ, t) > φ3(μ, t), for th AVE (1).
φ2(μ, t) > φ4(μ, t) > φ3(μ, t) > φ1(μ, t), for th AVE (2).
whereas for the “time” aspect, the order of numerical performance from good to bad is
φ2(μ, t) > φ4(μ, t) > φ3(μ, t) > φ1(μ, t), for th AVE (1).
φ2(μ, t) > φ3(μ, t) > φ4(μ, t) > φ1(μ, t), for th AVE (2).
In other words,φ2(μ, t) is the best choice of smoothing function to work with the proposed smoothing-type algorithm, meanwhile it also best approximate the function
|t|. This is a very interesting discovery which may be helpful in other contexts. One of future directions is to check whether such phenomenon occurs in other types of algorithms.
Acknowledgements The author B. Saheya’s work is supported by Natural Science Foundation of Inner Mongolia (Award No. 2014MS0119). The author J.-S. Chen’s work is supported by Ministry of Science and Technology, Taiwan.
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