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.

Ó2012 Elsevier Inc. All rights reserved.

1. Introduction

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

minx^TQxþc^Tx over x2S; ð1Þ

whereQis annnsymmetric matrix,cis a vector inRⁿandSis the binary discrete setf0;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!Rdefined as

/_FBða;bÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a²þb² q

ðaþbÞ 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 ðaþbÞ; ð3Þ

0096-3003/$ - see front matterÓ2012 Elsevier Inc. All rights reserved.

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

⇑Corresponding author.

E-mail addresses:jschen@math.ntnu.edu.tw(J.-S. Chen),697400011@ntnu.edu.tw(J.-F. Li),jwu_dora@mail.dlut.edu.cn(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

wherep>1 is an arbitrary fixed real number andkða;bÞk_pdenotes thep-norm ofða;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 ofða;bÞappeared in the FB function by a more generalp- norm. The function/_p is still an NCP-function, which naturally induces another NCP-functionw_p:R²!R_þgiven by

wpða;bÞ:¼1

2j/pða;bÞj²: ð4Þ

For any givenp>1, the functionw_pis shown to possess all favorable properties of the FB functionw_FB. Traditionally, in the continuation approach for BQP, one needs to utilize the fact that

x2 f0;1gⁿ()xi¼x²_i; i¼1;2;. . .;n: ð5Þ To the contrast, our proposed continuous optimization approach arises from the complementarity condition formulation of 01 vectorx2 f0;1gⁿ, which includes the equivalence(5)with redundant constraints

06xi61; i¼1;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 constraints16xi61;i¼1;2;. . .;n, respectively, to the objective function, and then construct a global smoothing function. Since the generalized Fischer–Burmeister functionw_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 thatS¼ f1;1gⁿ. By a transformationz¼ ðxþeÞ=2 for the variablexand the unit vector e inR, we can extend the conclusions to the caseS¼ 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 constraintt2 fa;bgwitha;b2Ris equivalent to a com- plementarity condition (or equilibrium constraint), i.e.,

t2 fa;bg () taP0; btP0; ðtaÞðtbÞ ¼0:

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

min fðxÞ

s:t: ð1þxi;1xiÞ ¼0; i¼1;2;. . .;n;

1þxiP0; 1xiP0; i¼1;2;. . .;n:

ð6Þ

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

ð1þxi;1xiÞ ¼0; 1þxiP0; 1xiP0; ()/ð1þxi;1xiÞ ¼0: ð7Þ Thus we reformulate the original BQP problem, which together with(6) and (7), as the following continuous optimization problem:

min fðxÞ

s:t: /ð1þxi;1xiÞ ¼0; i¼1;2;. . .;n

16xi61; i¼1;2;. . .;n:

ð8Þ

Accordingly, the global minimizer of (8) is the solution of (1). Note that although the box constraints 16xi61;i¼1;2;. . .;nin(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 ¼minfa;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

/pð1þxi;1xiÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j1þxij^pþ j1xij^p qp

2¼0; i¼1;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 fðxÞ

s:t: /pð1þxi;1xiÞ ¼0; i¼1;2;. . .;n

16xi61; i¼1;2;. . .;n:

ð9Þ

We also note that using the equivalence thatxi2 f1;1g ()x²_i ¼1 gives another another type of continuous optimization:

min fðxÞ

s:t: x²_i ¼1; i¼1;2;. . .;n

16xi61; i¼1;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/_pð1þx_i;1x_iÞ ¼0;i¼1;2;. . .;nhave incorporated the equivalent formulation x²_i ¼1;i¼1;2;. . .;n, ofx2 f1;1gwith its relaxation formulation16xi61;i¼1;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 fromRtoRþ. Suppose f;g satisfy (i) f⁰ðxÞ>0and g⁰ðxÞ<0for all x2 ða;bÞ,

(ii) f⁰⁰ðxÞ<0and g⁰⁰ðxÞ<0for all x2 ða;bÞ, (iii)ðfgÞ⁰ðaÞ<0and fðaÞPgðaÞ.

ThenðfgÞ⁰ðxÞ<0for all x2 ða;bÞ.

