Applied Mathematics and Computation

A continuation approach for solving binary quadratic program based on a class of NCP-functions

Jein-Shan Chen

^a,⇑,1

, Jing-Fan Li

^a

, Jia Wu

^b

aDepartment of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan, ROC

bSchool of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o

Keywords:

Nonlinear complementarity problem Generalized Fischer–Burmeister function Binary quadratic program

a b s t r a c t

In the paper, we consider a continuation approach for the binary quadratic program (BQP) based on a class of NCP-functions. More specifically, we recast the BQP as an equivalent minimization and then seeks its global minimizer via a global continuation method. Such approach had been considered in[11]which is based on the Fischer–Burmeister function.

We investigate this continuation approach again by using a more general function, called the generalized Fischer–Burmeister function. However, the theoretical background for such extension can not be easily carried over. Indeed, it needs some subtle analysis.

Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we consider the following binary quadratic program (BQP)

min x^TQx þ c^Tx over x 2 S; ð1Þ

where Q is an n  n symmetric matrix, c is a vector in Rⁿand S is the binary discrete set f0; 1gⁿ. It is known that BQP is NP-hard and has a variety of applications in computer science, operations research and engineering, see[1,3,8,13,14]and references therein. There have been proposed several continuous approaches for solving BQP[9,12,15]which often need to cooperate with branch and bound algorithms or some heuristic strategies to generate an exact or approximate solution.

In[10], another type of continuous approach was proposed which is to reformulate BQP as an equivalent mathematical pro- gramming problem with equilibrium constraints (MPEC) and then consider an effective algorithm to find its global solution.

In this approach, many NCP-functions are employed to convert equilibrium constraints into a collection of quasi-linear equality constraints. Among others, the Fischer–Burmeister function /_FB:R²! R defined as

/_FBða; bÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a²þ b² q

 ða þ bÞ ð2Þ

is a popular one. In this paper, we investigate this continuation approach again by using a more general function /_p:R²! R, called the generalized Fischer–Burmeister function and defined by

/pða; bÞ :¼ kða; bÞk_p ða þ bÞ; ð3Þ

0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.amc.2012.10.033

⇑Corresponding author.

E-mail addresses:[email protected](J.-S. Chen),[email protected](J.-F. Li),[email protected](J. Wu).

1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is supported by National Science Council of Taiwan.

Applied Mathematics and Computation 219 (2012) 3975–3992

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Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c

where p > 1 is an arbitrary fixed real number and kða; bÞk_pdenotes the p-norm of ða; bÞ, i.e., kða; bÞk_p¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj^pþ jbj^p pp

. In other words, in the generalized FB function /_p, we replace the 2-norm of ða; bÞ appeared in the FB function by a more general p- norm. The function /_p is still an NCP-function, which naturally induces another NCP-function w_p:R²! Rþgiven by

wpða; bÞ :¼1

2j/pða; bÞj²: ð4Þ

For any given p > 1, the function w_pis shown to possess all favorable properties of the FB function w_FB. Traditionally, in the continuation approach for BQP, one needs to utilize the fact that

x 2 f0; 1gⁿ() xi¼ x²_i; i ¼ 1; 2; . . . ; n: ð5Þ

To the contrast, our proposed continuous optimization approach arises from the complementarity condition formulation of 0  1 vector x 2 f0; 1gⁿ, which includes the equivalence(5)with redundant constraints

0 6 xi61; i ¼ 1; 2; . . . ; n:

so that it can generate an integer feasible solution. For finding the global minimizer of our continuous optimization problem, we employ the similar way as in[10,11]. In summary, the method is to add a quadratic penalty term associated with its equi- librium constraints and a logarithmic barrier term associated with box constraints 1 6 xi61; i ¼ 1; 2; . . . ; n, respectively, to the objective function, and then construct a global smoothing function. Since the generalized Fischer–Burmeister function w_p is quasi-linear, the quadratic penalty for equilibrium constraints will make the convexity of the global smoothing function more stronger. Particularly, we have shown that the global smoothing function is strictly convex in the whole domain for barrier parameter large enough or in a subset of its domain for penalty parameter large enough. According to the feature above, we use a global continuation algorithm defined in[11]via a sequence of unconstrained minimization for this function with varying penalty and barrier parameters. Although the idea is brought from[11], as will be seen, the theoretical back- ground for such extension can not be easily carried over. Indeed, it needs some subtle analysis for extending the background materials. Without loss of generally, in this paper we consider the case that S ¼ f1; 1gⁿ. By a transformation z ¼ ðx þ eÞ=2 for the variable x and the unit vector e in R, we can extend the conclusions to the case S ¼ f0; 1gⁿ.

