Some Special Cases of Optimizing over the efficient set of generalized convex multiobjective programming problem

Some Special Cases of Optimizing overthe Efficient Set of a Generalized Convex Multiobjective Programming Problem Tran Ngoc Thang School of Applied Mathematics and Informatics Hanoi Univ

Some Special Cases of Optimizing over

the Efficient Set of a Generalized Convex

Multiobjective Programming Problem

Tran Ngoc Thang

School of Applied Mathematics and Informatics

Hanoi University of Science and Technology

Email: thang.tranngoc@hust.edu.vn

Tran Thi Hue

Faculty of Management Information System The Banking Academy of Vietnam Email: huett@bav.edu.vn

Abstract—Optimizing over the efficient set is a very hard

and interesting task in global optimization, in which local

optima are in general different from global optima At the

same time, this problem has some important applications

in finance, economics, engineering, and other fields In this

article, we investigate some special cases of optimization

problems over the efficient set of a generalized convex

multiobjective programming problem Preliminary

com-putational experiments are reported and show that the

proposed algorithms can work well.

AMS Subject Classification: 90 C29; 90 C26

Keywords: Global optimization, Efficient set,

General-ized convexity, Multiobjective programming problem

I INTRODUCTION

The generalized convex multiobjective programming

problem (GM OP ) is given as follows

Minf (x) = (f1(x), , fm(x))T s.t x ∈ X ,

where X ⊂ Rn is a nonempty convex compact set

and fi, i = 1, , m, are generalized convex functions

on X In the case m = 2, problem (GM OP ) is

called a generalized convex biobjective programming

problem The special cases of problem(GM OP ), where

fi, i = 1, , m are linear (resp convex), called a linear multiobjective programming problem (resp a con-vex multiobjective programming problem), have received

special attention in the literature (see the survey in [21] and references therein) However, to the best of our knowledge, there is little result about numerical methods

in the nonconvex case where fi, i = 1, , m, are nonconvex (see [4], [16], )

The main problem in this paper is formulated as

whereΦ : X → R is a continuous function and XE is the efficient solution set for problem(GM OP ), i.e

XE= {x0

∈ X | 6 ∃x ∈ X : f (x0

) ≥ f (x), f (x0

) 6= f (x)}

As usual, the notationy1 ≥ y2, where y1, y2

∈ Rm, is used to indicatey1

i ≥ y2

i for alli = 1, , m

It is well-known that, in general, the set XE is non-convex and given implicitly as the form of a standard mathematical programming problem, even in the case

m = 2, the objective functions f1, f2 are linear and the feasible set X is polyhedral Hence, problem (PX)

is a global optimization problem and belongs to

NP-hard problem class This problem has many applications

in economics, finance, engineering, and other fields

Recently this problem has a great deal of attention

from researchers (for instance, see [1], [5], [6], [7],

[8], [12], [15], [16] [17], [19] and references therein)

Like problem (GM OP ), there is only few numerical

algorithms to solve problem (PX) in the nonconvex

case (see [1], [16], ) In this article, simple convex

programming procedures are proposed for solving three

special cases of problem(PX) where XE is the efficient

solution set for problem (GM OP ) in the nonconvex

case These special-case procedures require quite little

computational effort in comparison to ones required by

algorithms for the general problem(PX)

In Section 2, the theoretical preliminaries are presented

to analyze three special cases of optimization over the

efficient solution set of problem (GM OP ) Section 3

proposes the algorithms to solve these cases, including

some computational experiments to illustrate the

algo-rithms Some conclusions are given in the last section

II THEORETICALPRELIMINARIES

First, recall that a differentiable function h : X → R

is called pseudoconvex on X if

h∇h(x2

), x1− x2i ≥ 0 ⇒ h(x1

) − h(x2

) ≥ 0 for allx1, x2∈ X For example, by Proposition 5.20 in

[2], the fractional functionr(x)/l(x), where r : Rn→ R

is convex on X and l : Rn → R is linear such that

l(x) > 0 for all x ∈ X , is a pseudoconvex function

By the definition in [10, tr 132], a function h is called

quasiconvex on X if

h(x1

) − h(x2

) ≤ 0 ⇒ h(λx1

+ (1 − λ)x2

) ≤ h(x2

), for all x1, x2 ∈ X and 0 ≤ λ ≤ 1 If h is quasiconvex

theng := −h is quasiconcave.