Proof. To achieve our result, we need to verify two things: (i)ðfgÞ⁰ðaÞ<0 and (ii)ðfgÞ⁰ðxÞis decreasing onx2 ða;bÞ. We pro- ceed these verifications as below.

(i) From the assumptions and the chain rule, it is clear that ðfgÞ⁰ðaÞ ¼f⁰ðaÞgðaÞ þfðaÞg⁰ðaÞ<0:

(ii) SinceðfgÞ⁰ðxÞ ¼f⁰ðxÞgðxÞ þfðxÞg⁰ðxÞ, we see that in order to showðfgÞ⁰ðxÞis decreasing onx2 ða;bÞ, it is enough to argue bothf⁰ðxÞgðxÞandfðxÞg⁰ðxÞare decreasing onða;bÞ. We look into the first term first. Note that

f⁰ðxÞgðxÞ

ð Þ⁰¼f⁰⁰ðxÞgðxÞ þf⁰ðxÞg⁰ðxÞ60 8x2 ða;bÞ;

becausef⁰⁰ðxÞ<0;gðxÞP0;f⁰ðxÞ>0 andg⁰ðxÞ<0. This claims thatf⁰ðxÞgðxÞis decreasing onx2 ða;bÞ. The decreasing of fðxÞg⁰ðxÞoverða;bÞcan 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. Letw_pbe defined as in(4). Then,w⁰⁰_pð1þt;1tÞis positive at t¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p for all pP2.

Proof. For symmetry, we only prove the case oft¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p . First, from direct computations and simplifying the expression ofw⁰⁰_p, we have

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

w⁰⁰_pð1þt;1tÞ ¼ ð1þtÞ^pþ ð1tÞ^p1p

ð1þtÞ²ð1tÞ²ð1þtÞ^pþ ð1tÞ^p2Fðp;tÞ; ð11Þ whereFðp;tÞ ¼f0ðp;tÞ½f1ðp;tÞ þf2ðp;tÞ þf3ðp;tÞ þf4ðp;tÞwith

f0ðp;tÞ ¼ ð1 þtÞ^pþ ð1tÞ^p1p; f1ðp;tÞ ¼ ð1tÞ²ðtþ1Þ^2p; f2ðp;tÞ ¼ ðtþ1Þ²ð1tÞ^2p;

f3ðp;tÞ ¼ ð2t²þ4p6Þðtþ1Þ^pð1tÞ^p; f4ðp;tÞ ¼ ð88pÞð1t²Þ^p:

Since the first term on the right side of(11)is always positive for allpP2, it suffices to show thatF p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

>0 for all pP2. However, it is very hard to claim this fact directly. Our strategy is to construct a functionA:R!Rsuch that

AðpÞ6F p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

8pP2: ð12Þ

The special feature forAðpÞis that it is easier to verifyAðpÞP0 for allpP2 so that our goal could be reached. Now, we pro- ceed the proof by carrying out the aforementioned two steps.

Step (1): Construct a functionAðÞsatisfying(12). Indeed, the functionFð;Þis composed off0;f1;f2;f3andf4, so for eachfi, we will construct a corresponding piecewise functionaisuch thataiðpÞ6fi p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1 p

fori¼0;1;2;3;4. Then, combining them together to build up the functionAðÞ. For making the reader understand more easier, we will give some pictures during the process of proof.

(i) First, we explain how to set upa0ðpÞ. Notice that the second derivative off0with respect topis positive att¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p for allpP2;f0is strictly convex att¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1 p

for allpP2 (the detailed arguments are provided in Appendix A).

Hence, we consider a real piecewise function defined as a0ðpÞ ¼

1

8ðp2Þ þ2²³ if 26p68 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p 6þ8ð2²³Þ;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p þ1 if pP8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1

p 6þ8ð2²³Þ:

8<

:

Fig. 1depicts the relation betweena0ðpÞandf0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

. Besides, the following facts a0ð2Þ ¼f0 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

lim

p!2^þa⁰₀ðpÞ< d dpf0 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

a⁰⁰₀ðpÞ ¼0< d² dp²f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

Fig. 1.The graphs ofa0andf0.