2. Continuous formulation based onUpfunction

In this section we will reformulate(1)as an equivalent continuous optimization based on the /_pfunction. As will be seen, the following equivalence plays a key role which says that a binary constraint t 2 fa; bg with a; b 2 R is equivalent to a com- plementarity condition (or equilibrium constraint), i.e.,

t 2 fa; bg () t  a P 0; b  t P 0; ðt  aÞðt  bÞ ¼ 0:

With this, the unconstrained BQP problem in(1)can be recast as a mathematical programming problem with equilibrium constraints (MPEC)

min f ðxÞ

s:t: ð1 þ xi;1  xiÞ ¼ 0; i ¼ 1; 2; . . . ; n;

1 þ xiP0; 1  xiP0; i ¼ 1; 2; . . . ; n:

ð6Þ

In fact, given any NCP-function / : R  R ! R, the property of NCP-functions (see[6]) yields that the equilibrium constraint in(6)is indeed equivalent to an equality constraint associated with /:

ð1 þ xi;1  xiÞ ¼ 0; 1 þ xiP0; 1  xiP0; () /ð1 þ xi;1  xiÞ ¼ 0: ð7Þ

Thus we reformulate the original BQP problem, which together with(6) and (7), as the following continuous optimization problem:

min f ðxÞ

s:t: /ð1 þ xi;1  xiÞ ¼ 0; i ¼ 1; 2; . . . ; n

1 6 xi61; i ¼ 1; 2; . . . ; n:

ð8Þ

Accordingly, the global minimizer of (8) is the solution of (1). Note that although the box constraints

1 6 xi61; i ¼ 1; 2; . . . ; n in(8)are indeed redundant, we keep them on purpose. Actually, we shall see that such constraints play a crucial role in the construction of a global smoothing function for problem(8)as was shown in[9,10]. Generally speaking, most NCP-functions are non-differentiable, such as the popular Fischer–Burmeister function in(2), the generalized Fischer–Burmeister function in(3), as well as the minimum function

/_minða; bÞ ¼ minfa; bg:

However, it is very interesting to observe that, when specializing / in(8)as the generalized Fischer–Burmeister function, we can reach smooth constraint functions

/pð1 þ xi;1  xiÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j1 þ xij^pþ j1  xij^p qp

 2 ¼ 0; i ¼ 1; 2; . . . ; n

and consequently some usual nonlinear programming solvers can be employed to design an effective algorithm for solving problem(8). In view of this, we in this paper pay attention to the following equivalent continuous formulations reformulated by the generalized Fischer–Burmeister function:

min f ðxÞ

s:t: /pð1 þ xi;1  xiÞ ¼ 0; i ¼ 1; 2; . . . ; n

1 6 xi61; i ¼ 1; 2; . . . ; n:

ð9Þ

We also note that using the equivalence that xi2 f1; 1g () x²_i ¼ 1 gives another another type of continuous optimization:

min f ðxÞ

s:t: x²_i ¼ 1; i ¼ 1; 2; . . . ; n

1 6 xi61; i ¼ 1; 2; . . . ; n:

ð10Þ

The formulation of(10)looks simple and friendly at first glance, nonetheless, the following remarkable advantages explain why we still stick to the smooth constrained optimization problem(9):

(i) The quasi-linearity of generalized Fischer–Burmeister function implies that it feasible set tends to be convex.

(ii) The equality constraint conditions /_pð1 þ xi;1  x_iÞ ¼ 0; i ¼ 1; 2; . . . ; n have incorporated the equivalent formulation x²_i ¼ 1; i ¼ 1; 2; . . . ; n, of x 2 f1; 1g with its relaxation formulation 1 6 xi61; i ¼ 1; 2; . . . ; n, which indicates that, when solving(9)with a penalty function method, an implicit interior point constraint is additionally imposed on.

(iii) FromProposition 2.1as below, we see that the quadratic penalty function of equality constraints is strictly convex in a very large region when the penalty parameter is large enough.

These advantages have great contributions to searching for an optimal solution or a favorable suboptimal solution of(1), which will be shown later. Before we prove the main proposition, we first introduce several technical lemmas which are important for building up the background materials of our extension.

Lemma 2.1. Let f ; g be real-valued functions from R to Rþ. Suppose f ; g satisfy

(i) f⁰ðxÞ > 0 and g⁰ðxÞ < 0 for all x 2 ða; bÞ, (ii) f⁰⁰ðxÞ < 0 and g⁰⁰ðxÞ < 0 for all x 2 ða; bÞ, (iii) ðfgÞ⁰ðaÞ < 0 and f ðaÞ P gðaÞ.

Then ðfgÞ⁰ðxÞ < 0 for all x 2 ða; bÞ.

Proof. To achieve our result, we need to verify two things: (i) ðfgÞ⁰ðaÞ < 0 and (ii) ðfgÞ⁰ðxÞ is decreasing on x 2 ða; bÞ. We pro- ceed these verifications as below.

(i) From the assumptions and the chain rule, it is clear that ðfgÞ⁰ðaÞ ¼ f⁰ðaÞgðaÞ þ f ðaÞg⁰ðaÞ < 0:

(ii) Since ðfgÞ⁰ðxÞ ¼ f⁰ðxÞgðxÞ þ f ðxÞg⁰ðxÞ, we see that in order to show ðfgÞ⁰ðxÞ is decreasing on x 2 ða; bÞ, it is enough to argue both f⁰ðxÞgðxÞ and f ðxÞg⁰ðxÞ are decreasing on ða; bÞ. We look into the first term first. Note that

f⁰ðxÞgðxÞ

ð Þ⁰¼ f⁰⁰ðxÞgðxÞ þ f⁰ðxÞg⁰ðxÞ 6 0 8x 2 ða; bÞ;

because f⁰⁰ðxÞ < 0; gðxÞ P 0; f⁰ðxÞ > 0 and g⁰ðxÞ < 0. This claims that f⁰ðxÞgðxÞ is decreasing on x 2 ða; bÞ. The decreasing of f ðxÞg⁰ðxÞ over ða; bÞ can be concluded similarly.