In the caseh is differentiable, if h is quasiconvex on

X , we have h(x1) − h(x2) ≤ 0 ⇒ h∇h(x2), x1− x2

for allx1, x2∈ X , where ∇h(x2) is the gradient vector

ofh at x2 (see Theorem 9.1.4 in [10])

A vector function f (x) = (f1(x), , fm(x))T is

called convex (resp., pseudoconvex, quasiconvex) onX if its component functionsfi(x), i = 1, , m, are convex (resp., pseudoconvex, quasiconvex) functions onX Recall that a vector function f is called scalarly pseudoconvex on X if Pm

i=1λifi is pseudoconvex on

X for every λ = (λ1, , λm) ≥ 0 (see [16])

By definition, iff is convex then it is scalarly pseudo-convex, andf is scalarly pseudoconvex then it is pseudo-convex Hence, the convex multiobjective programming problem is a special case of problem(GM OP ).

Example II.1 Consider the vector function f (x) over the setX = {x ∈ R2| Ax ≥ b, x ≥ 0}, where

f (x) = −x

2− 0.6x1+ 0.5x2

x1− x2− 2 ,

x2+ x1

x2− x1+ 2



and

A =







−2 −1





, b =







2 6

−10







We have

λTf (x) = λ1(x

2+ 0.6x1− 0.5x2) + λ2(x2+ x1)

It is easily seen that

r(x) = λ1(x2

1+ 0.6x1− 0.5x2) + λ2(x2

2+ x1)

is convex becauseλ1, λ2≥ 0, and l(x) = x2−x1+2 > 0

for all x ∈ X Therefore, by Proposition 5.20 in [2], the functionλTf (x) = r(x)/l(x) is pseudoconvex , i.e f (x)

is scalarly pseudoconvex.

Let

Y = {y ∈ Rm|y = f (x) for some x ∈ X }

As usual, the set Y is said to be the outcome set for

problem(GM OP )

Let yI

i = min{yi | y ∈ Y}, i = 1, , m It is

clear that yI

i is also the optimal value of the following

programming problem

min fi(x) s.t x ∈ X (Pi) For each i ∈ {1, , m}, if fi is pseudoconvex, we can

apply convex programming algorithms to solve problem

(Pi) (Remark 2.3 in [3])

The point yI = (yI

1, , yI

m) is called the ideal point

of the set Y Notice that the ideal point yI need not

belong toY

Fig 1 The ideal point y I

Consider a generalized convex biobjective

program-ming problem

Minf (x) = (f1(x), f2(x))T s.t x ∈ X , (GBOP )

where X := {x ∈ Rn | gj(x) ≤ 0, j = 1, , k},

the functions gj(x), j = 1, , k, are differentiable

quasiconvex on Rm and the objective function f is

scalarly pseudoconvex on X

Now we will describe the three sets of conditions

associated with three special cases of problem (PX)

under consideration

Case 1 The feasible set XE is the efficient solution

set of problem (GM OP ), the ideal point yI belongs

to the outcome setY and the objective function Φ(x) of problem(PX) is pseudoconvex on X

Case 2 The feasible setXE is the efficient solution set

of problem (GBOP ) and the objective function Φ(x) has the form asΦ(x) = ϕ(f (x)) where ϕ : Y → R and

with λ = (λ1, λ2)T

∈ R2 This case could happen

in certain common situations, for instance, when the objective function of problem(PX) represents the linear composition of the criteria fi(x), i ∈ {1, 2} with the weighted coefficientsλi, i ∈ {1, 2}

Case 3 The feasible setXE is the efficient solution set

of problem (GBOP ) and the objective function Φ(x) has the form as Φ(x) = ϕ(f (x)) where ϕ : Y → R is quasiconcave and decreasing monotonic

Let Q ⊂ Rm be a nonempty set A point q0 ∈ Q

is called an efficient point of the set Q if there is no

q ∈ Q such that q0 ≥ q and q0 6= q The set of all efficient points ofQ is denoted by MinQ It is clear that

a point q0 ∈ MinQ if Q ∩ (q0

− Rm +) = {q0}, where

Rm+ = {y ∈ Rm|yi≥ 0, i = 1, , m}

Since the functions fi, i = 1, , m, are continuous and X ⊂ Rn is a nonempty compact set, the outcome setY is also compact set in Rm Therefore, the efficient setMinY is nonempty [9] Let

The setYEis called the efficient outcome set for problem

(GM OP ) By definition, it is easy to see that

The relationship between the efficient solution setXE

and the efficient setMinY is described as follows

Proposition II.1.

i) For anyy0∈ MinY, if x0∈ X satisfies f (x0) ≤ y0, thenx0∈ X .

ii) For anyx ∈ XE, ify = f (x ), then y ∈ MinY.

