R E S E A R C H
Open Access
Optimality for
E
-
[, ]
convex
multi-objective programming problems
Tarek Emam
**Correspondence:
[email protected] Department of Mathematics, Faculty of Science, University of Hail, Hail, Kingdom of Saudi Arabia Department of Mathematics, Faculty of Science, Suez University, Suez, Egypt
Abstract
In this paper, we are interested in deriving the sufficient and necessary conditions for an optimal solution of special classes of programming problems. These classes involve generalizedE-[0, 1] convex functions. The characterization of efficient solutions forE-[0, 1] convex multi-objective programming problems is obtained. Finally, sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solution are derived.
Keywords: E-[0, 1] convex functions; optimal solutions; multi-objective problems; properly efficient solutions
1 Introduction
The study of multi-objective programming problems was very active in recent years. The weak minimum (weakly efficient, weak Pareto) solution is an important concept in math-ematical models, economics, decision theory, optimal control, and game theory (see, for example, [–]). In most works, the assumption of convexity was made for the objective functions. The extension of convexity is an area of active current research in the field of optimization theory. Various relaxations of convexity were possible, and were called generalized convex functions. The definition of generalized convex functions has occu-pied the attention of a number of mathematicians; for an overview of generalized con-vex functions we refer to [–]. A significant generalization of concon-vexity is the concept ofE-[, ] convexity [].E-[, ] convexity depends on the effect of an operatorEon the range of the function and the closed unit interval [, ]. Inspired and motivated by above reasons, the purpose of this paper is to formulate the problems which involve general-izedE-[, ] convex functions. The paper is organized as follows. In Section , we define generalizedE-[, ] convex functions, which are called pseudoE-[, ] convex functions, and obtain sufficient and necessary conditions for an optimal solution ofE-[, ] convex programming problems. In Section , we consider the Mond-Weir type dual and gener-alize its results under theE-[, ] convexity assumptions. In Section , we formulate the multi-objective programming problem which involvesE-[, ] convex functions. An effi-cient solution for the problem considered is characterized by weighting, andε-constraint approaches. At the end of this paper, we obtain sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solution for problems involving generalizedE-[, ] convex functions. Let us survey, briefly, the definitions and some re-sults as regardsE-[, ] convexity.
Definition [] A real valued functionf :M⊆Rn→Ris said to be aE-[, ] convex function onMwith respect toE:R×[, ]→R, ifMis a convex set and, for eachx,y∈M andλ,λ∈[, ],λ+λ= ,
f(λx+λy)≤E
f(x),λ
+Ef(y),λ
.
Iff(λx+λy)≥E(f(x),λ) +E(f(y),λ), thenfis called aE-[, ] concave function onM. If the inequality signs in the previous two inequalities are strict, thenf is called strictly E-[, ] convex and strictlyE-[, ] concave, respectively.
EveryE-[, ] convex function with respect toE:R×[, ]→Ris a convex function if E(f(x),λ) =λf(x). LetE:R×[, ]→Rbe a mapping such thatE(t,λ) = ( +λ)t,t∈R,λ∈
[, ], then the functionh(x) =ki=aifi(x) isE-[, ] convex onMforai≥,i= , , . . . ,k, if the functionsfi:Rn→Rare allE-[, ] convex on a convex setM⊆Rn. LetE:R×[, ]→ Rbe a mapping such thatE(t,λ) =min{λt,t},t∈R,λ∈[, ], then a numerical function f :M⊂Rn→R+defined on a convex setM⊆RnisE-[, ] convex if and only if itsepi(f) is convex. LetBbe an open convex subset ofRnand letE:R×[, ]→Rbe a mapping such thatE(t,λ) =min{λ,t},t∈R,λ∈[, ], thenf is continuous onBiff isE-[, ] convex onB. Iff :Rn→Ris a differentiableE-[, ] convex function aty∈Mwith respect to E:R×[, ]→Rsuch thatE(t,λ) =min{λt,t},t∈R,λ∈[, ], then, for eachx∈M, we have (x–y)∇f(y)≤f(x) –f(y). For more details as regardsE-[, ] convex functions, see [].
Definition [] A real valued functionf :M⊆Rn→Ris said to be a quasiE-[, ] convex function onMwith respect toE:R×[, ]→R, ifMis a convex set and, for eachx,y∈M andλ,λ∈[, ],λ+λ= ,
f(λx+λy)≤max
Ef(x),λ
,Ef(y),λ
.
Iff(λx+λy)≥min{E(f(x),λ),E(f(y),λ)}, thenfis called a quasiE-[, ] concave func-tion onM. If the inequality signs in the previous two inequalities are strict, thenfis called strictly quasiE-[, ] convex and strictly quasiE-[, ] concave, respectively.