Thus, from all the above, the proof is complete. h

The conclusion of next lemma is simple and neat, however, its arguments are very tedious. Indeed the main idea behind is approximation.

Lemma 2.2. Let w_pbe defined as in(4). Then, w⁰⁰_pð1 þ t; 1  tÞ is positive at t ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p for all p P 2.

Proof. For symmetry, we only prove the case of t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p . First, from direct computations and simplifying the expression of w⁰⁰_p, we have

J.-S. Chen et al. / Applied Mathematics and Computation 219 (2012) 3975–3992 3977

w⁰⁰_pð1 þ t; 1  tÞ ¼ ð1 þ tÞ^pþ ð1  tÞ^p¹p

ð1 þ tÞ²ð1  tÞ²ð1 þ tÞ^pþ ð1  tÞ^p² Fðp; tÞ; ð11Þ

where Fðp; tÞ ¼ f0ðp; tÞ f½1ðp; tÞ þ f2ðp; tÞ þ f3ðp; tÞ þ f4ðp; tÞ with f0ðp; tÞ ¼ ð1 þ tÞ ^pþ ð1  tÞ^p¹p;

f1ðp; tÞ ¼ ð1  tÞ²ðt þ 1Þ^2p; f2ðp; tÞ ¼ ðt þ 1Þ²ð1  tÞ^2p;

f3ðp; tÞ ¼ ð2t²þ 4p  6Þðt þ 1Þ^pð1  tÞ^p; f4ðp; tÞ ¼ ð8  8pÞð1  t²Þ^p:

Since the first term on the right side of(11)is always positive for all p P 2, it suffices to show that F p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

>0 for all p P 2. However, it is very hard to claim this fact directly. Our strategy is to construct a function A : R ! R such that

AðpÞ 6 F p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

8p P 2: ð12Þ

The special feature for AðpÞ is that it is easier to verify AðpÞ P 0 for all p P 2 so that our goal could be reached. Now, we pro- ceed the proof by carrying out the aforementioned two steps.

Step (1): Construct a function AðÞ satisfying(12). Indeed, the function Fð; Þ is composed of f0;f1;f2;f3and f4, so for each fi, we will construct a corresponding piecewise function aisuch that aiðpÞ 6 fi p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³ 1 p

 

for i ¼ 0; 1; 2; 3; 4. Then, combining them together to build up the function AðÞ. For making the reader understand more easier, we will give some pictures during the process of proof.

(i) First, we explain how to set up a0ðpÞ. Notice that the second derivative of f0with respect to p is positive at t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p for all p P 2; f0is strictly convex at t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³ 1 p

for all p P 2 (the detailed arguments are provided in Appendix A).

Hence, we consider a real piecewise function defined as

a0ðpÞ ¼

1

8ðp  2Þ þ 2²³ if 2 6 p 6 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p  6 þ 8ð2²³Þ;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p þ 1 if p P 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p  6 þ 8ð2²³Þ:

8<

:

Fig. 1depicts the relation between a0ðpÞ and f0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

. Besides, the following facts

a0ð2Þ ¼ f0 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

lim

p!2^þa⁰₀ðpÞ < d dpf0 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

a⁰⁰₀ðpÞ ¼ 0 < d² dp²f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

Fig. 1. The graphs of a0and f0.

indicate the first part of function a0ðpÞ is less than f0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

for 2 < p 6 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 6 þ 8ð2²³Þ. On the other hand, an- other fact

p!1limf0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 q

þ 1

says that the second part of function a0ðpÞ is less than or equal to f0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

for p P 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 6 þ 8ð2²³Þ. Thus, we conclude that

a0ðpÞ 6 f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

8p P 2:

(ii) Secondly, we consider a quadratic function defined as

a1ðpÞ ¼ 1  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q ²

1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q ⁴

ln 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

ðp  1Þ²

þ 1  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q ²

1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q ⁴

1  ln 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

: Fig. 2depicts the relation between a1ðpÞ and f1 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³ 1

 p 

. Again, using the following facts

a1ð2Þ ¼ f1 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

a⁰₁ð2Þ ¼ d dpf1 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

a⁰⁰₁ðpÞ 6 d² dp²f1 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

8p P 2;

we immediately achieve

a1ðpÞ 6 f1 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

8p P 2:

(iii) Thirdly, we consider a function defined as

a2ðpÞ ¼

¹₅ðp  2Þ þ 1  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p ⁴

1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p ²

if 2 6 p 6 12 þ 20 2 ¹³ 2²³

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p 40ð2¹³Þ  40  10ð2²³Þ

; 0

if p P 12 þ 20 2 ¹³ 2²³

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p 40ð2¹³Þ  40  10ð2²³Þ : 8>

>>

>>

>>

<

>>

>>

>>

>:

Fig. 2. The graphs of a1and f1.

J.-S. Chen et al. / Applied Mathematics and Computation 219 (2012) 3975–3992 3979

Fig. 3depicts the relation between a2ðpÞ and f2 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

. We observe that the function f2is positive and convex on p P 2, then the following facts

a2ð2Þ ¼ f2 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

lim

p!2^þa⁰₂ðpÞ < d dpf2 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

a⁰⁰₂ðpÞ ¼ 0 < d² dp²f2 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

8p > 2

yield a2ðpÞ 6 f2 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

for all p P 2.

