Multi-objective Optimization Problem with Bounded Parameters

DOI:10.1051/ro/2014023 www.rairo-ro.org

MULTI-OBJECTIVE OPTIMIZATION PROBLEM WITH BOUNDED PARAMETERS

Ajay Kumar Bhurjee

and Geetanjali Panda

Abstract. In this paper, we propose a nonlinear multi-objective op- timization problem whose parameters in the objective functions and constraints vary in between some lower and upper bounds. Existence of the efficient solution of this model is studied and gradient based as well as gradient free optimality conditions are derived. The theoretical developments are illustrated through numerical examples.

Keywords. Multi-objective optimization problem, efficient solution, optimality condition, interval valued convex function.

Mathematics Subject Classification. 90C25, 90C29, 90C30.

1. Introduction

In most of the real-life optimization models, the parameters in the objective function and constraints are not known exactly due to the presence of improper information in the data set. The lower and upper bounds of the parameters can be estimated from the historical data. In other words, the parameters are not fixed and assumed to lie in closed intervals. In that case, the objective and constraint functions map from real space to the set of intervals. So these functions are interval valued functions. In this paper, we address these type optimization models with several conflicting objective functions and call these optimization models as multi- objective interval optimization problem, in short (M IOP). Such type situation appears in production planning, portfolio selection, transportation models etc.

Example 1 explains such a model related to portfolio optimization.

Received September 5, 2013. Accepted April 11, 2014.

1 Department of Mathematics, Indian Institute of Technology Kharagpur, 721302, India.

geetanjali@maths.iitkgp.ernet.in

Article published by EDP Sciences c EDP Sciences, ROADEF, SMAI 2014

A.K. BHURJEE AND G. PANDA

Example 1. Consider a portfolio management problem which hasn number of risky assets. Due to uncertainty in the market, the returns of the assets cannot be predicted exactly. From the historical data one can estimate the upper and lower bound of the parameters of the return in a fixed time period. Hence the expected return can be described in terms of interval parameters for a fixed time period. Let x_j be the proportion of the total fund invested onjth asset. Return of jth asset lies betweenr^L_j and r_j^R, (r^L_j < r^R_j), which is found from previous data. Since the return of all assets are lying in intervals, so the standard deviation and correlation between any two of them will also lie in intervals. SupposeQM = (Qij)_n×ndenotes then×nsymmetric covariance interval matrix (Qij= [q^L_ij, q^R_ij]) corresponding to ith andjth asset. In this circumstance, the expected return and the variance of the resulting portfoliox = (x₁, x₂, . . . , x_n)^T, are

i[r^L_i, r_i^R]x_i and

i,j[q_ij^L, q_ij^R]x_ix_j, respectively.

In order to maximize the expected return and minimize the risk factor of the portfolio simultaneously, it is necessary to solve the interval multi-objective opti- mization problem,

min

⎛

⎜⎝−

i

[r^L_i, r_i^R]x_i,

⎛

⎝

i,j

[q_ij^L, q_ij^R]x_ix_j

⎞

⎠

12⎞

⎟⎠

subject to

i

x_i= 1, x_i≥0, i= 1,2, . . . , n.

This is a nonlinear multi-objective interval optimization model.

Multi-objective optimization problems with interval parameters are studied by many researchers during last two decades (see [1,3,4,8,9,11]). The optimiza- tion models of all these papers have linear interval valued functions. Some re- cent developments in the area of nonlinear multi-objective interval optimization problem(M IOP) are due to Dunwei Gonget al.[2], Soareset al.[10] and Wu [13].

Dunwei Gong et al. [2] have suggested an interactive evolutionary algorithm for (M IOP). This algorithm periodically provides a set of non-dominated solutions.

GL Soareset al.[10] have considered a robust multi-objective optimization prob- lem with interval valued function as an inclusion of real valued function. In both the papers conditions for the existence of solution of (M IOP) has not been stud- ied. Wu [13] has studied the conditions for the existence of solution of a (M IOP) whose objective functions are interval valued functions and all constraints are real valued functions. This paper discusses the conditions for the existence of solution of (M IOP) model whose objective functions as well as constraints are nonlinear interval valued functions. For this purpose approach of this paper is different from above approaches. The existence results use the concept of convexity and differen- tiability of a general interval valued function and derives the sufficient optimality conditions for the existence ofI-Efficient solution of (M IOP).

