Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 145323, 7 pages http://dx.doi.org/10.1155/2015/145323
Research Article Multivariate Spectral Gradient Algorithm for Nonsmooth Convex Optimization Problems Yaping Hu School of Science, East China University of Science and Technology, Shanghai 200237, China Correspondence should be addressed to Yaping Hu;
[email protected] Received 20 April 2015; Revised 4 July 2015; Accepted 5 July 2015 Academic Editor: Dapeng P. Du Copyright Β© 2015 Yaping Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose an extended multivariate spectral gradient algorithm to solve the nonsmooth convex optimization problem. First, by using Moreau-Yosida regularization, we convert the original objective function to a continuously differentiable function; then we use approximate function and gradient values of the Moreau-Yosida regularization to substitute the corresponding exact values in the algorithm. The global convergence is proved under suitable assumptions. Numerical experiments are presented to show the effectiveness of this algorithm.
Proposition 1 (see Chapter XV, Theorem 4.1.4, [1]). The Moreau-Yosida regularization function πΉ is convex, finitevalued, and differentiable everywhere with gradient
1. Introduction Consider the unconstrained minimization problem min π (π₯) ,
π₯βRπ
(1)
where π : Rπ β R is a nonsmooth convex function. The Moreau-Yosida regularization [1] of π at π₯ β Rπ associated with π§ β Rπ is defined by 1 πΉ (π₯) = minπ {π (π§) + βπ§ β π₯β2 } , π§βR 2π
min πΉ (π₯)
1 (π₯ β π (π₯)) , π
(2)
(3)
and the original problem (1) are equivalent in the sense that the two corresponding solution sets coincidentally are the same. The following proposition shows some properties of the Moreau-Yosida regularization function πΉ(π₯).
(4)
where π (π₯) = arg minπ {π (π§) + π§βR
where ββ
β is the Euclidean norm and π is a positive parameter. The function minimized on the right-hand side is strongly convex and differentiable, so it has a unique minimizer for every π§ β Rπ . Under some reasonable conditions, the gradient function of πΉ(π₯) can be proved to be semismooth [2, 3], though generally πΉ(π₯) is not twice differentiable. It is widely known that the problem π₯βRπ
π (π₯) β‘ βπΉ (π₯) =
1 βπ§ β π₯β2 } 2π
(5)
is the unique minimizer in (2). Moreover, for all π₯, π¦ β Rπ , one has σ΅© σ΅© 1σ΅© σ΅©σ΅© σ΅©σ΅©π (π₯) β π (π¦)σ΅©σ΅©σ΅© β€ σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© . π
(6)
This proposition shows that the gradient function π : Rπ β Rπ is Lipschitz continuous with modulus 1/π. In this case, the gradient function π is differentiable almost everywhere by the Rademacher theorem; then the Bsubdifferential [4] of π at π₯ β Rπ is defined by ππ΅ π (π₯) = {π β RπΓπ : π = lim βπ (π₯π ) , π₯π β π·π } , π₯π β π₯
(7)
where π·π = {π₯ β Rπ : π is differentiable at π₯}, and the next property of BD-regularity holds [4β6].
2
Mathematical Problems in Engineering
Proposition 2. If π is BD-regular at π₯, then
Proposition 4. Let π be arbitrary positive number and let ππ (π₯, π) be a vector satisfying (9). Then, one gets
(i) all matrices π β ππ΅ π(π₯) are nonsingular;
(ii) there exists a neighborhood N of π₯ β Rπ , π
1 > 0, and π
2 > 0; for all π¦ β N, one has π
σ΅©σ΅© β1 σ΅©σ΅© σ΅©σ΅©π σ΅©σ΅© β€ π
2 , σ΅© σ΅©
2
π ππ β₯ π
1 βπβ ,
βπ β Rπ , π β ππ΅ π (π¦) .