(iv) Fourthly, we consider a real piecewise function defined as

a3ðpÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2¹³

p 24  12ð2²³Þ

þ 16ð2²³ 2¹³Þ  8

h i

p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2  2¹³

p ð24ð2²³Þ  48Þ þ 40ð2²³ 2¹³Þ þ 20 if 2 6 p 6⁵₂;

₃₁⁴2¹³þ₃₁⁴

ð2  2¹³Þ⁵²p þ⁷²₃₁2¹³þ⁷²₃₁

ð2  2¹³Þ⁵² if ⁵₂6p 6 18;

0 if p P 18:

8>

>>

>>

><

>>

>>

>>

:

Fig. 4depicts the relation between a3ðpÞ and f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

. The relation is clear from the picture, however, we need to go through three subcases to verify it mathematically.

If 2 6 p 6⁵₂, we compute f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

¼ 2ð2 ¹³Þ  8 þ 4p

ð2  2¹³Þ^p. Moreover, we have d

dpf3 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

¼ ð2  2¹³Þ^p 4 þ 2ð2 ¹³Þ  8 þ 4p

lnð2  2¹³Þ

h i

; d²

dp²f3 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

¼ ð2  2¹³Þ^plnð2  2¹³Þ 8 þ 2ð2 ¹³Þ  8 þ 4p

lnð2  2¹³Þ

h i

: Then, the following facts

a3ð2Þ ¼ f3 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

a3 5 2

 

¼ f3 5 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

lim

p!2^þa⁰₃ðpÞ 6 d dpf3 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

Fig. 3. The graphs of a2and f2.

and f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

being concave on 2; ⁵₂

imply a3ðpÞ 6 f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

under this case.

If⁵₂6p 6 18, using the facts that

a3

5 2

 

¼ f3

5 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

lim

p!⁵₂^þ

a⁰₃ð Þ 6p d dpf3

5 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

and a3ðpÞ ¼ f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

having only one solution at p ¼⁵₂, we obtain a3ðpÞ 6 f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

under this case.

If p P 18, knowing f3ðpÞ > 0 for all p, then it is clear that a3ðpÞ 6 f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

under this case.

(v) Finally, notice that the second derivative of f4with respect to p is positive at t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p for all p P^2þlnð22

1 3Þ lnð22¹³Þ , and negative for p 6^2þlnð22

1 3Þ

lnð22¹³Þ , so f4 is strictly convex at t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

for all p P^2þlnð22

1 3Þ

lnð22¹³Þ and strictly concave for all p 6^2þlnð22

13Þ

lnð22¹³Þ . Hence, we consider a real piecewise function defined as

a4ðpÞ ¼

¹⁵³₅₀p  8ð2  2¹³Þ²þ¹⁵³₂₅ if 2 6 p 6⁵₂;

⁴⁹⁷₅₀þ 16ð2  2¹³Þ²

h i

p þ⁵⁸³₂₅ 48ð2  2¹³Þ² if ⁵₂6p 6 3;

¹³₁₀p ¹³₅ if 3 6 p 6⁴⁹₁₃;

¹⁵₂ if p P⁴⁹₁₃:

8>

>>

>>

>>

><

>>

>>

>>

>>

:

Fig. 5depicts the relation between a4ðpÞ and f4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

. Again, we need to discuss several subcases to prove the rela- tion mathematically.

For 2 6 p 6⁵₂, the following facts

a4ð2Þ ¼ f4 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

lim

p!2^þa⁰₄ðpÞ < d dpf4 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

a⁰⁰₄ðpÞ ¼ 0 < d² dp²f4 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

yield the first part of function a4ðpÞ is less than f4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

under this case.

Fig. 4. The graphs of a3and f3.

J.-S. Chen et al. / Applied Mathematics and Computation 219 (2012) 3975–3992 3981

For⁵₂6p 6 3, using the following facts

a4ð3Þ < f4 3;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

p!3lim^a⁰₄ðpÞ > d dpf4 3;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

;

a⁰⁰₄ðpÞ ¼ 0 < d² dp²f4 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

:

ð13Þ

We have a4ðpÞ is less than f4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

under this case.

For 3 6 p 6⁴⁹₁₃, we know that

lim

p!3^þa⁰₄ðpÞ > d dpf4 3;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

:

This together with(13)gives a4ðpÞ is less than f4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

under this case.

For p P⁴⁹₁₃, from f4ðpÞ being strictly convex for all p 6^2þlnð22

1 3Þ

lnð22¹³Þ and being strictly concave for all p P^2þlnð22

1 3Þ

lnð22¹³Þ , we know d

dpf4

1 þ lnð2  2¹³Þ lnð2  2¹³Þ

!

¼ 0 and lim

p!1f4ðpÞ ¼ 0;

which lead to f4ðpÞ > ¹⁵₂ for all p 6 2. Thus, a4ðpÞ 6 f4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

under this case.

Now, we are ready to define a function A : R ! R satisfying(12). As the mentioned idea, the function is defined by AðpÞ ¼ a0ðpÞ a½ 1ðpÞ þ a2ðpÞ þ a3ðpÞ þ a4ðpÞ:

According to our constructions of aiðpÞ, it is clear that AðpÞ 6 F p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

for all p P 2.Fig. 6shows the relation between AðpÞ and F p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³ 1

 p 

.