indicate the first part of functiona0ðpÞis less thanf0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

for 2<p68 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

6þ8ð2²³Þ. On the other hand, an- other fact

p!1limf0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 q

þ1

says that the second part of functiona0ðpÞis less than or equal tof0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

forpP8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

6þ8ð2²³Þ. Thus, we conclude that

a0ðpÞ6f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

8pP2:

(ii) Secondly, we consider a quadratic function defined as a1ðpÞ ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q ²

1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q ⁴

ln 1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

ðp1Þ² þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q ²

1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q ⁴

1ln 1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

: Fig. 2depicts the relation betweena1ðpÞandf1 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1

p

. Again, using the following facts a1ð2Þ ¼f1 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

; a⁰₁ð2Þ ¼ d

dpf1 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

a⁰⁰₁ðpÞ6 d² dp²f1 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

8pP2;

we immediately achieve a1ðpÞ6f1 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

8pP2:

(iii) Thirdly, we consider a function defined as

a2ðpÞ ¼

¹₅ðp2Þ þ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p ⁴

1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p ²

if 26p612þ20 2 ¹³2²³

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p 40ð2¹³Þ 4010ð2²³Þ

; 0

if pP12þ20 2 ¹³2²³

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p 40ð2¹³Þ 4010ð2²³Þ : 8>

>>

>>

>>

<

>>

>>

>>

>:

Fig. 2.The graphs ofa1andf1.

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

Fig. 3depicts the relation betweena2ðpÞandf2 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

. We observe that the functionf2is positive and convex on pP2, then the following facts

a2ð2Þ ¼f2 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

lim

p!2^þa⁰₂ðpÞ< d dpf2 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

a⁰⁰₂ðpÞ ¼0< d² dp²f2 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

8p>2 yielda2ðpÞ6f2 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1

p

for allpP2.

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

a3ðpÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 22¹³

p 2412ð2²³Þ

þ16ð2²³2¹³Þ 8

h i

p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

22¹³

p ð24ð2²³Þ 48Þ þ40ð2²³2¹³Þ þ20 if 26p6⁵

2; ₃₁⁴2¹³þ₃₁⁴

ð22¹³Þ⁵²pþ⁷²₃₁2¹³þ⁷²₃₁

ð22¹³Þ⁵² if ⁵₂6p618;

0 if pP18:

8>

>>

>>

><

>>

>>

>>

:

Fig. 4depicts the relation betweena3ðpÞandf3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

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

If 26p6⁵

2, we computef3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

¼2ð2¹³Þ 8þ4p

ð22¹³Þ^p. Moreover, we have d

dpf3 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

¼ ð22¹³Þ^p 4þ2ð2¹³Þ 8þ4p

lnð22¹³Þ

h i

;

d² dp²f3 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

¼ ð22¹³Þ^plnð22¹³Þ8þ2ð2¹³Þ 8þ4p

lnð22¹³Þ

h i

:

Then, the following facts a3ð2Þ ¼f3 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

a3 5

2 ¼f3 5 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

lim

p!2^þa⁰₃ðpÞ6 d dpf3 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

Fig. 3.The graphs ofa2andf2.

andf3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

being concave on 2; ⁵₂

implya3ðpÞ6f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

under this case.

If⁵₂6p618, using the facts that a3

5 2 ¼f3

5 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

lim

p!⁵₂^þ

a⁰₃ð Þp 6 d dpf3

5 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

; anda3ðpÞ ¼f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1 p

having only one solution atp¼⁵₂, we obtaina3ðpÞ6f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

under this case.

IfpP18, knowingf3ðpÞ>0 for allp, then it is clear thata3ðpÞ6f3 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

under this case.