(8)
Instead of the corresponding exact values, we often use the approximate value of function πΉ(π₯) and gradient π(π₯) in the practical computation, because π(π₯) is difficult and sometimes impossible to be solved precisely. Suppose that, for any π > 0 and for each π₯ β Rπ , there exists an approximate vector ππ (π₯, π) β Rπ of the unique minimizer π(π₯) in (2) such that π (ππ (π₯, π)) +
1 σ΅©σ΅© π σ΅©2 σ΅©π (π₯, π) β π₯σ΅©σ΅©σ΅© β€ πΉ (π₯) + π. 2π σ΅©
(9)
The implementable algorithms to find such approximate vector ππ (π₯, π) β Rπ can be found, for example, in [7, 8]. The existence theorem of the approximate vector ππ (π₯, π) is presented as follows. Proposition 3 (see Lemma 2.1 in [7]). Let {π₯π } be generated according to the formula π₯π+1 = π₯π β πΌπ ππ ,
πππ π = 1, 2, . . . ,
(10)
where πΌπ > 0 is a stepsize and ππ is an approximate subgradient at π₯π ; that is, ππ β πππ π (π₯π ) = {π | π (π) β β¨π, π₯π β© β€ π (π₯π ) + ππ } , πππ π = 1, 2, . . . .
πππ π = 1, 2, . . . ,
(12)
(13)
(ii) Conversely, if (11) holds with ππ given by (13), then (12) holds: π₯π+1 = ππ (π₯π , ππ ). π
We use the approximate vector π (π₯, π) to define approximation function and gradient values of the Moreau-Yosida regularization, respectively, by πΉπ (π₯, π) = π (ππ (π₯, π)) + ππ (π₯, π) =
1 σ΅©σ΅© π σ΅©2 σ΅©π (π₯, π) β π₯σ΅©σ΅©σ΅© , 2π σ΅©
(14)
π
(π₯ β π (π₯, π)) . π
(17)
σ΅© σ΅©σ΅© π (18) σ΅©σ΅©π (π₯, π) β π (π₯)σ΅©σ΅©σ΅© β€ β2ππ. Algorithms which combine the proximal techniques with Moreau-Yosida regularization for solving the nonsmooth problem (1) have been proved to be effective [7, 9, 10], and also some trust region algorithms for solving (1) have been proposed in [5, 11, 12], and so forth. Recently, Yuan et al. [13, 14] and Li [15] have extended the spectral gradient method and conjugate gradient-type method to solve (1), respectively. Multivariate spectral gradient (MSG) method was first proposed by Han et al. [16] for optimization problems. This method has a nice property that it converges quadratically for objective function with positive definite diagonal Hessian matrix [16]. Further studies on such method for nonlinear equations and bound constrained optimization can be found, for instance, in [17, 18]. By using nonmonotone technique, some effective spectral gradient methods are presented in [13, 16, 17, 19]. In this paper, we extend the multivariate spectral gradient method by combining with a nonmonotone line search technique as well as the Moreau-Yosida regulation function to solve the nonsmooth problem (1) and do some numerical experiments to test its efficiency. The rest of this paper is organized as follows. In Section 2, we propose multivariate spectral gradient algorithm to solve (1). In Section 3, we prove the global convergence of the proposed algorithm; then some numerical results are presented in Section 4. Finally, we have a conclusion section.
2. Algorithm
then (11) holds with σ΅© σ΅©2 ππ = π (π₯π ) β π (π₯π+1 ) β πΌπ σ΅©σ΅©σ΅©ππ σ΅©σ΅©σ΅© β₯ 0.
2π σ΅© σ΅©σ΅© π σ΅©σ΅©π (π₯, π) β π (π₯)σ΅©σ΅©σ΅© β€ β , π
(16)
(11)
(i) If ππ satisfies ππ β ππ (π₯π+1 ) ,
πΉ (π₯) β€ πΉπ (π₯, π) β€ πΉ (π₯) + π,
(15)
The following proposition is crucial in the convergence analysis. The proof of this proposition can be found in [2].