Step (2): We will show that AðpÞ P 0 for all p P 2. Notice that AðpÞ is piecewise smooth, hence A⁰ðpÞ is a piecewise function. Indeed, the expression of A⁰ðpÞ looks very ugly and tedious, we display it Appendix B. Furthermore, we also present an approximate expression for A⁰ðpÞ in Appendix C which helps us understand the structure of A⁰ðpÞ. The key point is that from the expression of the A⁰ðpÞ, we can verify the following facts:

Að2Þ ¼ 0;

A 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 q

 6 þ 8ð2²³Þ

 

>A 5 2

 

>0;

and

A⁰ðpÞ < 0 if p 2 ð⁵₂;8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 6 þ 8ð2²³ÞÞ;

A⁰ðpÞ > 0 otherwise;

(

Fig. 5. The graphs of a4and f4.

with the exception of points of discontinuity. Thus, we conclude AðpÞ P 0 for all p P 2 and(12)is satisfied, which imply F p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³ 1 p

 

P0 for all p P 2. Then, the proof is complete. h

Lemma 2.3

(a) Let f be a convex function defied on a convex set C in Rⁿand g be a nondecreasing convex function defined on an interval I in R. Suppose fðCÞ # I. Then, the composite function g  f defined by ðg  f ÞðxÞ ¼ gðf ðxÞÞ is convex on C.

(b) Suppose /₁:U ! R is a twice continuously differentiable function with a compact set U 2 Rⁿand /₂:X ! R is a twice con- tinuously differentiable function such that the minimum eigenvalue of its Hessian matrixr²_xx/₂ðxÞ is greater than

e

(> 0) for all x 2 X, where X  U. Then there exists a constant ^b >0 such that /₁þ b/2is a strictly convex function on X for b > ^b.

Proof

(a) See[2, ChapIII, Lemma1.4].

(b) See[9, Theorem3.1]. h

Proposition 2.1. Let /_p and w_pbe defined as in(3) and (4), respectively. Then, for any fixed p P 2, the following hold.

(a) The function /_pð1 þ t; 1  tÞ is strictly convex for all t 2 R.

(b) The function w_pð1 þ t; 1  tÞ is strictly convex for all t R  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

h i

.

Proof

(a) It is known know that /_pis a convex function[4–6]. Note that f is a composition of /_pand an affine function. Thus, f is convex since it is a composition of a convex function and an affine function (the composition of two convex functions is not necessarily convex, however, our case does guarantee the convexity because one of them is affine).

(b) Due to the symmetry of w_pð1 þ t; 1  tÞ, it is enough to show that wpð1 þ t; 1  tÞ is strictly convex for t P ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p . To

proceed, we discuss two cases.

(i) If t P 1, the function w_pð1 þ t; 1  tÞ can be regard as a composite function of /pð1 þ t; 1  tÞ and hðÞ ¼ ðÞ². Because hðÞ is nondecreasing convex function on ½0; 1 and /_pð1 þ t; 1  tÞ is positive strictly convex for t P 1, fromLemma 2.3, we obtain wð1 þ t; 1  tÞ is strictly convex for t P 2.

(ii) If 1 > t P ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p , we know that

Fig. 6. The graphs of A and F.

J.-S. Chen et al. / Applied Mathematics and Computation 219 (2012) 3975–3992 3983

 w⁰_pð1 þ t; 1  tÞ ¼ /pð1 þ t; 1  tÞ/⁰_pð1 þ t; 1  tÞ;

 w⁰⁰pð1 þ t; 1  tÞ ¼ /h ⁰pð1 þ t; 1  tÞ/⁰pð1 þ t; 1  tÞ  /pð1 þ t; 1  tÞ/⁰⁰pð1 þ t; 1  tÞi :

Then, it suffices to show that w⁰⁰_pð1 þ t; 1  tÞ < 0 for p P 2. To this end, we compute the third derivative of /_pð1 þ t; 1  tÞ with respect to t and prove that it is negative. To see this,

/⁰⁰⁰_pð1 þ t; 1  tÞ ¼4 ð1 þ tÞ ^pþ ð1  tÞ^p1pð1 þ tÞ^pð1  tÞ^pðp  1Þ

ð1 þ tÞ³ðt  1Þ³ð1 þ tÞ^pþ ð1  tÞ^p3  Tðp; tÞ; ð14Þ

where T is a real valued function defined by

Tðp; tÞ ¼ ð1 þ tÞ^pð2p  1  3tÞ  ð1  tÞ^pð2p þ 3t  1Þ:

It is not hard to verify the first term of the right side of(14)is always negative for all p P 2. Thus, we only need to show Tðp; tÞ > 0 for all p P 2 which is equivalent to verifying Tð2; tÞ > 0 and Tðp; tÞ > Tð2; tÞ for all p > 2. These can be done as below.

(i) Because Tð2; tÞ ¼ 6t  6t³, it is clear Tð2; tÞ > 0.