(v) Finally, notice that the second derivative off4with respect topis positive att¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p for allpP^2þlnð22

1 3Þ lnð22¹3Þ , and negative for p6^2þlnð22

1 3Þ

lnð22¹³Þ , so f4 is strictly convex at t¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

for all pP^2þlnð22

1 3Þ

lnð22¹³Þ and strictly concave for all p6^2þlnð22

13Þ

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

a4ðpÞ ¼

¹⁵³₅₀p8ð22¹³Þ²þ¹⁵³₂₅ if 26p6⁵

2; ⁴⁹⁷₅₀þ16ð22¹³Þ²

h i

pþ⁵⁸³₂₅48ð22¹³Þ² if ⁵₂6p63;

¹³₁₀p¹³₅ if 36p6⁴⁹

13; ¹⁵₂ if pP⁴⁹₁₃: 8>

>>

>>

>>

><

>>

>>

>>

>>

:

Fig. 5depicts the relation betweena4ðpÞandf4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

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

For 26p6⁵

2, the following facts a4ð2Þ ¼f4 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

lim

p!2^þa⁰₄ðpÞ< d dpf4 2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

a⁰⁰₄ðpÞ ¼0< d² dp²f4 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

yield the first part of functiona4ðpÞis less thanf4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

under this case.

Fig. 4.The graphs ofa3andf3.

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

For⁵₂6p63, using the following facts a4ð3Þ<f4 3;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

p!3lima⁰₄ðpÞ> d dpf4 3;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

;

a⁰⁰₄ðpÞ ¼0< d² dp²f4 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

:

ð13Þ

We havea4ðpÞis less thanf4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

under this case.

For 36p6⁴⁹

13, we know that lim

p!3^þa⁰₄ðpÞ> d dpf4 3;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

:

This together with(13)givesa4ðpÞis less thanf4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

under this case.

ForpP⁴⁹₁₃, fromf4ðpÞbeing strictly convex for allp6^2þlnð22

1 3Þ

lnð22¹3Þ and being strictly concave for allpP^2þlnð22

1 3Þ

lnð22¹3Þ , we know d

dpf4

1þlnð22¹³Þ lnð22¹³Þ

!

¼0 and lim

p!1f4ðpÞ ¼0;

which lead tof4ðpÞ>¹⁵₂ for allp62. Thus,a4ðpÞ6f4 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

under this case.

Now, we are ready to define a functionA:R!Rsatisfying(12). As the mentioned idea, the function is defined by AðpÞ ¼a0ðpÞ½a1ðpÞ þa2ðpÞ þa3ðpÞ þa4ðpÞ:

According to our constructions ofaiðpÞ, it is clear thatAðpÞ6F p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

for allpP2.Fig. 6shows the relation between AðpÞandF p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1

p

.

Step (2): We will show thatAðpÞP0 for allpP2. Notice thatAðpÞis piecewise smooth, hence A⁰ðpÞis a piecewise function. Indeed, the expression ofA⁰ðpÞlooks very ugly and tedious, we display it Appendix B. Furthermore, we also present an approximate expression forA⁰ðpÞin Appendix C which helps us understand the structure ofA⁰ðpÞ. The key point is that from the expression of theA⁰ðpÞ, we can verify the following facts:

Að2Þ ¼0;

A 8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 q

6þ8ð2²³Þ

>A 5

2 >0;

and

A⁰ðpÞ<0 if p2 ð⁵₂;8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

6þ8ð2²³ÞÞ; A⁰ðpÞ>0 otherwise;

(

Fig. 5.The graphs ofa4andf4.

with the exception of points of discontinuity. Thus, we concludeAðpÞP0 for allpP2 and(12)is satisfied, which imply F p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1 p

P0 for allpP2. Then, the proof is complete. h

Lemma 2.3

(a) Let f be a convex function defied on a convex set C inRⁿand g be a nondecreasing convex function defined on an interval I in R. Suppose fðCÞ#I. Then, the composite function gf defined byðgfÞðxÞ ¼gðfðxÞÞis convex on C.

(b)Suppose/₁:U!Ris a twice continuously differentiable function with a compact setU2Rⁿand/₂:X!Ris a twice con- tinuously differentiable function such that the minimum eigenvalue of its Hessian matrixr²_xx/₂ðxÞis greater than

e

(>0) for all x2X, whereXU. Then there exists a constantb^>0such that/₁þb/₂is a strictly convex function onXforb>^b.

Proof

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

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

Proposition 2.1. Let/_p andw_pbe defined as in(3) and (4), respectively. Then, for any fixed pP2, the following hold.

(a) The function/_pð1þt;1tÞis strictly convex for all t2R.