In this section, we present the multivariate spectral gradient algorithm to solve the nonsmooth convex unconstrained optimization problem (1). Our approach is using the tool of the Moreau-Yosida regularization to smoothen the nonsmooth function and then make use of the approximate values of function πΉ and gradient π in multivariate spectral gradient algorithm. We first recall the multivariate spectral gradient algorithm [16] for smooth optimization problem: min {π (π₯) | π₯ β Rπ } ,
(19)
π
where π : R β R is continuously differentiable and its gradient is denoted by π. Let π₯π be the current iteration; multivariate spectral gradient algorithm is defined by π₯π+1 = π₯π β diag {
1 1 1 , , . . . , π } ππ , ππ π1π π2π
(20)
where ππ is the gradient vector of π at π₯π and diag{π1π , π2π , . . . , πππ } is solved by minimizing σ΅©σ΅©σ΅©diag {π1 , π2 , . . . , ππ } π β π’ σ΅©σ΅©σ΅© (21) σ΅©σ΅© πβ1 πβ1 σ΅© σ΅©2
Mathematical Problems in Engineering
3
with respect to {ππ }ππ=1 , where π πβ1 = π₯π β π₯πβ1 , π’πβ1 = ππ β ππβ1 . Denote the πth element of π π and π¦π by π ππ and π¦ππ , respectively. We present the following multivariate spectral gradient (MSG) algorithm. π
Algorithm 5. Set π₯0 β R , π β (0, 1), π½ > 0, π > 0, πΎ β₯ 0, πΏ > 0, π β [0, 1], π β (0, 1), πΈ0 = 1, and π0 β (0, 1]; {ππ } is a strictly decreasing sequence with limπ β 0 ππ = 0, π := 0. Step 1. Set π0 = π0 . Calculate πΉπ (π₯0 , π0 ) by (14) as well as ππ (π₯0 , π0 ) by (15). Let π½0 = πΉπ (π₯0 , π0 ), π0 = βππ (π₯0 , π0 ). Step 2. Stop if βππ (π₯π , ππ )β = 0. Otherwise, go to Step 3. Step 3. Choose ππ+1 satisfying 0 < ππ βππ (π₯π , ππ )β2 }; find πΌπ which satisfies
ππ+1
β€ π
πΉπ (π₯π + πΌπ ππ , ππ+1 ) β π½π β€ ππΌπ ππ (π₯π , ππ ) ππ ,
min{ππ ,
role in manipulating the degree of nonmonotonicity in the nonmonotone line search technique, with π = 0 yielding a strictly monotone scheme and with π = 1 yielding π½π = πΆπ , where πΆπ =
1 π π βπΉ (π₯π , ππ ) π + 1 π=0
(25)
is the average function value. (iii) From Step 6, we can obtain that 1 σ΅© σ΅© σ΅© σ΅© min {π, } σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© β€ σ΅©σ΅©σ΅©ππ σ΅©σ΅©σ΅© πΏ 1 1 σ΅© σ΅© β€ max { , } σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© ; π πΏ
(26)
then there is a positive constant π such that, for all π, (22)
where πΌπ = π½2βππ and ππ is the smallest nonnegative integer such that (22) holds.
π σ΅© σ΅©2 ππ (π₯π , ππ ) ππ β€ β π σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© ,
(27)
which shows that the proposed multivariate spectral gradient algorithm possesses the sufficient descent property.
Step 4. Let π₯π+1 = π₯π + πΌπ ππ . Stop if βππ (π₯π+1 , ππ+1 )β = 0.
3. Global Convergence
Step 5. Update π½π+1 by the following formula: πΈπ+1 = ππΈπ + 1, π½π+1 =
ππΈπ π½π + πΉπ (π₯π + πΌπ ππ , ππ+1 ) . πΈπ+1
(23)
Step 6. Compute the search direction ππ+1 by the following: (a) If π¦ππ /π ππ > 0, then set πππ+1 = π¦ππ /π ππ ; otherwise set πππ+1 = π ππ π¦π /π ππ π π for π = 1, 2, . . . , π, where π¦π = ππ (π₯π+1 , ππ+1 ) β ππ (π₯π , ππ ) + πΎπ π , π π = π₯π+1 β π₯π . (b) If πππ+1 β€ π or πππ+1 β₯ 1/π, then set πππ+1 = πΏ for π = 1, 2, . . . , π. Let ππ+1 = βdiag{1/π1π+1 , 1/π2π+1 , . . . , 1/πππ+1 }ππ (π₯π+1 , ππ+1 ). Step 7. Set π := π + 1; go back to Step 2. Remarks. (i) The definition of ππ+1 = π(βππ (π₯π , ππ )β2 ) in Algorithm 5, together with (15) and Proposition 3, deduces that σ΅©2 σ΅© σ΅©2 σ΅© ππ+1 = π (σ΅©σ΅©σ΅©π₯π β ππ (π₯π , ππ )σ΅©σ΅©σ΅© ) = π (σ΅©σ΅©σ΅©π₯π β π₯π+1 σ΅©σ΅©σ΅© ) σ΅© σ΅©2 = π (πΌπ2 σ΅©σ΅©σ΅©ππ σ΅©σ΅©σ΅© ) ;
(24)
then, with the decreasing property of ππ+1 , the assumed condition ππ = π(πΌπ2 βππ β2 ) in Lemma 7 holds. (ii) From the nonmonotone line search technique (22), we can see that π½π+1 is a convex combination of the function value πΉπ (π₯π+1 , ππ+1 ) and π½π . Also π½π is a convex combination of the function values πΉπ (π₯π , ππ ), . . ., πΉπ (π₯1 , π1 ), πΉπ (π₯0 , π0 ) as π½0 = πΉπ (π₯0 , π0 ). π is a positive value that plays an important
In this section, we provide a global convergence analysis for the multivariate spectral gradient algorithm. To begin with, we make the following assumptions which have been given in [5, 12β14]. Assumption A. (i) πΉ is bounded from below. (ii) The sequence {ππ }, ππ β ππ΅ π(π₯π ), is bounded; that is, there exists a constant π > 0 such that, for all π, σ΅©σ΅© σ΅©σ΅© σ΅©σ΅©ππ σ΅©σ΅© β€ π.