Proof: i) Since y0∈ MinY, by definition, we have

(y0

− Rm

+) ∩ Y = y0

Moreover,f (x0

) ∈ (y0

− Rm +) and

f (x0) ∈ Y because x0∈ X and f (x0) ≤ y0 Therefore,

f (x0) = y0 ∈ MinY Combined this fact with (3) and

(4), we implyx0∈ XE

ii) This fact follows immediately from (3) and (4)

Fig 2 The efficient set MinY

Let

Z = Y + Rm

+ = {z ∈ Rm | z ≥ y for some y ∈ Y}

It is clear that Z is a nonempty, full-dimension closed

set but it is nonconvex in general

Fig 3 The set Z

The following interesting property ofZ (see Theorem 3.2 in [20]) will be used in the sequel

Proposition II.2. MinZ = MinY.

Now we will consider the first special case where the ideal pointyI belongs to the outcome set Y

Proposition II.3 If yI ∈ Y then MinY = {yI} Proof: SinceyI ∈ Y, there exists xI ∈ X such that

yI = f (xI), i.e yI

i = fi(xI) for all i = 1, 2, , m For each i ∈ {1, 2, , m}, since yI

i is the optimal value

of problem (Pi), xI is an optimal solution of problem (Pi) Hence, xI ∈ Argmin{fi(x) | x ∈ X } for all

i = 1, , m By definition, xI is an efficient solution

of problem (GM OP ), i.e xI ∈ XE From Proposition II.1(ii), it implies yI ∈ MinY Since yI

i is the optimal value of problem(Pi) for i = 1, 2, , m, by definition

of efficient points,yI is the only efficient point ofY Let

Xid= {x ∈ X | fi(x) ≤ yI

i, i = 1, 2, , m}

By [10, Theorem 9.3.5], for each i = 1, 2, , m,

if fi is the pseudoconvex function, fi is quasiconvex Therefore,Xidis a convex set because every lower level set of continuous quasiconvex functions is convex The following assertion provides a property to detect whether

yI belongs to Y

Proposition II.4 IfXid is not empty then yI ∈ Y and

XE= Xid Otherwise,yI does not belong to Y Proof: By definition, if Xid = ∅, yI 6∈ Z Since

Y ⊆ Z, yI 6∈ Y Otherwise, Xid is not empty, yI ∈ Z Therefore, yI ∈ Y because yI

i is the optimal value of problem (Pi) for i = 1, 2, , m By Proposition II.3 and Proposition II.1(i), we getMinY = {yI} and XE= {x ∈ X | f (x) ≤ yI} = Xid

In the next two cases, we consider problem (PX) where X is the efficient solution set of problem

(GBOP ) and Φ(x) has the form as Φ(x) = ϕ(f (x))

where ϕ : Y → R Then the outcome-space

reformula-tion of problem (PX) can be given by

Combining (4) and Proposition II.2, problem (PY) can

be rewritten as follows

Therefore, instead of solving problem (PY), we solve

problem(PZ)

Proposition II.5 If f is scalarly pseudoconvex, the set

Z = f (X) + R2

+ is a convex set inR2.