Every quasiE-[, ] convex function with respect toE:R×[, ]→Ris a convex func-tion ifE(f(x),λ) =λf(x). LetE:R×[, ]→Rbe a mapping such thatE(f(x),λ) =f(λx) for eachx∈M,λ∈[, ], thenf(ni=λixi)≤max≤i≤nE(f(xi),λi) for eachxi∈M,λi≥,
n
i=λi= , iff :Rn→RisE-[, ] convex on a convex setM⊆Rn. LetE:R×[, ]→R be a mapping such thatE(t,λ) =min{λ,t},t∈R,λ∈[, ], then the level setLE-[,]α is a
convex set for eachα∈Riff :Rn→Ris quasiE-[, ] convex on a convex setM⊆Rn. LetE:R×[, ]→Rbe a mapping such thatE(t,λ) =max{λ,t},t∈R,λ∈[, ], and let
α=minxminλE(f(x),λ), then the level setLE-[,]α is a convex set if and only iff is quasi
E-[, ] convex. Iff :Rn→Ris a differentiable quasiE-[, ] convex function aty∈M with respect toE:R×[, ]→Rsuch thatE(t,λ) =min{λ,t},t∈R,λ∈[, ], then, for eachx∈M, we have (x–y)∇f(y)≤. For more details as regards quasiE-[, ] convex functions, see [].
2 GeneralizedE-[0, 1]convex programming problems
Definition A real valued functionf :M⊆Rn→Ris said to be a pseudoE-[, ] convex function on a convex setM⊆Rnif there exists a strictly positive functionb:Rn×Rn→R such that
Ef(x),λ
<Ef(y),λ
⇒ f(λx+λy)≤E
f(y),λ
–λλb(x,y)
for allx,y∈Mandλ,λ∈[, ],λ+λ= .
Remark Every pseudoE-[, ] convex function with respect toE:R×[, ]→Ris con-vex function ifE(t,λ) =min{t,λ},t∈R,λ∈[, ].
Proposition Let E:R×[, ]→R be a map such that E(t,λ) =max{t,λ},t∈R,λ∈[, ]. A convex function f :Rn→R on a convex set M⊆Rnis a pseudo E-[, ]convex function on M.
Proof LetE(f(x),λ) <E(f(y),λ). Sincef is a convex function on a convex setM⊆Rn, for allx,y∈Mandλ,λ∈[, ],λ+λ= , we have
f(λx+λy)≤λf(x) +λf(y)≤λE
f(x),λ
+λE
f(y),λ
.
That is,
f(λx+λy)≤E
f(y),λ
+λ
Ef(x),λ
–Ef(y),λ
≤Ef(y),λ
+λλ
Ef(x),λ
–Ef(y),λ
=Ef(y),λ
–λλ
Ef(y),λ
–Ef(x),λ
=Ef(y),λ
–λλb(x,y),
sinceb(x,y) =E(f(y),λ) –E(f(x),λ) > . This is the required result. Theorem Let E:R×[, ]→R be a map such that E(t,λ) =min{t,λ},t∈R,λ∈[, ] and M⊆Rnbe a convex set.If f :Rn→R is a differentiable pseudo E-[, ]convex function at y∈M,then(x–y)∇f(y) < ,for each x∈M.
Proof Sincef :Rn→Ris a differentiable pseudoE-[, ] convex function aty∈M,
Ef(x),λ
<Ef(y),λ
⇒ f(λx+λy)≤E
f(y),λ
–λλb(x,y)≤f(y) –λλb(x,y)
for eachx∈Mandλ,λ∈[, ],λ+λ= . That is,
Ef(x),λ
<Ef(y),λ
⇒ fy+λ(x–y)≤f(y) –λλb(x,y)
⇒ f(y) +λ(x–y)∇f(y) + O
Dividing the above inequality byλ> and lettingλ→, we get
(x–y)∇f(y)≤–b(x,y) <
for eachx∈M.
Remark LetE:R×[, ]→Rbe a map such thatE(t,λ) =min{t,λ},t∈R,λ∈[, ], and M⊆Rnbe a convex set. Iff:Rn→Ris a differentiable pseudoE-[, ] convex function at y∈M, then (x–y)∇f(y)≥⇒E(f(x),λ)≥E(f(y),λ), for eachx∈Mandλ,λ∈[, ],
λ+λ= .
Lemma Let E:R×[, ]→R be a mapping such that E(t,λ) =λmin{t,λ},t∈R,λ∈
[, ].If gi:Rn→R is an E-[, ]convex function on Rn,i= , , . . . ,m,then the set M={x∈ Rn:gi(x)≤,i= , , . . . ,m}is convex set.
Proof Sincegi(x), i= , , . . . ,m, are E-[, ] convex functions with respect to E(t,λ) =
λmin{t,λ}, for eachx,y∈Mandλ,λ∈[, ],λ+λ= ,
gi(λx+λy)≤E
gi(x),λ
+Egi(y),λ
=λmin
gi(x),λ
+λmin
gi(y),λ
≤λgi(x) +λgi(y)≤, i= , , . . . ,m,
henceλx+λy∈Mfor allλ,λ∈[, ],λ+λ= . This means thatMis convex set.
Lemma Let E:R×[, ]→R be a mapping such that E(t,λ) =min{t,λ},t∈R,λ∈[, ]. If gi:Rn→R is a quasi E-[, ]convex function on Rn,i= , , . . . ,m,then the set M={x∈ Rn:gi(x)≤,i= , , . . . ,m}is convex set.