(28)
The following two lemmas play crucial roles in establishing the convergence theorem for the proposed algorithm. By using (26) and (27) and Assumption A, similar to Lemma 1.1 in [20], we can get the next lemma which shows that Algorithm 5 is well defined. The proof ideas of this lemma and Lemma 1.1 in [20] are similar, hence omitted. Lemma 6. Let {πΉπ (π₯π , ππ )} be the sequence generated by Algorithm 5. Suppose that Assumption A holds and πΆπ is defined by (25). Then one has πΉπ (π₯π , ππ ) β€ π½π β€ πΆπ for all π. Also, there exists a stepsize πΌπ satisfying the nonmonotone line search condition. Lemma 7. Let {(π₯π , ππ )} be the sequence generated by Algorithm 5. Suppose that Assumption A and ππ = π(πΌπ2 βππ β2 ) hold. Then, for all π, one has πΌπ β₯ π0 ,
(29)
where π0 > 0 is a constant. Proof (Proof by Contradiction). Let πΌπ satisfy the nonmonotone Armijo-type line search (22). Assume on the contrary
4
Mathematical Problems in Engineering π
ππΌπσΈ ππ (π₯π , ππ ) ππ < πΉπ (π₯π + πΌπσΈ ππ , ππ+1 )
that lim inf π β β πΌπ = 0 does hold; then there exists a subsequence {πΌπ }πΎσΈ such that πΌπ β 0 as π β β. From the nonmonotone line search rule (22), πΌπσΈ = πΌπ /2 satisfies π
πΉπ (π₯π + πΌπσΈ ππ , ππ+1 ) β π½π > ππΌπσΈ ππ (π₯π , ππ ) ππ ;
β πΉπ (π₯π , ππ ) β€ πΉ (π₯π + πΌπσΈ ππ ) β πΉ (π₯π ) + ππ+1
(30)
= πΌπσΈ πππ π (π₯π )
together with πΉπ (π₯π , ππ ) β€ π½π β€ πΆπ in Lemma 6, we have π
πΉ
(π₯π + πΌπσΈ ππ , ππ+1 ) β πΉπ β₯πΉ
π
+
(π₯π , ππ )
(π₯π + πΌπσΈ ππ , ππ+1 ) β π½π
>
ππΌπσΈ ππ
π
(π₯π , ππ ) ππ .