Proof: Letw be an arbitrary point in the boundary¯

∂Z of the set Z By geometry, there exists ¯y ∈ MinZ

such that y ≤ ¯¯ w From Proposition II.2, we imply that

¯

y ∈ MinY Hence, by Proposition II.1(ii), there exists

¯

x ∈ X such that ¯y = f (¯x) and ¯x is an efficient solution

of problem(GM OP )

Let J = {j ∈ {1, , k} | gj(¯x) = 0} and s = |J|

For any vector a ∈ Rs, we denote aJ := {aj, j ∈ J}

Since x ∈ X¯ E, by [11, Corollary 3.1.6], there exist a

vector ¯λ ∈ R2

+\ {0} and a vector ¯µJ ∈ Rs

+, such that

¯

λT∇f (¯x) + ¯µT∇gJ(¯x) = 0 which means

¯

λT∇f (¯x) = −¯µT

J∇gJ(¯x) (5) Since µ¯J ≥ 0, gJ(x) ≤ 0 for all x ∈ X and

gJ(¯x) = 0, we have ¯µTgJ(x) − ¯µTgJ(¯x) ≤ 0 for

all x ∈ X Combined this fact with the condition

gj, j = 1, , k, are differentiable quasiconvex and (1),

we get µT

J∇gJ(¯x), x − ¯x ≤ 0 for all x ∈ X Thus, by

(5), one has ¯λT∇f (¯x), x − ¯x ≥ 0 or

T

f (¯x)
, x − ¯x ≥ 0 ∀x ∈ X (6) Moreover, ¯λTf is pseudoconvex on X because f is

scalarly pseudoconvex onX Therefore, (6) implies that

¯

λTf (x) − ¯λTf (¯x) ≥ 0 ∀x ∈ X ,

i.e ¯λ, y − ¯y ≥ 0 for all y ∈ Y For i = 1, 2, set ˆ

λi= ¯λi if ¯i= ¯wi and ˆλi = 0 if ¯yi 6= ¯wi It is easy to check that

Dˆλ, ¯w − ¯yE= 0 and ˆλ ≥ 0, ˆλ 6= 0 (7) Hence, D ˆλ, y − ¯wE ≥ 0 for all y ∈ Z Set H( ¯w) = {y ∈ Rm | Dˆλ, y − ¯wE

≥ 0} Then Z ⊂ H( ¯w) for all

¯

w ∈ ∂Z By [14, Theorem 6.20], we imply that Z is a convex set

Since Z is a nonempty convex subset in R2 by Proposition II.5, it is well known [13] that the efficient set MinZ is homeomorphic to a nonempty closed interval

ofR By geometry, it is easily seen that the problem

min{y2 : y ∈ Z, y1= yI

1} (PS) has a unique optimal solutionyS and the problem

min{y1 : y ∈ Z, y2= yI

2} (PE) has a unique optimal solution yE Since Z is convex, problems(PS) and (PE) are convex programming prob-lems If yI ∈ Y then, by Propositions II.3 and II.4, yI

is the only optimal solution to problem(PZ) and Xid is the optimal solution set of problem(PX)

Fig 4 The efficient curve MinZ

If yI 6∈ Y then yS 6= yE and the efficient setMinZ

is a curve on the boundary of Z with starting point yS

and the end pointy such that

y1E> yS1 andyS2 > yE2 (8) Note that we also get the efficient solutionsxS, xE∈ XE

such that yS = f (xS) and yE = f (xE) while solving

problems (PS) and (PE) For the convenience, xS, xE

is called to be the efficient solutions respect to yS, yE,

respectively

In the second case, we consider problem(PZ) where

ϕ(y) = hλ, yi Direct computation shows that the

equa-tion of the line throughyS andyE ishc, yi = α, where







y E

1 −y S

1, 1

y S

2 −y E 2

 ,

E 1

y E

1 −y S 1

E 2

y S

2 −y E 2

(9)

From (8), it is easily seen that the vector c is strictly

positive Now, let

˜

Z = {y ∈ Z | hc, yi ≤ α}

and

Γ = ∂ ˜Z \ (yS

, yE), where (yS, yE) = {y = tyS + (1 − t)yE | 0 < t < 1}

and∂ ˜Z is the boundary of the set ˜Z

Fig 5 The convex set ˜ Z

It is clear that ˜Z is a compact convex set because Z

is convex By the definition and geometry, we can see

thatΓ contains the set of all extreme points of ˜Z and

Consider the following convex problem

minhλ, yi s.t y ∈ ˜Z (CP0) that has the explicit reformulation as follows

s.t f (x) − y ≤ 0

x ∈ X ,

hc, yi ≤ α,

(CP1)

where the vector c ∈ R2 and the real number α is determined by (9)

Proposition II.6 Suppose that (x∗, y∗) is an optimal solution of problem (CP1) Then x∗ is an optimal solution of problem(PX).