Development of the paper is explained in several sections. Section2 discusses some prerequisites on interval analysis. In Section3, sufficient optimality condition

for (M IOP) and the existence of solution is studied. The theoretical developments are illustrated in several numerical examples.

The following notations are used throughout the paper.

Bold capital letters denote closed intervals; I(R)= The set of all closed intervals in R; (I(R))^k= The product space I(R)×I(R)×. . .×I(R)

ktimes

. C^k_v= k-dimensional column vector, whose elements are intervals; C^k_v ∈(I(R))^k, C^k_v = (C1,C2, . . . ,Ck)^T, Cj = [c^L_j, c^R_j]; c^L_j, c^R_j ∈ R, j ∈ Λ_k, Λ_k = {1,2, . . . , k}; The degenerate interval ˆη is denoted by ˆη= [η, η], whereη is a real number.

2. Preliminaries

For two real vectorsa= (a₁, a₂, . . . , a_n)^T, b= (b₁, b₂, . . . , b_n)^T inRⁿ, we denote avb⇔a_i≥b_i; avb⇔a_i≤b_i; a >_vb⇔a_i > b_i; a <_vb⇔a_i< b_i, i∈Λ_n. Let∗ ∈ {+,−,·, /}be a binary operation on the set of real numbers. The binary operationbetween two intervalsA= [a^L, a^R] andB= [b^L, b^R] inI(R), denoted by AB is the set {a∗b : a ∈ A, b ∈ B}. In case of division (AB), it is assumed that 0∈/ B. These interval operations can also be expressed in terms of parameters. Any point inAmay be expressed asa(t) =a^L+t(a^R−a^L), t∈[0,1].

An intervalAis said to be positive interval ifa(t) is positive for everyt. Algebraic operations of intervals may be explained in parametric form as follows.

AB={a(t₁)∗b(t₂)|t₁, t₂∈[0,1]} (2.1) An interval vectorC^k_v∈(I(R))^k may be expressed in terms of parameters as

C^k_v =

c(t)|c(t) = (c₁(t₁), c₂(t₂), . . . , c_k(t_k))^T, where t= (t₁, t₂, . . . , t_k)^T, c_j(t_j)∈Cj,

c_j(t_j) =c^L_j +t_j(c^R_j −c^L_j), t_j ∈[0,1], j∈Λ_k

The set of intervalsI(R) is not a totally order set. Partial ordering between two intervals can be represented in parametric form. ForA,B∈I(R),

ABifa(t)≤b(t), ∀ t∈[0,1] andA≺B ifa(t)< b(t), ∀t∈[0,1] (2.2) A=B if a(t) =b(t), ∀t∈[0,1]. (2.3) Note that A B is not same as BA 0. For example [2,5] [3,7], but [3,7][2,5] = [−2,5]0.

Next we summarize interval valued function and some of its properties below which are due to [1].

A.K. BHURJEE AND G. PANDA

Forc(t)∈ C^k_v,let f_c(t) :Rⁿ → R. For a given interval vectorC^k_v, an interval valued functionFC^k_v :Rⁿ→I(R) can be expressed in the parametric form as

F_C^k_v(x) =

f_c(t)(x)f_c(t):Rⁿ →R, c(t)∈C^k_v

. (2.4)

For every fixed x, if f_c(t)(x) is continuous in t then min_t∈[0,1]kf_c(t)(x) and max_t∈[0,1]kf_c(t)(x) exist. In that case

F_C^k_v(x) =

t∈[0,1]min^kf_c(t)(x), max

t∈[0,1]^kf_c(t)(x)

.

Iff_c(t)(x) is linear intthen min_t∈[0,1]kf_c(t)(x) and max_t∈[0,1]kf_c(t)(x) exist in the set of vertices of C^k_v. Iff_c(t)(x) is monotonically increasing in t, then F_C^k_v(x) = [f_c(0)(x), f_c(1)(x)]. The partial derivatives ofFC^k_v :Rⁿ→I(R) atx^∗ is calculated as follows.