(b) The functionw_pð1þt;1tÞis strictly convex for all t R ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

h i

. Proof

(a) It is known know that/_pis a convex function[4–6]. Note thatfis a composition of/_pand an affine function. Thus,fis 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 ofw_pð1þt;1tÞ, it is enough to show thatw_pð1þt;1tÞis strictly convex fortP ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p . To

proceed, we discuss two cases.

(i) If tP1, the function w_pð1þt;1tÞ can be regard as a composite function of /_pð1þt;1tÞ and hðÞ ¼ ðÞ². BecausehðÞis nondecreasing convex function on½0;1and/_pð1þt;1tÞis positive strictly convex fortP1, fromLemma 2.3, we obtainwð1þt;1tÞis strictly convex fortP2.

(ii) If 1>tP ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p , we know that

Fig. 6.The graphs ofAandF.

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

w⁰_pð1þt;1tÞ ¼ /pð1þt;1tÞ/⁰_pð1þt;1tÞ;

w⁰⁰_pð1þt;1tÞ ¼ /h ⁰_pð1þt;1tÞ/⁰_pð1þt;1tÞ /_pð1þt;1tÞ/⁰⁰_pð1þt;1tÞi :

Then, it suffices to show thatw⁰⁰_pð1þt;1tÞ<0 forpP2. To this end, we compute the third derivative of/_pð1þt;1tÞ with respect totand prove that it is negative. To see this,

/⁰⁰⁰_pð1þt;1tÞ ¼4ð1þtÞ^pþ ð1tÞ^p1pð1þtÞ^pð1tÞ^pðp1Þ

ð1þtÞ³ðt1Þ³ð1þtÞ^pþ ð1tÞ^p3 Tðp;tÞ; ð14Þ whereTis a real valued function defined by

Tðp;tÞ ¼ ð1þtÞ^pð2p13tÞ ð1tÞ^pð2pþ3t1Þ:

It is not hard to verify the first term of the right side of(14)is always negative for allpP2. Thus, we only need to show Tðp;tÞ>0 for allpP2 which is equivalent to verifyingTð2;tÞ>0 andTðp;tÞ>Tð2;tÞfor allp>2. These can be done as below.

(i) BecauseTð2;tÞ ¼6t6t³, it is clearTð2;tÞ>0.

(ii) To show thatTðp;tÞ>Tð2;tÞforp>2, we first argue that

ð1þtÞ^p>ð1tÞ^p1ð2pþ3t1Þ 8p>2; ð15Þ It is equivalent to show that ^ð1þtÞp

ð1tÞ^p1ð2pþ3t1Þis greater than 1 for allp>2. Therefore, we consider the derivative of the following function with respect topas follows:

d dp

ð1þtÞ^p

ð1tÞ^p1ð2pþ3t1Þ¼ ð1þtÞ^p

ð1tÞ^p1ð2pþ3t1Þ² ð1½ 3t2pÞlnð1tÞ þ ð2pþ3t1Þlnð1þtÞ 2: ð16Þ Observing both terms of the right side of(16) are positive for allp>2 and using ^ð1þtÞp

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

2p13t>1t 8p>2: ð17Þ

Combining(15) and (17), we haveTðp;tÞPTð2;tÞ. Hence, /⁰⁰⁰_pð1þt;1tÞ<0 8pP2:

Then, applyingLemma 2.1gives the desired result for which we setfðtÞ ¼ /_pð1þt;1tÞandgðtÞ ¼/⁰_pð1þt;1tÞ. h The result ofProposition 2.1(b) could be improved under some sense. More specifically, the interval wherew_pð1þt;1tÞ is strictly convex varies as long aspchanges. We originally wish to figure out the exact interval wherew_pð1þt;1tÞis strictly convex for eachp. However, it is very hard to find a closed form dependingpto 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 ofw_pð1þt;1tÞfor differentp.

may be not possible in our opinion). To compromise, we try to find such an appropriate common interval for allpP2 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_pð1þxi;1xiÞfor the equality constraints and the logarithmic barrierPn

i¼1½lnð1þxiÞ þlnð1xiÞof the box con- straints into the(9). Construct a global smoothing function

/ðx;

a

;

s

Þ ¼fðxÞ þ

a

^X

n

i¼1

w_pð1þxi;1xiÞ

s

^X

n

i¼1

lnð1þxiÞ þlnð1xiÞ

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

Þonð1;1Þⁿwhen the barrier parameter is large enough, and the strictly convexity of function/ðx;

a

;

s

Þin a large subset of its domain for all

s

>0.