Proof: It is well known that a convex programming

problem with the linear objective function has an optimal solution which belongs to the extreme point set of the feasible solution set [18] Therefore, problem(CP0) has

an optimal solution y∗ ∈ Γ This fact and (10) implies thaty∗∈ MinZ Since MinZ ⊂ ˜Z, it implies that y∗ is

an optimal solution of problem(PZ)

Since MinZ = YE = MinY, by definition, we have

hλ, y∗i ≤ hλ, yi for all y ∈ YE andy∗∈ MinY Then

hλ, y∗i ≤ hλ, f (x)i, ∀x ∈ XE (11) Since (x∗, y∗) is a feasible solution of problem (CP1),

we have f (x∗) ≤ y∗ By Proposition II.1, x∗ ∈ XE Furthermore, we have f (x∗) ∈ Y and y∗ ∈ MinY The definition of efficient points infers thaty∗= f (x∗) Combining this fact and (11), we getΦ(x) ≥ Φ(x∗) for all x ∈ XE which means x∗ is an optimal solution of problem(PX)

In the last case, we consider problem(PZ) where the function ϕ : Y → R is quasiconcave and decreasing monotonic The following assertion presents the special property of the optimal solution to problem(PZ)

Fig 6 The illustration to Case 3

Proposition II.7 If the function ϕ is quasiconcave

and decreasing monotonic then the optimal solution of

problem (PZ) is attained at either yS or yE.

Proof: Let Z△ = conv{yS, yI, yE}, where

conv{yS, yI, yE} stands for the convex hull of the points

{yS, yI, yE} Since MinZ ⊂ Z△, we have

min{ϕ(y) | y ∈ MinZ} ≥ min{ϕ(y) | y ∈ Z△} (12)

It is obvious that the optimal solution of the problem

min{ϕ(y) | y ∈ Z△}, where the objective function ϕ

is quasiconcave, belongs to the extreme point set ofZ△

[18] Therefore,

Argmin{ϕ(y) | y ∈ Z△} ∈ {yS, yI, yE}

Moreover, sinceϕ is also decreasing, we have

Argmin{ϕ(y) | y ∈ Z△} ∈ {yS, yE}

Since yS, yE∈ MinZ, this fact and (12) imply

min{ϕ(y) | y ∈ MinZ} = min{ϕ(y) | y ∈ Z△}

andArgmin{ϕ(y) | y ∈ MinZ} ∈ {yS, yE}

III PROCEDURES ANDCOMPUTINGEXPERIMENTS

Case 1 The feasible set XE is the efficient solution

set of problem (GM OP ), the ideal point yI belongs

to the outcome set Y and the objective function Φ(x) of

problem (P ) is pseudoconvex on X

By Proposition II.4, to detect whether the ideal point

yI belongs to Y and solve problem (PX) in this case,

we solve the following problem

min Φ(x) s.t x ∈ Xid (CPid) Since Φ(x) is pseudoconvex on X and Xid is convex,

we can apply convex programming algorithms to solve problem(CPid) (Remark 2.3 in [3]) The procedure for this case is described as follows

Procedure 1.

Step 1 For eachi = 1, , m, find the optimal value yI

i

of problem(Pi)

Step 2 Solve problem(CPid)

If problem(CPid) is not feasible Then STOP (Case 1

does not apply)

Else Find an optimal solution x∗ to problem (CPid) STOP (x∗ is an optimal solution to problem(PX)) Below we present a numerical example to illustrate Procedure 1

Example III.1 Consider the problem(PX), where XE

is the efficient solution set to the following problem

Min (f1(x), f2(x)) = (0.5x2

1− x1+ 0.3x2, x2

2+ x1)

s.t. x2

1+ x2

2− 4x1− 4x2≤ −6

−x1+ x2≥ α

x1+ x2≥ 2

x1≥ 1

andΦ(x) = min{0.5x2

+ x2+ 0.2; 2x2− 4.6x1+ 5.8}.

• In the case α = 0:

Step 1 Solving problems(P1) and (P2), we obtain the ideal pointyI = (−0.2000, 2.0000).

Step 2 Solving problem(CPid), we can find an optimal solution x∗ = (1.0000, 1.0000) Then x∗ is the optimal solution to problem (PX) and Φ(x∗) = 1.7000 is the optimal value of problem(P )

• In the case α = −1:

Step 1 Solving problems (P1) and (P2), we obtain the

ideal pointyI = (−0.2299, 1.8917).