π
(32)
πππ π (π’π ) ππ + ππ+1
β€ πΌπσΈ πππ π (π₯π ) +
(31)
By (28) and (31) and Proposition 4 and using Taylorβs formula, there is
1 2
2 (πΌπσΈ )
π σΈ 2 σ΅©σ΅© σ΅©σ΅©2 (πΌπ ) σ΅©σ΅©ππ σ΅©σ΅© 2
+ ππ+1 , where π’π β (π₯π , π₯π+1 ). From (32) and Proposition 4, we have π
(ππ (π₯π , ππ ) β π (π₯π )) ππ β (1 β π) ππ (π₯π , ππ ) ππ β ππ+1 /πΌπσΈ 2 πΌπ ] = πΌπσΈ > [ σ΅©σ΅© σ΅©σ΅©2 2 π σ΅©σ΅©ππ σ΅©σ΅© σ΅© σ΅©2 σ΅© σ΅© σ΅© σ΅©2 π (1 β π) σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© β β2ππ /π σ΅©σ΅©σ΅©ππ σ΅©σ΅©σ΅© β ππ /πΌπσΈ 2 π (πΌπ ) π (1 β π) σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© 2 β₯[ β π (πΌπ )] ] β = [ 2 σ΅©σ΅© σ΅©σ΅©2 σ΅© σ΅© σ΅©σ΅©ππ σ΅©σ΅© βπ π π σ΅©σ΅©ππ σ΅©σ΅© σ΅© σ΅© β₯[
π (1 β π) (max {1/π, 1/πΏ})
2
β
(33)
π (πΌπ ) 2 β π (πΌπ )] , βπ π
where the second inequality follows from (26), Part 3 in Proposition 4, and ππ+1 β€ ππ , the equality follows from ππ = π(πΌπ2 βππ β2 ), and the last inequality follows from (27). Dividing each side by πΌπ and letting π β β in the above inequality, we can deduce that 2π (1 β π) 1 1 ) = + β, β₯ lim ( 2 2 π β β (max {1/π, 1/πΏ}) π πΌπ
Therefore, it follows from the definition of π½π+1 and (23) that π½π+1 = β€
(34)
which is impossible, so the conclusion is obtained.
sequence {π₯π }β π=0 has accumulation point, and every accumulation point of {π₯π }β π=0 is optimal solution of problem (1). Proof. Suppose that there exist π0 > 0 and π0 > 0 such that σ΅©σ΅© π σ΅© (36) σ΅©σ΅©π (π₯π , ππ )σ΅©σ΅©σ΅© β₯ π0 , βπ > π0 . From (22), (26), and (29), we get π
πΉπ (π₯π + πΌπ ππ , ππ+1 ) β π½π β€ ππΌπ ππ (π₯π , ππ ) ππ 1 σ΅© σ΅©2 β€ β ππΌπ min {π, } σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© πΏ 1 β€ β ππ0 π0 min {π, } , βπ > π0 . πΏ
(38)
ππ0 π0 min {π, 1/πΏ} . πΈπ+1
By Assumption A, πΉ is bounded from below. Further by Proposition 4, πΉ(π₯π ) β€ πΉπ (π₯π , ππ ) for all π, we see that πΉπ (π₯π , ππ ) is bounded from below. Together with πΉπ (π₯π , ππ ) β€ π½π for all π from Lemma 6, it shows that π½π is also bounded from below. By (38), we obtain β
β π=π0
ππ0 π0 min {π, 1/πΏ} < β. πΈπ+1
(39)
On the other hand, the definition of πΈπ+1 implies that πΈπ+1 β€ π + 2, and it follows that β
(37)
ππΈπ π½π + π½π β ππ0 π0 min {π, 1/πΏ} πΈπ+1
β€ π½π β
By using the above lemmas, we are now ready to prove the global convergence of Algorithm 5. Theorem 8. Let {π₯π } be generated by Algorithm 5 and suppose that the conditions of Lemma 7 hold. Then one has σ΅© σ΅© lim σ΅©σ΅©π (π₯π )σ΅©σ΅©σ΅© = 0; (35) πββ σ΅©
ππΈπ π½π + πΉπ (π₯π + πΌπ ππ , ππ+1 ) πΈπ+1
β π=π0
β ππ0 π0 min {π, 1/πΏ} ππ0 π0 min {π, 1/πΏ} β₯ β πΈπ+1 π+2 π=π
= + β.
0
(40)
Mathematical Problems in Engineering
5
This is a contradiction. Therefore, we should have σ΅© σ΅© lim σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© = 0. πββ σ΅©
Table 1: Test problems.
(41)
From (17) in Proposition 4 together with ππ as π β β, which comes from the definition of ππ and limπ β 0 ππ = 0 in Algorithm 5, we obtain σ΅© σ΅© lim σ΅©σ΅©π (π₯π )σ΅©σ΅©σ΅© = 0.
πββ σ΅©
(42)
Set π₯β as an accumulation point of sequence {π₯π }β π=0 ; there is a convergent subsequence {π₯ππ }β π=0 such that lim π₯ππ = π₯β .
πββ
(43)
Nr. 1 2 3 4 5 6 7 8 9 10 11 12
Problems Rosenbrock Crescent CB2 CB3 DEM QL LQ Mifflin 1 Mifflin 2 Wolfe Rosen-Suzuki Shor
Dim. 2 2 2 2 2 2 2 2 2 2 4 5
πops (π₯) 0 0 1.9522245 2.0 β3 7.20 β1.4142136 β1.0 β1.0 β8.0 β44 22.600162
From (4) we know that π(π₯π ) = (π₯π βπ(π₯π ))/π. Consequently, (42) and (43) show that π₯β = π(π₯β ). Hence, π₯β is an optimal solution of problem (1).