∂F_C^k_v(x^∗)

∂x_i =

∂f_c(t)(x^∗)

∂x_i

for everyt∈[0,1]^k, c(t)∈C^k_v

.

If ^∂fc(t)_∂x^(x∗)

i is continuous in tthen

∂FC^k_v(x^∗)

∂x_i =

t∈[0,1]min^k

∂f_c(t)(x^∗)

∂x_i , max

t∈[0,1]^k

∂f_c(t)(x^∗)

∂x_i

.

The gradient of an interval valued function, FC^k_v : Rⁿ → I(R) at x= x^∗ is an interval vector,

∇F_C^k_v(x^∗) =

∂FC^k_v(x^∗)

∂x₁ ,∂FC^k_v(x^∗)

∂x₂ , . . . ,∂FC^k_v(x^∗)

∂x_n T

.

Using the representation of the partial ordering and interval valued function we can define interval valued convex function as follows.

Definition 2.1. Interval valued convex function.

SupposeD ⊆Rⁿ is a convex set. For givenC^k_v ∈(I(R))^k, the interval valued function F_C^k_v : D → I(R) is said to be convex with respect to if for every x₁, x₂∈D and 0≤λ≤1,

FC^k_v(λx₁+ (1−λ)x₂)λFC^k_v(x₁)⊕(1−λ)FC^k_v(x₂).

Remark 2.2. From (2.2) and definition of interval valued convex function, one may observe thatF_C^k_v is convex with respect tomeansf_c(t)(λx₁+ (1−λ)x₂)≤ λf_c(t)(x₁) + (1−λ)f_c(t)(x₂), for all t ∈ [0,1]^k. So we can conclude that FC^k_v is convex with respect toif and only iff_c(t)(x) is a convex function onD for every t∈[0,1]^k.

The following separation theorem is needed to prove the existence of the solution of (M IOP).

Proposition 2.3 [6]. Let f be a m-dimensional convex vector function on the convex set Γ ⊂Rⁿ. Then either

(I) f(x) <_v 0 has a solution x∈Γ or (II) p^Tf(x)≥ 0 for all x∈ Γ for some pv 0, p∈R^m but never both.

3. Methodology

We propose a general multi-objective interval optimization problem as, (M IOP) min F(x)

subject to G^p_Dmp

v (x)(or)Bp, p∈Λ_q, (3.5) where F(x) =

F¹_Ckv1(x),F²_Ckv2(x), . . . ,F^m_Ckm v (x)

T

, Fⁱ_Cki v ,G^p_Dmp

v : Rⁿ → I(R), i∈Λ_m, partial orderings in the constraints (3.5) are as defined in (2.2).

Using expression (2.4), the interval valued functions Fⁱ_Cki

v can be represented in the parametric form as

Fⁱ_Ckiv (x) =

f_cⁱ_i(ti)(x)|f_cⁱ_i(ti):Rⁿ→R, c_i(t_i)∈C^k_vⁱ

.

Using Expression (2.4) and Inequality (2.2) the constraints of (M IOP) can be expressed as

x∈Rⁿ|G^p_Dmp

v (x)Bp

=

x∈Rⁿ|g_d^p

p(t_p)(x)≤b(t_p)

, (3.6)

whereg^p_d

p(t_p):Rⁿ→R, d_p(t_p)∈D^mv^p, b(t_p)∈Bp.

Throughout this section, we considert= (t₁, t₂, . . . , t_m)^T,t_i = (t¹_i, t²_i. . . , t^k_iⁱ)^T, t^j_i ∈[0,1],j∈Λ_ki,i∈Λ_m,t_p∈[0,1].

The feasible set for (M IOP) can be expressed as the set, S=

x∈Rⁿ|G^p_Dmp

v (x)(or)Bp, p∈Λ_q

=

p∈Λq

x∈Rⁿ|g_d^p_p(t

p)(x)≤(or≥)bp(t_p), t_p∈[0,1]

.

Using expression (2.4), (M IOP) can be rewritten as minx∈S

(f_c¹_1(t1)(x), f_c²_2(t2)(x), . . . , f_c^m_m(tm)(x))|ci(t_i)∈C^k_vⁱ, i∈Λ_m

.