Proof Sincegi(x),i= , , . . . ,m, are quasiE-[, ] convex functions with respect toE(t,λ) = min{t,λ}, for eachx,y∈Mandλ,λ∈[, ],λ+λ= ,
gi(λx+λy)≤max
Egi(x),λ
,Egi(y),λ
≤maxgi(x),gi(y)
≤, i= , , . . . ,m,
henceλx+λy∈Mfor allλ,λ∈[, ],λ+λ= . This means thatMis convex set.
Now, we discuss the necessary and sufficient conditions for a feasible solution to be an optimal solution forE-[, ] convex programming problems. Consider the following E-[, ] convex programming problem:
min f(x)
(P)¯ subject to
Heref :Rn→Randg
i:Rn→R,i= , , . . . ,m, areE-[, ] convex functions with respect toE:R×[, ]→R.
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =λmin{t,λ},t∈R,λ∈
[, ].Suppose that there exists a feasible solution x∗ for(P),¯ and f, g are differentiable E-[, ]convex functions with respect to the same E at x∗.If there is u∈Rmand u≥such that(x∗,u)satisfies the following conditions:
∇fx∗+∇uTgx∗= ,
uTgx∗= , gx∗≤,
()
then x∗is an optimal solution for problem(P).¯
Proof For eachx∈M, we have
f(x) –fx∗≥x–x∗∇fx∗= –x–x∗∇uTgx∗
≥–uTg(x) –gx∗= –uTg(x)≥,
where the above inequalities hold becausef,gareE-[, ] convex atx∗with respect to the sameE(see Theorem . in []). Thus,x∗is the minimizer off(x) under the constraint g(x)≤, which implies thatx∗is an optimal solution for problem (P).¯ Remark [] In Theorem above, sinceu≥,g(x∗)≤, anduT∇g(x∗) = , we have
uigi
x∗= , i= , , . . . ,m. ()
IfI(x∗) ={i:gi(x∗) = }andJ={i:gi(x∗) < }, thenI∪J={, , . . . ,m}, and () givesui= fori∈J. It is obvious then, from the proof of Theorem , thatE-[, ] convexity ofgI at x∗ is all that is needed instead of theE-[, ] convexity ofg atx∗as was assumed in the theorem above.
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =min{t,λ},t∈R,λ∈
[, ].Suppose that there exist a feasible solution x∗ for(P)¯ and scalars,ui≥,i∈I(x∗), such that()of Theoremholds.If f is pseudo E-[, ]convex,and gI are quasi E-[, ] convex at x∗∈M,then E(f(x∗),λ),λ∈[, ]is an optimal solution in the objective space of problem(P).¯
Proof SinceE(gI(x),λ)≤E(gI(x∗),λ) = ,ui≥,λ,λ∈[, ],λ+λ= , andgI are quasiE-[, ] convex atx∗, we have
x–x∗ i∈I(x∗)
ui
∇gi
x∗T≤, ∀x∈M, ()
by using the above inequality in (), and pseudoE-[, ] convexity off atx∗, we obtain
x–x∗∇fx∗T≥ ⇒ Ef(x),λ
≥Efx∗,λ
⇒ f(x)≥Efx∗,λ
.
The next two theorems use the idea proposed by Mahajan and Vartak [].
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =min{t,λ},t∈R,λ∈
[, ].Suppose that there exist a feasible solution x∗ for(P)¯ and scalars,ui≥,i∈I(x∗), such that()of Theoremholds.If f is pseudo E-[, ]convex,and uT
IgI is quasi E-[, ] convex at x∗∈M,then E(f(x∗),λ),λ∈[, ]is an optimal solution in the objective space of problem(P).¯
Proof The proof of this theorem is similar to the proof of Theorem except that the argument to get the inequality () is as follows: SinceE(gI(x),λ)≤E(gI(x∗),λ),uI≥,
λ,λ∈[, ],λ+λ= , we obtain
uTI EgI(x),λ
≤ =uTI EgI
x∗,λ
for allx∈M. QuasiE-[, ] convexity ofuT
IgIatx∗yields
x–x∗∇uTI gI
x∗≤, ∀x∈M.
We can proceed as in the above theorem to prove thatE(f(x∗),λ) is an optimal solution
in the objective space of problem (P).¯
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =min{t,λ},t∈R,λ∈
[, ].Suppose that there exists a feasible point x∗ for(P)¯ and the numerical function f + uTIgI is pseudo E-[, ]convex at x∗.If there is a scalar u∈Rm such that(x∗,u)satisfies the conditions()of Theorem,then E(f(x∗),λ),λ∈[, ],is an optimal solution in the objective space of problem(P).¯
Proof The proof of this theorem is similar to the proof of Theorem except that the ar-guments are as follows: () can be written as
∇fx∗+∇uT IgI
x∗= .
This can be rewritten in the form
x–x∗∇f+uTIgI
x∗≤, ∀x∈M,
which gives
Ef+uTIgI
x∗,λ
≤Ef+uTIgI
(x),λ
, ∀x∈M,
becausef +uT
IgIis pseudoE-[, ] convex atx∗,i.e.,
Ef+uTIgI
x∗,λ
≤f(x) +uTIgI
(x), ∀x∈M.