Step 2 Solving problem (CPid), we can find that it is

not feasible It means that the ideal point yI does not

belong to Y.

Case 2 The feasible setXEis the efficient solution set of

problem (GBOP ) and the objective function Φ(x) has

the form as (2).

In this case, the procedure for solving problem(PX)

is established by Proposition II.4 and Proposition II.6

Recall that if yI ∈ Y, Xid = Argmin(PX) Therefore,

we can obtain an optimal solution of problem(PX) by

solving the following convex programming problem

min he, xi s.t x ∈ Xid, (CPid

2 ) wheree = (1, , 1) ∈ Rn

Procedure 2.

Step 1 For each i = 1, 2, find the optimal value yI

i of problem(Pi)

Step 2 Solve problem(CPid

2 )

If problem (CPid

2 ) is not feasible Then Go to Step 3

(yI 6∈ Y)

Else Find an optimal solutionx∗to the problem(CPid

2 )

STOP (x∗ is an optimal solution to problem(PX))

Step 3 Solve problem(PS) and problem (PE) to find the

efficient pointsyS, yE and the efficient solutionsxS, xE

respect to yS, yE, respectively

Step 4 Solve problem(CP1) to find an optimal solution

(x∗, y∗) STOP (x∗ is an optimal solution to(PX))

Below are some examples to illustrate Procedure 2

Example III.2 Consider problem (PX), where XE is

the efficient solution set to problem (GBOP ) with

f (x) = x2

+ 1, f (x) = (x − 3)2

+ 1,

X =x ∈ R | (x1− 1) + (x2− 2) ≤ 1, 2x1− x2≤ 1 ,

andΦ(x) = λ1f1(x) + λ2f2(x).

It is easily seen that the function f is scalarly pseudo-convex on X because f1, f2are convex on X Therefore,

we can apply Procedure 2 to solve this problem.

Step 1 The optimal value of the problems(P1) and (P2), respectively, isyI

1= 1.0000 and yI

2= 1.0000.

Step 2 Solving problem (CPid

2 ), we can find that it is not feasible Then go to Step 3.

Step 3 Solve problems(PS) and (PE) to obtain

yS = (1.9412, 1.0000), yE= (1.0000, 1.9326)

and

xS = (0.9612, 2.9991), xE= (0.0011, 2.0435)

respect to yS, yE, respectively.

Step 4 For each λ = (λ1, λ2) ∈ R2

, solve problem

(CP1

) to find the optimal solution (x∗, y∗) Then x∗ is

an optimal solution and Φ(x∗) is the optimal value of

(PX) The computational results are shown in Table I.

(0.0, 1.0) (0.9612, 2.9991) (1.9412, 1.0000) 1.0000 (0.2, 0.8) (0.4460, 2.8325) (1.1989, 1.0281) 1.0622 (0.5, 0.5) (0.2929, 2.7071) (1.0858, 1.0858) 1.0858 (0.8, 0.2) (0.1675, 2.5540) (1.0208, 1.1989) 1.0622 (1.0, 0.0) (0.0011, 2.0435) (1.0000, 1.9326) 1.0000 (−0.2, 0.8) (0.9654, 2.9992) (1.9412, 1.0000) 0.4118 (0.8, −0.2) (0.0011, 2.0435) (1.0000, 1.9326) 0.4146

TABLE I

C OMPUTATIONAL RESULTS OF E XAMPLE III.2

Example III.3 Consider problem(PX), where Φ(x) =

λ1f1(x) + λ2f2(x) and XE is the efficient solution set

to problem (GBOP ) with

f (x) = −x

2− 0.6x1+ 0.5x2

x2+ x1



andX =x ∈ R | Ax ≥ b, x ≥ 0 , where

A =









−2 −1







 , b =









2 6

−10









By Example II.1, it is verified that f is scalarly

pseudoconvex on X Therefore, we can apply Procedure

2 to solve this problem.

Step 1 The optimal value of the problems(P1) and (P2),

respectively, is yI

1= −0.4167 and yI

2 = 1.5400.

Step 2 Solving problem (CPid

2 ), we can find that it is not feasible Then go to Step 3.