Since (M IOP) has several interval valued conflicting objective functions, so ex- act solution of (M IOP) may not exist which minimizes all objective functions

A.K. BHURJEE AND G. PANDA

simultaneously. Like general multi-objective problem, solution of (M IOP) is a compromise/ Pareto optimal/ efficient solution. For everyx, the objective value of (M IOP) is an interval vector. So, for any two different feasible points xand y, the objective values F(x) andF(y) can be compared componentwise like real vectors. We denote this by

F(x)vF(y)⇔Fⁱ_Ckiv (x)Fⁱ_Ckiv (y)∀i∈Λ_m.

A partial ordering can not compare all intervals. Due to the complexities associ- ated with partial orderings, involved at different stages of (M IOP), it is difficult to derive the efficient solution of (M IOP) directly like general vector optimiza- tion problem. To avoid these complications, (M IOP) is transformed to a general optimization problem in the subsequent sections. Some gradient free and gradient based results are established to study the existence of efficient solution of (M IOP) through the solution of the transformed problem. We accept the partial ordering as defined in (2.2) to prove these results and call an efficient solution of (M IOP) with respect toas I-Efficient solution. (Several partial orderings inI(R) exist in literature (see [5,7]). The results of this paper are based on the partial ordering in the parametric form. However, similar theory may be developed with respect to any other partial ordering.) Like vector optimization problem,I-Efficient solution and properlyI−Efficient solution of (M IOP) may be defined as follows.

Definition 3.1. x^∗ ∈ S is called an I-Efficient solution of (M IOP) if there is nox∈Swith

Fⁱ_Cki

v (x)Fⁱ_Cki

v (x^∗), i∈Λ_m and for at least onej=i, F^j

C^kjv

(x)≺F^j

C^kjv

(x^∗) (3.7) Definition 3.2. x^∗ ∈S is called a properly I-Efficient solution of (M IOP) if x^∗ ∈S is anI-Efficient solution and there is a positive degenerate interval ˆη so that for somei∈Λ_mand for everyx∈S withFⁱ_Cki

v (x)≺Fⁱ_Cki

v (x^∗), at least one j=iexists withF^j

C^kjv

(x^∗)≺F^j

C^kjv

(x) and Fⁱ_Cki

v (x^∗)Fⁱ_Cki

v (x) F^j

C^kjv

(x)F^j

C^kjv

(x^∗) η.ˆ (3.8)

Example 2. Consider the problem min_(x1,x2)∈S{F1(x₁, x₂), F₂(x₁, x₂)}, where F₁(x₁, x₂) = [1,2]x₁ ⊕[1,3], F₂(x₁, x₂) = [1,3]x₂⊕[1,2] and S = {(x₁, x₂) ∈ Rⁿ| x²₁+x²₂≤1}.

(x^∗₁, x^∗₂) = (−^√₂³,−¹₂) is properlyI-Efficient solution of (M IOP).

F₁(x^∗₁, x^∗₂) = [1−√

3,3− ^√₂³], F₂(x^∗₁, x^∗₂) = [−¹₂,³₂]. Any point of the set S is (xⁿ₁, xⁿ₂) = (−^√2n−1_n ,−1 + _n¹), n∈ N. Substituting (xⁿ₁, xⁿ₂) in place of x, the

interval inequalities (3.7) becomes F₁(xⁿ₁, xⁿ₂)

1−√

3,3−

√3 2

andF₂(xⁿ₁, xⁿ₂)≺

−1 2,3

2

· (3.9)

Using the partial orderingand≺stated in (2.2) and interval operations stated in (2.1), (3.9) reduces to the following system of real inequalities.

√2n−1

n ≥

√3

2 and n >2, n∈N (3.10)

The system (3.10) has no solution. Consequently (3.7) has no solution. Hence by Definition 3.1, (−^√₂³,−¹₂) is anI-Efficient solution of (M IOP).