It follows, by using the definition ofI, that
Efx∗,λ
≤f(x), ∀x∈M.
Theorem (Necessary optimality criteria) Let E:R×[, ]→R be a mapping such that E(t,λ) =λmin{t,λ},t∈R,λ∈[, ].Assume that x∗is an optimal solution for problem(P)¯ and there exists a feasible point x for(P)¯ such that gi(x) < ,i= , , . . . ,m.If gi,i∈I(x∗),is E-[, ]convex at x∗∈M,then there exist scalars ui≥,i∈I(x∗),such that(x∗,ui)satisfies
∇fx∗+ i∈I(x∗)
ui∇gi
x∗= . ()
Proof We desire to show that
x–x∗∇gI
x∗≤ ⇒ x–x∗∇fx∗≥. ()
The result will follow as in [] by applying Farkas’ lemma. Assume () does not hold,i.e., there existsx∈Rnsuch that
x–x∗∇gI
x∗≤ ⇒ x–x∗∇fx∗< . ()
Since by the assumed Slater-type condition,
gi(x˜) –gi
x∗< , i∈Ix∗,
and fromE-[, ] convexity ofgiatx∗, we get
˜
x–x∗T∇gi
x∗< , i∈Ix∗. ()
Therefore from () and ()
x–x∗+ρx˜–x∗T∇gi
x∗< , i∈Ix∗,∀ρ> .
Hence for some positiveβsmall enough
gi
x∗+βx–x∗+ρx˜–x∗<gi
x∗= , i∈Ix∗.
Similarly, fori∈/I(x∗),gi(x∗) < , and forβ> small enough,
gi
x∗+βx–x∗+ρx˜–x∗≤, i∈/Ix∗.
Thus, forβsufficiently small and allρ> ,x∗+β[(x–x∗) +ρ(x˜–x∗)] is feasible for problem (P). For sufficiently small¯ ρ> () gives
fx∗+βx–x∗+ρx˜–x∗<fx∗, ()
which contradicts the optimality ofx∗for (P). Hence, the system () has no solution. The¯ result then follows from an application of Farkas’ lemma, namely
∇fx∗+ i∈I(x∗)
ui∇gi
x∗= , u≥.
3 Duality inE-[0, 1]convexity
We consider the Wolfe type dual and generalized its results under theE-[, ] convexity assumptions. Consider the following Wolfe type dual of problem (P):¯
max ψ(y,u) =f(y) +uTg(y)
(D)¯ subject to
∇f(y) +uT∇g(y) = , u≥,
wheref,gare differentiable functions defined onRn. We now prove the following duality theorems, relating problem (P) and (¯ D).¯
Theorem (Weak duality) Let E:R×[, ]→R be a map such that E(t,λ) =λmin{t,λ}, t∈R,λ∈[, ],and let there exist a feasible solution x for(P)¯ and(y,u),a feasible solution for(D).¯ If f,g are E-[, ]convex functions at y,then f(x)≮f(y) +uTg(y).
Proof Letxbe a feasible solution for (P) and (¯ y,u) be a feasible solution for (D). Suppose¯ contrary to the resultf(x) <f(y) +uTg(y), then
f(x) +uTg(x) <f(y) +uTg(y) or
f(x) +uTg(x) –f(y) –uTg(y) < . ()
E-[, ] convexity off,gaty, implies that
f(x) –f(y)≥(x–y)T∇f(y) and
uTg(x) –g(y)≥uT(x–y)T∇g(y),
and combining the above two inequalities gives
f(x) –f(y) +uTg(x) –uTg(y)≥(x–y)T∇f(y) +uT∇g(y),
and by using inequality (), we get
(x–y)T∇f(y) +uT∇g(y)< ,
which contradicts the constraint∇f(y) +uT∇g(y) = of (D).¯ Theorem (Strong duality) Let x∗be an optimal solution for(P)¯ and g satisfy the Kuhn-Tucker constraint qualification at x∗.Then,there exists u∗∈Rm,such that (x∗,u∗)is a
feasible solution for(D)¯ and the(P)¯ -objective at x∗equals the(D)¯ -objective at(x∗,u∗).If f, g are E-[, ]convex functions at x∗with respect to E:R×[, ]→R such that E(t,λ) =
λmin{t,λ},t∈R,λ∈[, ],then(x∗,u∗)is an optimal solution for problem(D).¯
Proof Sincegsatisfies the Kuhn-Tucker constraint qualification atx∗, there existsu∗∈Rm, such that the following Kuhn-Tucker conditions are satisfied:
u∗Tgx∗= , ()
gx∗≤, ()
u∗≥. ()
() and () show that (x∗,u∗) a feasible solution for (D). Also, () shows that the¯ (P)-objective at¯ x∗ equal to the (D)-objective at (¯ x∗,u∗). Now, fromE-[, ] convexity of f andg, we have
ψx∗,u∗–ψ(x,u) =fx∗+u∗Tgx∗–f(x) –uTg(x)
=fx∗–f(x) –uTg(x), by ()
≥x∗–xT∇f(x) –uTg(x)
= –x∗–xTuT∇g(x) –uTg(x), by ()
≥–uTgx∗–g(x)–uTg(x)
= –uTgx∗≥
for each feasible point (x,u) of (D). Hence, (¯ x∗,u∗) is an optimal solution for problem (D).¯
Example LetE-[, ] :R×[, ]→Rbe defined asE(t,λ) =λ√
t, wheret∈R, andλ∈
[, ]. Consider the problem (P)¯
min f(x,y) = (y–x)
s.t. (x,y)∈M=(x,y)∈R:x+y≤, ≤y≤,x≥,
wheref isE-[, ] convex function on convex setM. Formulate the dual problem (D) as¯ follows:
max f(y) +uTg(y)
s.t.∇f(y) +uT∇g(y) = , u≥.