Proposition 3.1. Let/ðx;

a

;

s

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

(a) There exists a constant^

s

>0such that if

s

>^

s

and

a

>0;/ðx;

a

;

s

Þis strictly convex onð1;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 x2 ð1;1Þⁿj jx_ij>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

;i¼1;2;. . .;n

n o

.

Proof

(a) LetX¼ ð1;1Þⁿand denote /_aðxÞ:¼fðxÞ þ

a

^X

n

i¼1

w_pð1þxi;1xiÞ;

/bðxÞ:¼ Xⁿ

i¼1

½lnð1þxiÞ þlnð1xiÞ:

Then the expression of the Hessian matrix of/_bðxÞat anyx2Xis given by r²xx/bðxÞ ¼diag 1

ð1x1Þ²þ 1

ð1þx1Þ²; ; 1

ð1xnÞ²þ 1 ð1þxnÞ²

!

;

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 ofw_pð1þt;1tÞwith a fixedp.

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

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

1 ð1x_iÞ²þ_ð1þx¹

iÞ² has minimum 2 at point xi¼0, and so every diagonal element of r²_xx/_bðxÞis at least 2. Thus, by letting U¼ ½1;1ⁿ;

e

¼2 and usingLemma 2.3(b) yield the desired result.

(b) Set/_a¼fðxÞand /_b¼Xⁿ

i¼1

w_pð1þxi;1xiÞ

s a

Xⁿ

i¼1

½lnð1þxiÞ þlnð1xiÞ: From the proof ofLemma 2.2, it follows that

r²xx

Xⁿ

i¼1

w_pð1xi;1xiÞ

!

¼diagw⁰⁰_pð1x1;1þx1Þ; ;w⁰⁰_pð1xn;1þxnÞ

;

wherew⁰⁰_pð1xi;1þxiÞcan be found in(11). Now takingfðtÞ ¼ /_pð1þt;1tÞ;gðtÞ ¼/⁰_pð1þt;1tÞand applying the proof (ii) ofLemma 2.1, we have

w⁰⁰_pð1t;1þtÞ>w⁰⁰₂ð1t;1þtÞ 8p>2:

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

w⁰⁰_bð1t;1þtÞ ¼2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2t²þ2Þ³ q

8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2t²þ2Þ³

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

Therefore, the above two inequalities imply

w⁰⁰_pð1t;1þtÞ>0:0004 8jtj>0:51 and 8pP2:

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

i¼1½lnð1þxiÞ þlnð1xiÞis positive definite, we obtain that every diagonal element ofr²_xx/_bis at least 0:0004. Now taking

U¼ ½1;1ⁿ; X¼D and

e

¼0:0004

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

Þto 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 sequencef

a

kgand a decreasing barrier parameter sequencef

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

i¼1½lnð1þxiÞ þlnð1xiÞwill become weak, but the strict con- vexity of/ðx;

a

k;

s

kÞcan still be guaranteed due to the increasing of the penalty parameter

a

k. This means that for eachk2IN, 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;1Þand

>0. Select a starting point^x⁰and setk¼0.

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

Step 2 If ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i¼1w_pð1x^k_i;1x^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;k¼kþ1 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

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

(a) For each k2IN, 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 ofx^kfor eachk2IN. LetX1¼ ³₄;³₄

. Since/ðx;

a

k;

s

kÞis continuous andX1is a compact set, there exist two real numbersL1andU1such that

L16/ðx;

a

k;

s

kÞ6U1 8x2X1:

On the other hand, we note that/ðx;

a

_k;

s

kÞ ! þ1whenxi₀!1orxi₀!1^þfor somei02 f1;2;. . .;ng. Hence, the continuity of function/ðx;

a

k;

s

kÞimplies that there exists andwith 0<d<1=4 such that

/ðx;

a

k;

s

kÞPU1 8x2 ð1;ð 1þd [ ½1d;1ÞÞⁿ: ð20Þ LetX¼ ½1þd;1dⁿ. Again,/ðx;

a

k;

s

kÞbeing continuous on a compact setXimplies that there exists an^x2Xsuch that for eachk2IN