Again

F₁(x^∗₁, x^∗₂)F₁(xⁿ₁, xⁿ₂) F₂(xⁿ₁, xⁿ₂)F₂(x^∗₁, x^∗₂) =

−2−√ 3 + _n¹√

2n−1

32 +_n³ ,2−^√₂³+_n²√ 2n−1

1 n−⁷₂

· For a sequence of intervals{[a^L_n, a^R_n]}, lim_n→∞[a^L_n, a^R_n] = [lim_n→∞a^L_n,lim_n→∞a^R_n] (see [12]). Hence

n→∞lim

F₁(x^∗₁, x^∗₂)F₁(xⁿ₁, xⁿ₂) F₂(xⁿ₁, xⁿ₂)F₂(x^∗₁, x^∗₂)=

−4−2√ 3

3 ,

√3−4 7

· Consider ˆη =

4−2√

3 3,⁴⁻²₃^√3

. Then we can write ^F_F^{1(x∗1,x∗2) F1(xn1,xn2)}

2(xⁿ₁,xⁿ₂) F2(x^∗₁,x^∗₂) η. Fromˆ Definition 3.2,x^∗= (−^√₂³,−¹₂) is a properlyI-Efficient solution of (M IOP).

Optimality conditions for the existence ofI-Efficient solution of (M IOP) are established in the following subsections.

3.1. Optimality condition without using derivative

As discussed earlier, it is difficult to find the efficient solution of (M IOP) di- rectly due to the complexities arising in the partial ordering in the set of intervals.

To address this difficulty, we construct a deterministic form of (M IOP) using some transformations as follows and prove that an optimal solution of the transformed problem is anI-Efficient solution of (M IOP) in the subsequent theorems.

Consider a vector valued weight function w (or w(t)) = (w₁(t₁), w₂(t₂). . . , w_m(t_m))^T,w_i(t_i) >0, andμ_i >0,i ∈ Λ_m, and construct an optimization problem

(M IOP_w^μ) : min

x∈SΨ(x), (3.11)

where Ψ(x) =

i∈Λmμ_iψ_i(x), and ψ_i(x) =

kiw_i(t_i)f_cⁱ

i(ti)(x) dt_i, dt_i = dt¹_idt²_i. . .dt^k_iⁱ and

ki = ₁

0

₁

0 . . . ₁ 0

(ki times)

.

A.K. BHURJEE AND G. PANDA

Note. w_i(t_i) may be treated as a preference weight function, which has to be provided by the decision maker. Different preference functions can be provided to estimate the Pareto optimal value of the model. For everyi,w_i(t_i) = 1 indicates that the investor’s natural attitude is to estimate the mean. If₁

0 w_i(t_i)dt_i= 1 for everyithen the investor’s inclination is to estimate in between the optimistic and pessimistic optimal value. In the subsequent results we will see that any selection ofw_i with positive value can provide anI-Efficient solution.

Theorem 3.3. If x^∗ ∈S is an optimal solution of (M IOP_w^μ) for some w >_v 0 andμ >_v0, thenx^∗ is an I-Efficient solution of (M IOP).

Proof.Letx^∗∈Sbe an optimal solution of (M IOP_w^μ),w >_v0 andμ >_v 0. Assume thatx^∗is not anI-Efficient solution of (M IOP). Then by Definition3.1, there is somex∈S satisfying (3.7). Using (2.2), the inequalities in (3.7) can be rewritten as follows.

For somexinS and eacht_i∈[0,1]^ki, f_cⁱ_i(ti)(x)≤f_cⁱ_i(ti)(x^∗), i∈Λ_mand f_c^j

j(tj)(x)< f_c^j

j(tj)(x^∗) for at least onej=i.

Since w_i(t_i) > 0 and μ_i > 0, the above relations imply that

i∈Λmμ_iψ_i(x) <

i∈Λmμ_iψ_i(x^∗). This is equivalent to Ψ(x) < Ψ(x^∗), which is impossible since x^∗ is the optimal solution of (M IOP_w^μ). Hence x^∗ is an I-Efficient solution of (M IOP).

Theorem 3.4. Ifx^∗ is anI-Efficient solution of(M IOP)with G^p_Dmp

v (x)Bp

andFⁱ_Cki

v ,G^p_Dmp

v , p∈Λ_q, i∈Λ_mare interval valued convex functions with respect tothen there exists a weight functionwv0such thatx^∗is an optimal solution of(M IOP_w^μ)for anyμ >_v0.