From the system ()-(), we have
y∗–x∗+u∗+u∗–u∗= ,
–y∗–x∗+u∗–u∗= ,
u∗x∗+y∗– = ,
u∗y∗– = ,
u∗ –y∗= ,
–u∗x∗= ,
x∗+y∗– ≤,
–y∗≤,
–x∗≤,
whereu∗i ≥,i= , , , . By solving this system, we conclude that (x∗,u∗) is the optimal solution of the dual problem (D) such that¯ x∗= (, ) andu∗= (, , , ). Hence,x∗= (, ) is the optimal solution of (P).¯
4 GeneralizedE-[0, 1]convex multi-objective programming problems
In this section, we formulate a multi-objective programming problem which involves E-[, ] convex functions. An efficient solution for the considered problem is characterized by weighting andε-constraint approaches. At the end of this section, we obtain sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solu-tion for this kind of problems. AnE-[, ] convex multi-objective programming problem is formulated as follows:
min f(x),f(x), . . . ,fk(x)
(P) subject to
x∈M=x∈Rn:gi(x)≤,i= , , . . . ,m
,
wherefj:Rn→R,j= , , . . . ,k, andgi:Rn→R,i= , , . . . ,m, areE-[, ] convex functions with respect toE:R×[, ]→R.
Definition [] A feasible solutionx∗for (P) is said to be an efficient solution for (P) if and only if there is no other feasiblexfor (P) such that, for somei∈ {, , . . . ,k},
fi(x) <fi
x∗, fj(x)≤fj
x∗ for allj=i.
Definition [] An efficient solution x∗ ∈M for (P) is a properly efficient solu-tion for (P) if there exists a scalar μ> such that for each i, i= , , . . . ,k, and each x ∈ M satisfying fi(x) < fi(x∗), there exists at least one j= i with fj(x) > fj(x∗), and [fi(x) –fi(x∗)]/[fj(x∗) –fj(x)]≤μ.
Lemma Let E:R×[, ]→R be a mapping such that E(t,λ) =λmin{t,λ},t∈R,λ∈
[, ].If f :Rn→Rk is an E-[, ]convex function on a convex set M⊆Rn,then the set A=x∈MA(x)is convex set such that
A(x) =z:z∈Rk,z>f(x) –fx∗, x∈M.
Proof Letz,z∈A, then for allx,x∈Mandλ,λ
∈[, ],λ+λ= , we have
λz+λz>λ
fx–fx∗+λ
fx–fx∗
=λf
x+λf
x–fx∗
≥λmin
fx,λ
+λmin
fx,λ
–fx∗
=Efx,λ
+Efx,λ
–fx∗
≥fλx+λx
sincef isE-[, ] convex function on convex setM. Thenλz+λz∈A, and henceAis
convex set.
4.1 Characterizing efficient solutions by weighting approach
To characterize an efficient solution for problem (P) by weighting approach [] let us scalarize problem (P) to get the form
(Pw) min k
j=
wjfj(x) subject tox∈M,
wherewj≥,j= , , . . . ,k,
k
j=wj= , andfj,j= , , . . . ,k, areE-[, ] convex functions with respect toE:R×[, ]→Rsuch thatE(t,λ) =λmin{t,λ},t∈R,λ∈[, ], on convex setM.
Theorem Ifx¯∈M is an efficient solution for problem(P),then there exists wj≥,j= , , . . . ,k,kj=wj= ,such thatx is an optimal solution for problem¯ (Pw).
Proof Letx¯∈Mbe an efficient solution for problem (P), then the systemfj(x) –fj(x¯) < , j= , , . . . ,k, has no solutionx∈M. Upon Lemma and applying the generalized Gordan theorem [], there existspj≥,j= , , . . . ,k, such thatpj[fj(x) –fj(x¯)]≥,j= , , . . . ,k, andkpj
j=pjfj(x)≥
pj
k j=pjfj(x¯).
Denotewj= pj
k
j=pj, thenwj≥,j= , , . . . ,k,
k
j=wj= , and
k
j=wjfj(x¯)≤
k
j=wjfj(x). Hencex¯is an optimal solution for problem (Pw). Theorem Ifx¯∈M is an optimal solution for(Pw¯)corresponding to w¯j, thenx is an¯ efficient solution for problem(P)if one of the following two conditions holds:
(i) w¯j> ,∀j= , , . . . ,k;
(ii) x¯is the unique solution of(Pw¯).