4. Numerical Results This section presents some numerical results from experiments using our multivariate spectral gradient algorithm for the given test nonsmooth problems which come from [21]. We also list the results of [14] (modified Polak-Ribi`erePolyak gradient method, MPRP) and [22] (proximal bundle method, PBL) to make a comparison with the result of Algorithm 5. All codes were written in MATLAB R2010a and were implemented on a PC with 2.8 GHz CPU, 2 GB of memory, and Windows 8. We set π½ = π = 1, π = 0.9, π = 10β10 , and πΎ = 0.01, and the parameter πΏ is chosen as 1 { { { { {σ΅© π β1 πΏ = {σ΅©σ΅©σ΅©π (π₯π , ππ )σ΅©σ΅©σ΅©σ΅© { { { { β5 {10
σ΅© σ΅© if σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© > 1, σ΅© σ΅© if 10β5 β€ σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© β€ 1, σ΅© σ΅© if σ΅©σ΅©σ΅©ππ (π₯π , ππ )σ΅©σ΅©σ΅© < 10β5 ;
(44)
then we adopt the termination condition βππ (π₯π , ππ )β β€ 10β10 . For subproblem (5), the classical PRP CG method (called subalgorithm) is used to solve it; the algorithm stops if βππ(π₯π )β β€ 10β4 or π(π₯π+1 )βπ(π₯π )+βππ(π₯π+1 )β2 ββππ(π₯π )β2 β€ 10β3 holds, where ππ(π₯π ) is the subgradient of π(π₯) at the point π₯π . The subalgorithm will also stop if the iteration number is larger than fifteen. In its line search, the Armijo line search technique is used and the step length is accepted if the search number is larger than five. Table 1 contains problem names, problem dimensions, and the optimal values. The summary of the test results is presented in Tables 23, where βNr.β denotes the name of the tested problem, βNFβ denotes the number of function evaluations, βNIβ denotes the number of iterations, and βπ(π₯)β denotes the function value at the final iteration.
The value of π controls the nonmonotonicity of line search which may affect the performance of the MSG algorithm. Table 2 shows the results for different parameter π, as well as different values of the parameter ππ ranging from 1/6(π + 2)6 to 1/2π2 on problem Rosenbrock, respectively. We can conclude from the table that the proposed algorithm works reasonably well for all the test cases. This table also illustrates that the value of π can influence the performance of the algorithm significantly if the value of π is within a certain range, and the choice π = 0.75 is better than π = 0. Then, we compare the performance of MSG to that of the algorithms MPRP and PBL. In this test, we fix ππ = 1/2π2 and π = 0.75. To illustrate the performance of each algorithm more specifically, we present three comparison results in terms of number of iterations, number of function evaluations, and the final objective function value in Table 3. The numerical results indicate that Algorithm 5 can successfully solve the test problems. From the number of iterations in Table 3, we see that Algorithm 5 performs best among these three methods, and the final function value obtained by Algorithm 5 is closer to the optimal function value than those obtained by MPRP and PBL. In a word, the numerical experiments show that the proposed algorithm provides an efficient approach to solve nonsmooth problems.
5. Conclusions We extend the multivariate spectral gradient algorithm to solve nonsmooth convex optimization problems. The proposed algorithm combines a nonmonotone line search technique and the idea of Moreau-Yosida regularization. The algorithm satisfies the sufficient descent property and its global convergence can be established. Numerical results show the efficiency of the proposed algorithm.