(ii) To show that Tðp; tÞ > Tð2; tÞ for p > 2, we first argue that

ð1 þ tÞ^p>ð1  tÞ^p1ð2p þ 3t  1Þ 8p > 2; ð15Þ

It is equivalent to show that ^ð1þtÞp

ð1tÞ^p1ð2pþ3t1Þis greater than 1 for all p > 2. Therefore, we consider the derivative of the following function with respect to p as follows:

d dp

ð1 þ tÞ^p

ð1  tÞ^p1ð2p þ 3t  1Þ¼ ð1 þ tÞ^p

ð1  tÞ^p1ð2p þ 3t  1Þ² ð1  3t  2pÞ lnð1  tÞ þ ð2p þ 3t  1Þ lnð1 þ tÞ  2½ : ð16Þ Observing both terms of the right side of(16) are positive for all p > 2 and using ^ð1þtÞp

ð1þ3tþ2pÞð1tÞ^p1>1 when p = 2, we can achieve(15). Secondly, we know that

2p  1  3t > 1  t 8p > 2: ð17Þ

Combining(15) and (17), we have Tðp; tÞ P Tð2; tÞ. Hence, /⁰⁰⁰_pð1 þ t; 1  tÞ < 0 8p P 2:

Then, applyingLemma 2.1gives the desired result for which we set f ðtÞ ¼ /_pð1 þ t; 1  tÞ and gðtÞ ¼ /⁰_pð1 þ t; 1  tÞ. h

The result ofProposition 2.1(b) could be improved under some sense. More specifically, the interval where w_pð1 þ t; 1  tÞ is strictly convex varies as long as p changes. We originally wish to figure out the exact interval where w_pð1 þ t; 1  tÞ is strictly convex for each p. However, it is very hard to find a closed form depending p to reflect this feature (indeed, it

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

p=1.1 p=1.5 p=2 p=3 p=10

Fig. 7. The graphs of wpð1 þ t; 1  tÞ for different p.

may be not possible in our opinion). To compromise, we try to find such an appropriate common interval for all p P 2 as shown inProposition 2.1(b). The following two figures (Figs. 7 and 8) depict the geometric view regarding what we just mentioned.

3. Global continuation algorithm for BQP

Due to the logarithmic barrier function being strictly convex andProposition 2.1, we now introduce the quadratic penalty Pn

i¼1w_pð1 þ xi;1  xiÞ for the equality constraints and the logarithmic barrier Pn

i¼1½lnð1 þ xiÞ þ lnð1  xiÞ of the box con- straints into the(9). Construct a global smoothing function

/ðx;

a

;

s

Þ ¼ f ðxÞ þ

a

^X

n

i¼1

w_pð1 þ xi;1  xiÞ 

s

^X

n

i¼1

lnð1 þ xiÞ þ lnð1  xiÞ

½  ð18Þ

where

s

>0 is a barrier parameter and

a

>0 is a penalty parameter. The next property indicates that the strictly convexity of function /ðx;

a

;

s

Þ on ð1; 1Þⁿwhen the barrier parameter is large enough, and the strictly convexity of function /ðx;

a

;

s

Þ in a large subset of its domain for all

s

>0.

Proposition 3.1. Let /ðx;

a

;

s

Þ be the function defined by(18). Then, the following hold.

(a) There exists a constant ^

s

>0 such that if

s

> ^

s

and

a

>0; /ðx;

a

;

s

Þ is strictly convex on ð1; 1Þⁿ.

(b) There exists a constant

a

^>0 such that if

a

> ^

a

and

s

>0; /ðx;

a

;

s

Þ is strictly convex on the set D :¼ x 2 ð1; 1Þⁿj jxij > ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³ 1 p

;i ¼ 1; 2; . . . ; n

n o

.

Proof

(a) Let X ¼ ð1; 1Þⁿand denote

/_aðxÞ :¼ f ðxÞ þ

a

^X

n

i¼1

w_pð1 þ xi;1  xiÞ;

/bðxÞ :¼ Xⁿ

i¼1

½lnð1 þ xiÞ þ lnð1  xiÞ:

Then the expression of the Hessian matrix of /_bðxÞ at any x 2 X is given by

r²xx/bðxÞ ¼ diag 1

ð1  x1Þ²þ 1

ð1 þ x1Þ²;   ; 1

ð1  xnÞ²þ 1 ð1 þ xnÞ²

!

;

0 1

2

3 0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2

y−axis x−axis

z−axis

Fig. 8. The graph of wpð1 þ t; 1  tÞ with a fixed p.

J.-S. Chen et al. / Applied Mathematics and Computation 219 (2012) 3975–3992 3985

where diagðxÞ denotes a diagonal matrix with the components of x as the diagonal elements. Moreover, the function

1 ð1xiÞ²þ ¹

ð1þxiÞ² has minimum 2 at point xi¼ 0, and so every diagonal element of r²_xx/_bðxÞ is at least 2. Thus, by letting U ¼ ½1; 1ⁿ;

e

¼ 2 and usingLemma 2.3(b) yield the desired result.

(b) Set /_a¼ f ðxÞ and

/_b¼Xⁿ

i¼1

w_pð1 þ xi;1  xiÞ 

s a

Xⁿ

i¼1

½lnð1 þ xiÞ þ lnð1  xiÞ:

From the proof ofLemma 2.2, it follows that

r²xx

Xⁿ

i¼1

w_pð1  xi;1  xiÞ

!