Step 3 Solve problems(PS) and (PE) to obtain

yS= (−0.2000, 1.5400), yE= (−0.4167, 8.3332)

and

xS = (0.4000, 2.4000), xE = (0.0000, 9.9997)

respect to yS, yE, respectively.

Step 4 For each λ = (λ1, λ2) ∈ R2, solve problem

(CP1) to find the optimal solution (x∗, y∗) Then x∗ is

an optimal solution and Φ(x∗) is the optimal value of

(PX) The computational results are shown in Table II.

(0.0, 1.0) (0.4000, 2.4000) (−0.2000, 1.5400) 1.5400

(0.2, 0.8) (0.4000, 2.4000) (−0.2000, 1.5400) 1.1920

(0.5, 0.5) (0.4000, 2.4000) (−0.2000, 1.5400) 0.6700

(0.8, 0.2) (0.0900, 2.8650) (−2.2870, 1.7378) 0.1180

(1.0, 0.0) (0.0000, 9.9997) (−0.4167, 8.3332) −0.4167

(−0.2, 0.8) (0.4000, 2.4000) (−0.2000, 1.5400) 1.2720

(0.8, −0.2) (0.0000, 9.9997) (−0.4167, 8.3332) − 2.0000

TABLE II

C OMPUTATIONAL RESULTS OF E XAMPLE III.3

Case 3 The feasible set XE is the efficient solution

set of problem (GBOP ) and the objective function Φ(x)

has the form as Φ(x) = ϕ(f (x)) where ϕ : Y → R is

quasiconcave and decreasing monotonic.

In this case, the procedure for solving problem is established by Proposition II.4 and Proposition II.7 Let

xopt be an optimal solution of problem(PX)

Procedure 3.

Step 1 For each i = 1, 2, find the optimal value yI

i of problem(Pi)

Step 2 Solve problem(CPid

2 )

If problem (CPid

2 ) is not feasible Then Go to Step 3

(yI 6∈ Y)

Else Find an optimal solutionx∗to the problem(CPid

2 ) STOP (x∗ is an optimal solution to problem(PX))

Step 3 Solve problem(PS) and problem (PE) to find the efficient pointsyS, yE and the efficient solutionsxS, xE

respect toyS, yE, respectively

Step 4 Ifϕ(yS) > ϕ(yE) Then xopt= xE Elsexopt=

xS STOP (xoptis an optimal solution to problem(PX))

We give below an example to illustrate Procedure 3

Example III.4 Consider problem (PX), where XE is the efficient solution set to problem (GBOP ) with

f1(x) = x2

1+ 2x1x2+ 3x2

2, f2(x) = x2

2− 0.5x1+ 0.3x2,

X =x ∈ R2| g1(x) ≤ 0, g2(x) ≤ 0 ,

g1(x) = 9(x1−2)2

+4(x2−3)2

−36, g2(x) = x1−2x2−3,

and Φ(x) = ϕ(f (x)) with ϕ(y) = −y2− y2.

It is easily verified that the function vector f is scalarly pseudoconvex on X because f1, f2 are convex on X

Moreover, the function ϕ is quasiconcave and decreasing monotonic on R2 Therefore, we can apply Procedure 3

to solve this problem.

Step 1 The optimal value of the problems(P1) and (P2), respectively, isyI

1= 2.2192 and yI

2= −1.2623.

Step 2 Solving problem (CPid

2 ), we can find that it is not feasible Then go to Step 3.

Step 3 Solve problem(PS) and problem (PE) to obtain

yS= (9.2894, −1.2623), yE= (2.2192, −0.4012)

and

xS = (2.7848, 0.2406), xE = (1.1432, 0.2892)

respect to yS, yE, respectively.

Step 4 Since ϕ(yS) < ϕ(yE), the optimal solution to

problem (PX) is xopt= xS = (2.7874, 0.2406) and the

optimal value of problem(PX) is Φ(xE) = −87.8812.

IV CONCLUSION

In this article, we have developed the simple convex

programming procedures for solving three special cases

of optimization problem over the efficient set (PX)

These special case procedures require quite little

com-putation effort in comparision to that required to solve

the general case because only some convex programming

problems need solving Therefore, they can be used as

screening devices to detect and solve these special cases

ACKNOWLEDGMENT This research is funded by Hanoi University of Science

and Technology under grant number T2016-TC-205

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