Proof For the proof see Chankong and Haimes [].
4.2 Characterizing efficient solutions by
ε
-constraint approachTheε-constraint approach is one of the common approaches for characterizing efficient solutions of multi-objective programming problems []. In the following we shall charac-terize an efficient solution for the multi-objectiveE-[, ] convex programming problem (P) in terms of an optimal solution of the following scalar problem:
min fq(x)
Pq(ε,E) subject tox∈M,
fj(x)≤E(εj,λj), j= , , . . . ,k,j=q.
Herefj,j= , , . . . ,k, areE-[, ] convex functions with respect toE:R×[, ]→Rsuch thatE(t,λ) =min{t,λ},t∈R,λ∈[, ], on the convex setM.
Proof Letx¯ be not optimal solution for Pq(ε¯,E¯) whereε¯j=fj(x¯), j= , , . . . ,k. So there existsx∈Msuch that
fq(x) <fq(x¯),
fj(x)≤ ¯E(ε¯j,λ¯j)≤ ¯εj=fj(x¯), j= , , . . . ,k,j=q,
sinceE¯(ε¯j,λ¯j) =min(ε¯j,λ¯j) and convexity ofM. This implies that the systemfj(x) –fj(x¯) < , j= , , . . . ,k, has a solutionx∈M. Thus,x¯is an inefficient solution for problem (P), which is a contradiction. Hencex¯is an optimal solution for problem Pq(ε¯,E¯). Theorem Letx¯∈M be an optimal solution,for all q ofPq(ε¯,E¯),whereε¯j=fj(x¯),j= , , . . . ,k.Thenx is an efficient solution for problem¯ (P).
Proof Sincex¯∈Mis an optimal solution for Pq(ε¯,E¯), whereε¯j=fj(x¯),j= , , . . . ,k, for each x∈M, we get
fq(x¯) <fq(x),
fj(x)≤ ¯E(ε¯j,λ¯j)≤ ¯εj=fj(x¯), j= , , . . . ,k,j=q,
whereE¯(ε¯j,λ¯j) =min(ε¯j,λ¯j). This implies the systemfj(x) –fj(x¯) < ,j= , , . . . ,k, has no solutionx∈M,i.e.,x¯is an efficient solution for problem (P).
4.3 Sufficient and necessary conditions for efficiency
In this section, we discuss the sufficient and necessary conditions for a feasible solutionx∗ to be efficient or properly efficient for problem (P) in the form of the following theorems.
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =λmin{t,λ},t∈R,λ∈
[, ].Suppose that there exist a feasible solution x∗for(P)and scalarsγi> ,i= , , . . . ,k, ui≥,i∈I(x∗),such that
k
i=
γi∇fi
x∗+ i∈I(x∗)
ui∇gi
x∗= . ()
If fi,i= , , . . . ,k,and gi,i∈I(x∗),are differentiable E-[, ]convex functions at x∗∈M, then x∗is a properly efficient solution for problem(P).
Proof Sincefi,i= , , . . . ,k, andgi,i∈I(x∗), are differentiableE-[, ] convex functions at x∗∈M, for anyx∈M, we have
k
i=
γifi(x) – k
i=
γifi
x∗≥x–x∗ k
i=
γi
∇fi
x∗T
= –x–x∗ i∈I(x∗)
ui
∇gi
x∗T
≥
k
i∈I(x∗) uigi
x∗– k
i∈I(x∗) uigi(x)
= – i∈I(x∗)
Thus, ki=γifi(x)≥
k
i=γifi(x∗), for all x∈M, which implies that x∗ is the minimizer ofki=γifi(x) under the constraintg(x)≤. Hence, from Theorem . of [],x∗ is a
properly efficient solution for problem (P).
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =λmin{t,λ},t∈R,λ∈
[, ].Suppose that there exist a feasible solution x∗for(P)and scalarsγi≥,i= , , . . . ,k,
k
i=γi= ,ui≥,i∈I(x∗),such that the triplet(x∗,γi,ui)satisfies()of Theorem.If
k
i=γifiis strictly E-[, ]convex,and gIis E-[, ]convex at x∗∈M,then x∗is an efficient solution for problem(P).
Proof Suppose thatx∗is not an efficient solution for (P), then there exist a feasiblex∈M and an indexrsuch that
fr(x) <fr
x∗,
fi(x)≤fi
x∗ for alli=r.
Sinceki=γifiis strictlyE-[, ] convex atx∗, the previous two inequalities lead to
≥ k
i=
γifi(x) – k
i=
γifi
x∗ ⇒ >x–x∗ k
i=
γi
∇fi
x∗T. ()
Also,E-[, ] convexity ofgi,i∈I(x∗), atx∗implies
x–x∗∇gi
x∗≤gi(x) –gi
x∗ ⇒ x–x∗∇gi
x∗≤, i∈Ix∗,
and, forui≥,i∈I(x∗), we get
x–x∗ i∈I(x∗)
ui
∇gi
x∗T≤. ()
Adding () and () contradicts (). Hence,x∗is an efficient solution for problem (P).