¼ diag w ⁰⁰pð1  x1;1 þ x1Þ;    ; w⁰⁰pð1  xn;1 þ xnÞ

;

where w⁰⁰_pð1  xi;1 þ xiÞ can be found in(11). Now taking f ðtÞ ¼ /_pð1 þ t; 1  tÞ; gðtÞ ¼ /⁰_pð1 þ t; 1  tÞ and applying the proof (ii) ofLemma 2.1, we have

w⁰⁰_pð1  t; 1 þ tÞ > w⁰⁰2ð1  t; 1 þ tÞ 8p > 2:

In addition, from[11](Lemma 3.1), we also have

w⁰⁰_bð1  t; 1 þ tÞ ¼2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2t²þ 2Þ³ q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ð2t²þ 2Þ³

q >0:0004 8jtj > 0:51:

Therefore, the above two inequalities imply

w⁰⁰_pð1  t; 1 þ tÞ > 0:0004 8jtj > 0:51 and 8p P 2:

This indicates that every diagonal element of r²_xxw_p is at least 0:0004. Using the fact that the Hessian matrix of

^s_aPn

i¼1½lnð1 þ xiÞ þ lnð1  xiÞ is positive definite, we obtain that every diagonal element ofr²_xx/_bis at least 0:0004. Now taking

U ¼ ½1; 1ⁿ; X ¼ D and

e

¼ 0:0004

and applyingLemma 2.3gives the desired conclusion. h

As remarked in[11], the result ofProposition 3.1offers motivation to use the function /ðx;

a

;

s

Þ to develop a global con- tinuation algorithm for the constrained optimization problem(9). This method will generate a global optimal solution or at least a desirable local solution via a sequence of unconstrained minimization

minx2Rⁿ/ðx;

a

k;

s

kÞ ð19Þ

with an increasing penalty parameter sequence f

a

kg and a decreasing barrier parameter sequence f

s

kg. Note that to ensure the strict convexity of /ðx;

a

k;

s

kÞ, we have to utilize a sufficiently large initial value

s

0to start with the algorithm. As the iteration goes on, the convexity of logarithmic barrier 

s

kPn

i¼1½lnð1 þ xiÞ þ lnð1  xiÞ will become weak, but the strict con- vexity of /ðx;

a

k;

s

kÞ can still be guaranteed due to the increasing of the penalty parameter

a

k. This means that for each k 2 IN, the minimization problem(19)can be easily solved if we have skillful technique to adjust the parameter

a

and

s

.

Algorithm 3.1

Step 0 Given parameters

a

0;

s

0,

r

1>1;

r

22 ð0; 1Þ and



>0. Select a starting point ^x⁰and set k ¼ 0.

Step 1 Solve the unconstrained minimization problem(19)with the starting point ^x^k, and denote by x^kits optimal solution.

Step 2 If ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i¼1w_pð1  x^k_i;1  x^k_iÞ q

6



, terminate the algorithm, else go to Step 3.

Step 3 Update the parameters

a

kþ1¼

r

1

a

kand

s

kþ1¼

r

2

s

k. Step 4 Set ^x^kþ1¼ ^x^k;k ¼ k þ 1 and go to Step 1.

IsAlgorithm 3.1well-defined? To answer this, we give an existence theorem of solution for the unconstrained minimi- zation problem(19). In fact, its proof can be found in[11]Lemma 3.2, we give a brief proof here for completeness.

Proposition 3.2. Let /ðx;

a

_k;

s

kÞ be the function defined as in(18). Then, the following hold.

(a) For each k 2 IN, the minimization problem(19)has a solution x^k.

(b) From (a), there exists an ^

s

such that the solution to problem(19)is unique when

s

k> ^

s

.

Proof

(a) We first show the existence of x^kfor each k 2 IN. Let X1¼  ³₄;³₄

. Since /ðx;

a

k;

s

kÞ is continuous and X1is a compact set, there exist two real numbers L1and U1such that

L16/ðx;

a

k;

s

kÞ 6 U1 8x 2 X1:

On the other hand, we note that /ðx;

a

_k;

s

kÞ ! þ1 when xi₀! 1^or xi₀! 1^þfor some i02 f1; 2; . . . ; ng. Hence, the continuity of function /ðx;

a

k;

s

kÞ implies that there exists an d with 0 < d < 1=4 such that

/ðx;

a

k;

s

kÞ P U1 8x 2 ð1; 1 þ d [ ½1  d; 1Þð Þⁿ: ð20Þ

Let X ¼ ½1 þ d; 1  dⁿ. Again, /ðx;

a

k;

s

kÞ being continuous on a compact set X implies that there exists an ^x 2 X such that for each k 2 IN

/ð^x;

a

k;

s

kÞ 6 /ðx;

a

k;

s

kÞ 8x 2 X:

Moreover, due to X1#X, we know

/ð^x;

a

k;

s

kÞ 6 U1: ð21Þ

Combining(20) and (21)yields that

/ð^x;

a

k;

s

kÞ 6 /ðx;

a

k;

s

kÞ 8x 2 ð1; 1Þⁿn X:

Thus, together with(20), it shows that ^x is exactly the desired solution x^k.

(b) From conclusion ofProposition 3.1(a), /ð^x;

a

k;

s

kÞ is strictly convex on ð1; 1Þⁿ. Hence x^kis unique. h

4. Numerical experiments

In this section, we report numerical results ofAlgorithm 3.1for solving the unconstrained binary quadratic programming problem. Our numerical experiments are carried out in Matlab (version 7.8) running on a PC Inter core 2 Q8200 of 2.33 GHz CPU and 2.00 GB Memory.