/ð^x;

a

k;

s

kÞ6/ðx;

a

k;

s

kÞ 8x2X:

Moreover, due toX1#X, we know

/ð^x;

a

k;

s

kÞ6U1: ð21Þ

Combining(20) and (21)yields that

/ð^x;

a

k;

s

kÞ6/ðx;

a

k;

s

kÞ 8x2 ð1;1ÞⁿnX:

Thus, together with(20), it shows that^xis exactly the desired solutionx^k.

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

a

k;

s

kÞis strictly convex onð1;1Þⁿ. Hencex^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 asx^ksatisfies the following criterion:

rx/ðx^k;

a

k;

s

kÞ

65:0e3:

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

a

0¼0;

r

1¼2;

r

2¼0:5;

e

¼1:0e3;

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^x⁰¼0:9ð1;1;. . .;1Þ^T2Rⁿis used for all test problems. To obtain an integer solutionx from the final iterate point^x ofAlgorithm 3.1, we let

x_i ¼ 1 ifj^x_i þ1j61:0e2 1 ifj^x_i 1j61:0e2 (

fori¼1;2;. . .;n:

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

s:t: zi2 f0;1g; i¼1;2;. . .;n:

To solve these problems with Algorithm 3.1, we use the formula z¼ ðxþeÞ=z to transform them into the following formulation

min ¹₄x^TQx¹₂x^TQe¹₄e^TQe s:t: xi2 f1;1g; i¼1;2;. . .;n:

The optimal values generated byAlgorithm 3.1with differentp(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 differentp, we adopt the performance profile introduced in [7]as a means. In particular, we regard Algorithm 3.1corresponding to a pas a solver and assume that there arenssolvers andnjtest problems from the OR-Library collectionJ. We are interested in using the optimal values calculated byAlgorithm 3.1as performance measure for differentp. For each problemjand solvers, let

tj;s:¼ the optimal value of problemjby solvers;

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 problemjby solverswith the best performance by any one of thenssolvers on this prob- lem, i.e., we employ the performance ratio

rj;s:¼

l

_j;s

minf

l

_j;s:s2 Sg¼maxftj;s:s2 Sg tj;s

;

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

q

_sð

s

Þ:¼1 nj

sizefj2 J :rj;s6

s

g;

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

Fig. 9shows the performance profile of the reciprocals of optimal values obtained byAlgorithm 3.1in the range of½1;1:04 for eight solvers on 50 test problems. The eight solvers correspond toAlgorithm 3.1withp¼1:1;p¼2;p¼4;p¼5;p¼10

;p¼20;p¼50 andp¼100, respectively. From this figure, we see thatAlgorithm 3.1are considerably efficient no matter which value ofpis chosen. In fact,Algorithm 3.1with the aforementionedpvalues can solve all the 50 test problems except forp¼5;20;100. Moreover,Algorithm 3.1withp¼4 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,p¼1:1 and p¼2 have a comparable performance withp¼4, please refer to Appendix D for more detailed numerical reports.

Appendix A

Here is the proof of the strictly convexity off₀ p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1 p

for allpP2.

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

dp²f0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

>0 for allp>2. In fact, d²

dp²f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

¼f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q ln f0ðp; ffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1 p

Þ^p

p² þa^plnðaÞ þb^plnðbÞ p f0ðp; ffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1 p

Þ

^p

2 64

3 75

2

þ 1

p³ f0 p; ffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p

^2p1MðpÞ;

ð22Þ wherea¼1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2¹³1 p

;b¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

p and

MðpÞ ¼ 2 f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

^2p

ln f0 p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2¹³1

q

^p

2pðaÞ^2pðlnaÞ

" #

þ p²22¹³p

ðlnbÞ²2p²22¹³p

ðlnaÞðlnbÞ

h i

þ p²22¹³p

ðlnaÞ²2p22¹³p

ðlnaÞ

h i

þ 2p22¹³p

ðlnbÞ 2pðb^2pÞðlnbÞ 2pðb^2pÞðlnbÞ

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 differentp.