Remark Similarly to Theorem , it can easily be seen thatx∗becomes a properly effi-cient solution for (P), in the above theorem, ifγi> , for alli= , , . . . ,k.
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =min{t,λ},t∈R,λ∈
[, ].Suppose that there exist a feasible solution x∗for(P)and scalarsγi> ,i= , , . . . ,k, ui≥,i∈I(x∗),such that()of Theoremholds.If
k
i=γifiis pseudo E-[, ]convex, and gI are quasi E-[, ]convex at x∗∈M,then E(f(x∗),λ),λ∈[, ]is a properly non-dominated solution in the objective space of problem(P).
Proof SinceE(gI(x),λ)≤E(gI(x∗),λ) = ,λ,λ∈[, ],λ+λ= , and from quasiE-[, ] convexity ofgIatx∗,uI≥, we get
x–x∗ i∈I(x∗)
ui
∇gi
by using the above inequality in (), and from pseudoE-[, ] convexity ofki=γifiatx∗, we get
x–x∗ k
i=
γi
∇fi
x∗T≥ ⇒ k
i=
γiE
fi(x),λ
≥
k
i=
γiE
fi
x∗,λ
⇒
k
i=
γifi(x)≥ k
i=
γiE
fi
x∗,λ
.
Hence,E(f(x∗),λ) is a properly nondominated solution in the objective space of problem
(P).
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =min{t,λ},t∈R,λ∈
[, ].Suppose that there exist a feasible solution x∗for(P)and scalarsγi≥,i= , , . . . ,k,
k
i=γi= ,ui≥,i∈I(x∗),such that()of Theoremholds.If
k
i=γifiis strictly pseudo E-[, ]convex and gI is quasi E-[, ]convex at x∗∈M,then E(f(x∗),λ),λ∈[, ]is a nondominated solution in the objective space of problem(P).
Proof Suppose thatE(f(x∗),λ) is dominated solution for (P), then there exist a feasiblex for (P) and an indexrsuch that
fr(x) <E
fr
x∗,λ
, fi(x)≤E
fi
x∗,λ
for alli=r.
SinceE(t,λ) =min{t,λ},t∈R,λ∈[, ], we have
Efr(x),λ
<Efr
x∗,λ
, Efi(x),λ
≤Efi
x∗,λ
, ∀i=r.
The strictly pseudoE-[, ] convexity ofki=γifiatx∗implies that
k
i=
γiE
fi(x),λ
≤
k
i=
γiE
fi
x∗,λ
⇒ x–x∗ k
i=
γi
∇fi
x∗T< .
Also, quasiE-[, ] convexity ofgIatx∗implies that
EgI(x),λ
≤EgI
x∗,λ
= ⇒ x–x∗∇gI
x∗≤.
The proof now follows along lines similar to Theorem . Remark Similarly to Theorem , it can easily be seen thatE(f(x∗),λ),λ∈[, ], be-comes a properly nondominated solution for (P), in the above theorem, ifγi> , for all i= , , . . . ,k.
Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =min{t,λ},t∈R,λ∈
[, ].Suppose that there exist a feasible solution x∗for(P)and scalarsγi> ,i= , , . . . ,k, ui≥,i∈I(x∗),such that()of Theoremholds.If
k
Proof The proof is similar to the proof of Theorem . Theorem Let E:R×[, ]→R be a mapping such that E(t,λ) =min{t,λ},t∈R,λ∈
[, ].Suppose that there exist a feasible solution x∗for(P)and scalarsγi≥,i= , , . . . ,k,
k
i=γi= ,ui≥,i∈I(x∗),such that()of Theoremholds.If I(x∗)=φ,
k
i=γifiis quasi E-[, ]convex and uIgIis strictly pseudo E-[, ]convex at x∗∈M,then E(f(x∗),λ),
λ∈[, ],is a nondominated solution in the objective space of problem(P).
Proof The proof is similar to the proof of Theorem . Remark Similarly to Theorem , it can easily be seen thatE(f(x∗),λ),λ∈[, ], be-comes a properly nondominated solution for (P), in the above theorem, ifγi> , for all i= , , . . . ,k.
Theorem (Necessary efficiency criteria) Let E:R×[, ]→R be a mapping such that E(t,λ) =λmin{t,λ},t∈R,λ∈[, ],and x∗be a properly efficient solution for problem(P). Assume that there exists a feasible point x for(P)such that gi(x) < ,i= , , . . . ,m,and each gi,i∈I(x∗),is E-[, ]convex at x∗∈M.Then there exist scalarsγi> ,i= , , . . . ,k and ui≥,i∈I(x∗),such that the triplet(x∗,γi,ui)satisfies
k
i=
γi∇fi
x∗+ i∈I(x∗)
ui∇gi
x∗= . ()
Proof Let the system
x–x∗T∇fq
x∗< ,
x–x∗T∇fi
x∗≤ for alli=q, ()
x–x∗T∇gi
x∗≤, i∈Ix∗,
have a solution for everyq= , , . . . ,k. Since, by the assumed Slater-type condition,
gi(x˜) –gi
x∗< , i∈Ix∗,
and fromE-[, ] convexity ofgiatx∗, we get
˜
x–x∗T∇gi
x∗< , i∈Ix∗. ()
Therefore from () and ()
x–x∗+ρx˜–x∗T∇gi
x∗< , ∀i∈Ix∗,ρ> .