In our numerical experiments, we employ BFGS algorithm with strong Wolfe–Powell line search to solve the uncon- strained minimization problem(19), and terminate the current iteration as long as x^ksatisfies the following criterion:

rx/ðx^k;

a

k;

s

kÞ

6 5:0e  3:

The values for the parameters involved inAlgorithm 3.1are chosen as follows:

a

0¼ 0;

r

1¼ 2;

r

2¼ 0:5;

e

¼ 1:0e  3;

and the initial barrier parameter

s

0varies with the scale of problems (here we choose its value the same as that in[11]). The starting point ^x⁰¼ 0:9ð1; 1; . . . ; 1Þ^T2 Rⁿis used for all test problems. To obtain an integer solution x from the final iterate point ^x ofAlgorithm 3.1, we let

x _i ¼ 1 if j^x _i þ 1j 6 1:0e  2 1 if j^x _i  1j 6 1:0e  2 (

for i ¼ 1; 2; . . . ; n:

The test problems are all from the OR-Library and have the following formulation max z^TQz

s:t: zi2 f0; 1g; i ¼ 1; 2; . . . ; n:

To solve these problems with Algorithm 3.1, we use the formula z ¼ ðx þ eÞ=z to transform them into the following formulation

 min ¹₄x^TQx ¹₂x^TQe ¹₄e^TQe s:t: xi2 f1; 1g; i ¼ 1; 2; . . . ; n:

The optimal values generated byAlgorithm 3.1with different p (p = 1.1, 2, 4, 5, 10, 20, 50, 100) are listed inTables 1–5(see Appendix D), where ‘–’ means that the algorithm fails to get an optimal solution when the maximum CPU time arrives.

Moreover, to present the objective evaluation and comparison of the performance ofAlgorithm 3.1with different p, we adopt the performance profile introduced in [7]as a means. In particular, we regard Algorithm 3.1corresponding to a p as a solver and assume that there are nssolvers and njtest problems from the OR-Library collection J . We are interested in using the optimal values calculated byAlgorithm 3.1as performance measure for different p. For each problem j and solver s, let

tj;s:¼ the optimal value of problem j by solver s;

l

_j;s:¼ 1 tj;s

:

J.-S. Chen et al. / Applied Mathematics and Computation 219 (2012) 3975–3992 3987

We compare the performance on problem j by solver s with the best performance by any one of the nssolvers on this prob- lem, i.e., we employ the performance ratio

rj;s:¼

l

_j;s

minf

l

_j;s:s 2 Sg¼maxftj;s:s 2 Sg tj;s

;

where S is the set of eight solvers. An overall assessment of each solver is obtained from

q

_sð

s

Þ :¼1 nj

sizefj 2 J : rj;s6

s

g;

which is called the performance profile of the reciprocal of optimal solution obtained byAlgorithm 3.1for solver s.

Fig. 9shows the performance profile of the reciprocals of optimal values obtained byAlgorithm 3.1in the range of ½1; 1:04

for eight solvers on 50 test problems. The eight solvers correspond toAlgorithm 3.1with p ¼ 1:1; p ¼ 2; p ¼ 4; p ¼ 5; p ¼ 10

;p ¼ 20; p ¼ 50 and p ¼ 100, respectively. From this figure, we see thatAlgorithm 3.1are considerably efficient no matter which value of p is chosen. In fact,Algorithm 3.1with the aforementioned p values can solve all the 50 test problems except for p ¼ 5; 20; 100. Moreover,Algorithm 3.1with p ¼ 4 has the best numerical performance (has the highest probability of being the optimal solver) and the probability of its being the winner on a given BQP is around 0.48. Besides, p ¼ 1:1 and p ¼ 2 have a comparable performance with p ¼ 4, please refer to Appendix D for more detailed numerical reports.

Appendix A

Here is the proof of the strictly convexity of f₀ p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 

for all p P 2.

To see this, it is enough to verify that ^d2

dp²f0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

>0 for all p > 2. In fact,

d² dp²f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

¼ f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q  ln f0ðp; ffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1 p

 Þ^p

p² þa^plnðaÞ þ b^plnðbÞ p f0ðp; ffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³ 1 p

Þ

 ^p

2 64

3 75

2

þ 1

p³ f0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 p 

 2p1 MðpÞ;

ð22Þ where a ¼ 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³ 1 p

;b ¼ 1  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

p and

MðpÞ ¼ 2 f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

 ^2p

ln f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³ 1

 q 

 ^p

 2pðaÞ^2pðln aÞ

" #

þ p²2  2¹³p

ðln bÞ² 2p²2  2¹³p

ðln aÞðln bÞ

h i

þ p²2  2¹³p

ðln aÞ² 2p 2  2 ¹³p

ðln aÞ

h i

þ 2p 2  2 ¹³p

ðln bÞ  2pðb^2pÞðln bÞ  2pðb^2pÞðln bÞ

h i

: ð23Þ

1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

The values of tau

The values of performance profile

p=1.1 p=2 p=4 p=5 p=10 p=20 p=50 p=100

Fig. 9. Performance profile of the reciprocals of optimal values byAlgorithm 3.1with different p.