Proof.Here S={x∈Rⁿ|G^p_Dmp

v (x)Bp, p∈Λ_q}= _p∈Λq{x∈Rⁿ|g_d^p_p(t p)(x)≤ b_p(t_p), t_p ∈ [0,1]}. Since G^p_D^mp

v is an interval valued convex function, so by Re- mark2.2,g_d^p_p(t

p)(x) is a convex function. HenceSis a convex set. Supposex^∗∈S is anI-Efficient solution of (M IOP). Then there exists nox∈S satisfying (3.7).

From Remark 2.2, we can conclude that f_cⁱ

i(ti) is convex on a convex set S for eacht_i. Since (3.7) has no solution, so using the concept of partial ordering and interval valued function in Section2, we can conclude that, for everyt_i in [0,1]^ki, the following system has no solution onS.

f_cⁱ_i(ti)(x)−f_cⁱ_i(ti)(x^∗)≤0∀i∈Λ_m andf_c^j

j(tj)(x)−f_c^j

j(tj)(x^∗)<0 for at least onej =i

If we denoteF(x, t) = (f_c¹_1(t1)(x)−f_c¹_1(t1)(x^∗), . . . , f_c^m_m(tm)(x)−f_c^m_m(tm)(x^∗))^T then above system implies that F(x, t) v 0 has no solution for everyt. This implies that F(x, t) <_v 0 has no solution for every t. Hence from Proposition 2.3, there

exists a real vector u = (u₁, u₂, . . . , u_m)^T, u v 0 such that u^TF(x, t) ≥ 0 is true for allx∈ S. Define w_i : [0,1]^ki →R₊∪ {0} by w_i(t_i) =u_i, i ∈Λ_m. Then u^TF(x, t)≥0 is same as

w(t)^TF(x)≥0,∀x∈S This implies that

i∈Λmw_i(t_i)f_cⁱ

i(ti)(x)≥

i∈Λmw_i(t_i)f_cⁱ

i(ti)(x^∗)∀x∈S.Hence forμ_i>0,

i∈Λmμ_iψ_i(x)≥

i∈Λmμ_iψ_i(x^∗)∀x∈S.This is equivalent toΨ(x)≥ Ψ(x^∗)∀x∈S, which implies thatx^∗is an optimal solution of (M IOP_w^μ) forwv0, μ >_v0.

Theorem 3.5. If x^∗ ∈ S is an optimal solution of (M IOP_w^μ), w >_v 0, w_i are continuous functions satisfying

kiw_i(t_i) dt_i= 1andμ >_v0 thenx^∗ is a properly I-Efficient solution of (M IOP).

Proof. Supposex^∗ is not a properly I-Efficient solution of (M IOP). So either x^∗is not anI-Efficient solution of (M IOP) orx^∗is anI-Efficient solution but does not satisfy the conditions in (3.8).

(I)Supposex^∗is not anI-Efficient solution of (M IOP). Then by Theorem3.3, x^∗ is not an optimal solution of (M IOP_w^μ) for any choice of w >_v 0 and μ >_v 0, which contradicts thatx^∗is the optimal solution of (M IOP_w^μ).

(II)Let x^∗ be anI-Efficient solution of (M IOP) but does not satisfy the con- dition for properlyI-Efficient solution (3.8). One can choose

η= (m−1) max

i,j max

ti,¯ti,tj,t¯j

μ_jw_j(t_j)w_j(¯t_j) μ_iw_i(t_i)w_i(¯t_i)

form≥2, i=j, t_i,t¯_i∈[0,1]^ki Sincew_i is continuous soη exists. Then from Definition3.2, for some i∈Λ_mand somex∈S withFⁱ_Cki

v (x)≺Fⁱ_Cki v (x^∗), Fⁱ_Cki

v (x^∗)Fⁱ_Cki

v (x) F^j

C^kjv

(x)F^j

C^kjv

(x^∗) η,ˆ ∀j∈Λ_m, i=j, (3.12) withF^j

C^kjv

(x^∗)≺F^j

C^kjv

(x) holds for ˆη= [η, η].