6
Mathematical Problems in Engineering Table 2: Results on Rosenbrock with different π and π. π=0 NI/NF/π(π₯) 30/46/1.581752e β 9 28/38/5.207744e β 9 29/37/1.502034e β 9 27/37/1.903969e β 9 27/36/4.859901e β 9
ππ 1/2π2 1/3(π + 2)3 1/4(π + 2)4 1/5(π + 2)5 1/6(π + 2)6
Time 1.794 1.420 1.388 1.451 1.376
π = 0.75 NI/NF/π(π₯) 29/30/7.778992e β 9 26/27/6.541087e β 9 27/28/5.112699e β 9 27/28/6.329141e β 9 27/28/6.073222e β 9
Time 1.076 1.023 1.030 1.092 1.025
Table 3: Numerical results for MSG/MPRP/PBL on problems 1β12. Nr. 1 2 3 4 5 6 7 8 9 10 11 12
MSG NI/NF/π(π₯) 29/30/7.778992e β 9 9/10/1.450669e β 5 9/10/1.9522245 4/9/2.000009 3/4/β2.999949 11/12/7.200000 3/4/β1.4142136 9/10/β0.9999638 12/13/β0.9999978 5/6/β7.999999 6/7/β43.99797 12/13/2.260017
MPRP NI/NF/π(π₯) 46/48/7.091824e β 7 11/13/6.735123e β 5 12/14/1.952225 2/6/2.000098 4/6/β2.999866 10/12/7.200011 2/3/β1.414214 4/6/β0.9919815 20/23/β0.9999925 β 28/58/β43.99986 33/91/22.60023
Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments The author would like to thank the anonymous referees for their valuable comments and suggestions which help a lot to improve the paper greatly. The author also thanks Professor Gong-lin Yuan for his kind offer of the source BB codes on nonsmooth problems. This work is supported by the National Natural Science Foundation of China (Grant no. 11161003).
References [1] J. B. Hiriart-Urruty and C. LemarΒ΄echal, Convex Analysis and Minimization Algorithms, Springer, Berlin, Germany, 1993. [2] M. Fukushima and L. Qi, βA globally and superlinearly convergent algorithm for nonsmooth convex minimization,β SIAM Journal on Optimization, vol. 6, no. 4, pp. 1106β1120, 1996. [3] L. Q. Qi and J. Sun, βA nonsmooth version of Newtonβs method,β Mathematical Programming, vol. 58, no. 3, pp. 353β367, 1993. [4] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, NY, USA, 1983. [5] S. Lu, Z. Wei, and L. Li, βA trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization,β Computational Optimization and Applications, vol. 51, no. 2, pp. 551β573, 2012.
PBL NI/NF/π(π₯) 42/45/0.381e β 6 18/20/0.679e β 6 32/34/1.9522245 14/16/2.0000000 17/19/β3.0000000 13/15/7.2000015 11/12/β1.4142136 66/68/β0.9999994 13/15/β1.0000000 43/46/β8.0000000 43/45/β43.999999 27/29/22.600162
πops (π₯) 0 0 1.9522245 2.0 β3 7.20 β1.4142136 β1.0 β1.0 β8.0 β44 22.600162
[6] L. Q. Qi, βConvergence analysis of some algorithms for solving nonsmooth equations,β Mathematics of Operations Research, vol. 18, no. 1, pp. 227β244, 1993. [7] R. Correa and C. LemarΒ΄echal, βConvergence of some algorithms for convex minimization,β Mathematical Programming, vol. 62, no. 1β3, pp. 261β275, 1993. [8] M. Fukushima, βA descent algorithm for nonsmooth convex optimization,β Mathematical Programming, vol. 30, no. 2, pp. 163β175, 1984. [9] J. R. Birge, L. Qi, and Z. Wei, βConvergence analysis of some methods for minimizing a nonsmooth convex function,β Journal of Optimization Theory and Applications, vol. 97, no. 2, pp. 357β383, 1998. [10] Z. Wei, L. Qi, and J. R. Birge, βA new method for nonsmooth convex optimization,β Journal of Inequalities and Applications, vol. 2, no. 2, pp. 157β179, 1998. [11] N. Sagara and M. Fukushima, βA trust region method for nonsmooth convex optimization,β Journal of Industrial and Management Optimization, vol. 1, no. 2, pp. 171β180, 2005. [12] G. Yuan, Z. Wei, and Z. Wang, βGradient trust region algorithm with limited memory BFGS update for nonsmooth convex minimization,β Computational Optimization and Applications, vol. 54, no. 1, pp. 45β64, 2013. [13] G. Yuan and Z. Wei, βThe Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems,β Mathematical Modelling and Analysis, vol. 17, no. 2, pp. 203β216, 2012. [14] G. Yuan, Z. Wei, and G. Li, βA modified Polak-Ribi`ere-Polyak conjugate gradient algorithm for nonsmooth convex programs,β Journal of Computational and Applied Mathematics, vol. 255, pp. 86β96, 2014.
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