Hence for some positiveβsmall enough
gi
x∗+βx–x∗+ρx˜–x∗<gi
x∗= , i∈Ix∗.
Similarly, fori∈/I(x∗),gi(x∗) < and forβ> small enough
gi
Thus, forβsufficiently small and allρ> ,x∗+β[(x–x∗) +ρ(x˜–x∗)] is feasible for problem (P). For sufficiently smallρ> () gives
fq
x∗+βx–x∗+ρx˜–x∗<fq
x∗. ()
Now, for allj=qsuch that
fj
x∗+βx–x∗+ρx˜–x∗>fj
x∗, ()
consider the ratio
N(β,ρ) D(β,ρ)=
[fq(x∗) –fq(x∗+β[(x–x∗) +ρ(x˜–x∗)])]/β
[fj(x∗+β[(x–x∗) +ρ(x˜–x∗)]) –fj(x∗)]/β. ()
From (),N(β,ρ)→–(x–x∗)T∇f
q(x∗) > . Similarly,D(β,ρ)→(x–x∗)T∇fj(x∗)≤; but, by (),D(β,ρ) > , soD(β,ρ)→. Thus, the ratio in () becomes unbounded, contra-dicting the proper efficiency ofx∗for (P). Hence, for eachq= , , . . . ,k, the system () has no solution. The result then follows from an application of Farkas’ lemma, namely
k
i=
γi∇fi
x∗+ i∈I(x∗)
ui∇gi
x∗= , u≥.
Theorem Assume that x∗is an efficient solution for problem(P)at which the Kuhn-Tucker constraint qualification is satisfied.Then,there exist scalarsγi≥,i= , , . . . ,k,
k
i=γi= ,uj≥,j= , , . . . ,m,such that
k
i=
γi∇fi
x∗+ m
j= uj∇gj
x∗= , m
j= ujgj
x∗= .
Proof Since every efficient solution is a weak minimum, by applying Theorem . of Weir and Mond [] forx∗, we see that there existγ ∈Rk,u∈Rmsuch that
γT∇fx∗+uT∇gx∗= , uTgx∗= ,
u≥, γ ≥, γTe= ,
wheree= (, , . . . , )∈Rk.
Example LetE-[, ] :R×[, ]→Rbe defined asE(t,λ) =λ√
t, wheret∈R, andλ∈
[, ]. Consider the problem:
min f(x,y) =x,
min f(x,y) = (y–x)
s.t. (x,y)∈M=(x,y)∈R:x+y≤, ≤y≤,x≥,
where f, and f areE-[, ] convex functions on convex setM. It is clear thatf(M) is R
-nonconvex set (see Figure (a)), but the image of the objective spacef(M) under the
Figure 1 Example 2 of bicriteriaE-[0, 1] convex problem. (a)f(M) isR2
-nonconvex.(b)The image off(M) underE-[0, 1] map isR2-convex.
(i) Formulate the weighting problem (Pw) as
min wx+w(y–x)
subject tox∈M,
wherew,w≥,w+w= .
It is clear that a point (,y)∈M, ≤y≤, is an optimal solution for (Pw) corresponding w= (w, ), <w≤, and a point (x, )∈M, ≤x≤ is an optimal solution for (Pw) corresponding tow= (,w), <w≤. Hence the set of efficient solutions of problem (P) can be described as
X=(x, )∈M: ≤x≤ and (,y)∈M: ≤y≤.
(ii) Formulate the problem Pq(ε) as
min x
subject to
(x,y)∈M,
(y–x)≤E(ε, )
and
min (y–x)
subject to
(x,y)∈M,
x≤E(ε, ).
It is easy to see that the points{(x, )∈M: ≤x≤ and (,y)∈M: ≤y≤}are optimal solutions corresponding to
(iii) Applying the Kuhn-Tucker conditions yields
γ
x∗– γ
y∗–x∗+u–u= ,
γ
y∗–x∗+u+u–u= ,
u
x∗+y∗– = ,
u
y∗– = ,
u
–y∗= ,
–ux∗=
and
x∗+y∗≤, y∗≥, y∗≤, x∗≥,
whereγi≥,i= , ,γ+γ= , andui≥,i= , , , . From this system we conclude that the set of efficient solutions can be described as
X=(x, )∈M: ≤x≤ and (,y)∈M: ≤y≤.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author is grateful to everyone who contributed, in one way or another, to the success of this paper. Also, UOH funds are gratefully acknowledged for a research grant that has been given so that this research could be conducted.
Received: 2 December 2014 Accepted: 23 April 2015
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![Figure 1 Example 2 of bicriteria Eunder-[0,1] convex problem. (a) f(M) is R2≧-nonconvex](https://thumb-us.123doks.com/thumbv2/123dok_us/257574.1025849/17.595.118.477.82.224/figure-example-of-bicriteria-eunder-convex-problem-nonconvex.webp)