(3.12) means for everyt_i,t¯_i, f_cⁱ

i(ti)(x^∗)−f_cⁱ

i(¯ti)(x) f_c^j

j(tj)(x)−f_c^j

j(¯tj)(x^∗)> η≥(m−1)

μ_jw_j(t_j)w_j(¯t_j) μ_iw_i(t_i)w_i(¯t_i)

∀ j∈Λ_m/{i},

which is

μ_iw_i(t_i)w_i(¯t_i)(f_cⁱ_i(ti)(x^∗)−f_cⁱ_i(¯ti)(x))>

(m−1)μ_jw_j(t_j)w_j(¯t_j)(f_c^j

j(tj)(x)−f_c^j

j(¯tj)(x^∗))

A.K. BHURJEE AND G. PANDA

So μ_i

ki

kj

w_i(t_i)w_i(¯t_i)(f_cⁱ_i(ti)(x^∗)−f_cⁱ_i(¯ti)(x))dt_idt_j>(m−1)μ_j

×

ki

kj

w_j(t_j)w_j(¯t_j)(f_c^j

j(tj)(x)−f_c^j

j(¯tj)(x^∗))dt_idt_j Since

kiw_i(t_i) dt_i= 1, after integrating the above inequality becomes μ_i(ψ_i(x^∗)−ψ_i(x))>(m−1)μ_j(ψ_j(x)−ψ_j(x^∗)) Hence

j∈Λm,j=i

μ_i(ψ_i(x^∗)−ψ_i(x))>(m−1)

j∈Λm,j=i

μ_j(ψ_j(x)−ψ_j(x^∗)) This implies μ_i(ψ_i(x^∗) − ψ_i(x)) >

j∈Λm,j=iμ_j(ψ_j(x) − ψ_j(x^∗)). Hence

j∈Λmμ_jψ_j(x^∗)>

j∈Λmμ_jψ_j(x).

That is,Ψ(x^∗)> Ψ(x). This contradicts the assumption thatx^∗ is the optimal solution of (M IOP_w^μ).

3.2. Optimality condition using derivative

For a feasible pointx^∗∈S,denoteJ(x^∗) ={p:G^p_Dmp

v (x^∗) =Bp, p∈Λ_q}. That is,J(x^∗) ={p:g_d^p_p(t

p)(x^∗) =b_p(t_p),∀t_p∈[0,1], p∈Λ_q}.

Theorem 3.6. SupposeFⁱ_Cki

v andG^p_Dmp

v are differentiable functions and convex with respect toatx^∗ ∈S of (M IOP), satisfying

i∈Λm

μ_i∇Fⁱ_Cki

v (x^∗)⊕

p∈J(x^∗)

ν_p∇G^p_Dmp

v (x^∗) =0, (3.13) whereμ_i >0, i∈Λ_m and for every p∈Λ_q,ν_p ≥0 if G^p_Dmp

v (x)Bp;ν_p ≤0 if G^p_Dmp

v (x)Bp; and ν_p is unrestricted if p∈ J(x^∗). Then x^∗ is an I-Efficient solution for(M IOP).

Proof. SupposeFⁱ_Cki

v and G^p_D^mp

v are convex at x^∗ with respect to . From Re- mark 2.2 it is true that f_cⁱ

i(ti) and g_d^p

p(t_p) are convex functions at x^∗ for every t_i, t_p. So for allx∈S and allt_i, t_p,

f_cⁱ_i(ti)(x)−f_cⁱ_i(ti)(x^∗)≥(x−x^∗)^T∇f_cⁱ_i(ti)(x^∗),∀i∈Λ_m (3.14) g_d^p

p(tp)(x)−g_d^p

p(tp)(x^∗)≥(x−x^∗)^T∇g_d^p_p(t

p)(x^∗),∀p∈Λ_q (3.15) From (2.3), Equation (3.13) becomes

i∈Λm

μ_i∇f_cⁱ_i(ti)(x^∗) +

p∈J(x^∗)

ν_p∇g_d^p_p(t

p)(x^∗) = 0, ∀t_i, t_p (3.16)