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Biconvex programming approach to optimization
over the weakly ecient set of a multiple objective ane
fractional problem
HoangQ. Tuyen, Le D. Muu
∗Department of Optimization & Control, Hanoi Institute of Mathematics, Box 631, Bo Ho, Hanoi, Viet Nam
Received 1 April 1999; received in revised form 1 October 2000; accepted 1 November 2000
Abstract
We formulate the problem of optimizinga convex function over the weakly ecient set of a multicriteria ane fractional program as a special biconvex problem. We propose a decomposition algorithm for solving the latter problem. The proposed algorithm is a branch-and-bound procedure taking into account the ane fractionality of the criterion functions. c2001 Published by Elsevier Science B.V.
Keywords:Ane fractional; Weakly ecient; Optimization over the weakly ecient set; Biconvex programming; Decomposition
1. Introduction and the problem statement
Consider the followingmulticriteria mathematical programmingproblem:
min{F(x) = (f1(x); : : : ; fp(x))|x∈X}; (VP)
where X⊂Rn is compact and each fi (i= 1; : : : ; p) is ane fractional on X.
Ane fractional functions are widely used as performance measures in some management situations, produc-tion planningand schedulingand the analysis of nancial enterprise [8,17]. Thus, multicriteria programming problems with ane fractional criterion functions are important and have wide applications.
We recall that a point x∈ X is called weakly ecient for Problem (VP) if whenever F(y)¡ F(x) and
y∈X then F(y) =F(x). By WE(F; X) we shall denote the set of all weakly ecient points. Here and subsequently for two vectors a= (a1; : : : ; ap) andb= (b1; : : : ; bp) the notation a ¡ b(resp. a6b) means that
This paper is supported in part by the Basic Program in Natural Sciences.
∗Correspondingauthor.
E-mail address:ldmuu@thevinh.ncst.ac.vn (L.D. Muu).
ai¡ bi (resp. ai6bi) for alli= 1; : : : ; p. A special case of the multicriteria ane fractional problem (VP) is
the multicriteria linear problem where X is polyhedral and each fi is linear. It is well known that even in
the linear case the weakly ecient set is not necessarily convex. Therefore, the computational eort required to generate all of the weakly ecient points becomes unmanageable and seems to grow exponentially with problem size.
In some situations, however, a real-valued function, say f, is available which acts as a criterion function for measuringthe importance of or for discriminatingamongthe ecient alternatives. The problem of nding a most preferred weakly ecient point (with respect to f) can be written as
min{f(x)|x∈WE(F; X)}: (P)
As an example of Problem (P), let us consider a domestic–foreign investment model which has pinvestment sources. Suppose that one can choose domestic or foreign investment from each investment source. Let
x= (x1; : : : ; xn) represent the vector of investment levels. Let (ai)Tx and (bi)Tx denote the domestic and
foreign investment prots resulting from ith investment source at the levelx. The overall goal of the decision maker is to determine a minimum cost-investment feasible plan xwhich is given by the functionf(x) :=dTx.
However for each investment source, the decision maker also wants to maintain a low domestic investment level relative to the foreign investment level at each investment project. This model leads to a minimization problem over the ecient set of a multiple objective linear fractional program where every criterion function
fi(x) is the ratio of the domestic and foreign investment prots, i.e.,fi(x) = (ai)Tx=(bi)Tx(i= 1; : : : ; p). Note
that for the multiple objective ane fractional program (VP), the weakly ecient set, in general, is much more easily handled than the ecient set. In fact, the weakly ecient set of (VP) is always compact while its ecient set may be neither open nor closed [8]. In response to this diculty, in the above model, instead of minimizingthe cost function f(x) over the ecient set of all feasible investment plans, the decision makers would minimize f(x) over the weakly ecient set rather than the ecient set. Since the weakly ecient set contains the ecient set, the minimum taken on the weakly ecient set, in general, is less than that taken on the ecient set.
Problem (P) can be considered as a direct development of optimization problem over the weakly ecient set of a multiple objective linear program. The latter problem has been considered by some authors [2,3– 6,20].
A main diculty of Problem (P) arises from the fact that the weakly ecient set, in general, is neither convex nor given explicitly as a constrained set of an ordinary mathematical programming problem. Because WE(F; X) is rarely a convex set, Problem (P) even with f linear, may have local extrema which are not global. Such an example can be easily found (see e.g. Fig. 1 in [7]).
Recently Problem (P) has been studied by Malivert [15]. It is shown in [15] that this problem can be proceeded by solvinga sequence of convex-constrained penalized problems of the form
min{f(x) +tkpw(x)|x∈X};
wheretk¿0, andpw is a certain penalty function representingthe weakly ecient set which is neither convex
nor dierentiable. So the penalized problems remain dicult global optimization ones.
This work was intended as an attempt to develop methods for globally solving Problem (P) when WE(F; X) is the weakly ecient set of a convex constrained multicriteria ane fractional program, and f is a convex function.
2. Biconvex programming formulation
Let A⊂Rp; B⊂Rn be two convex sets, and let g:A×B→R. The function g is said to be biconvex on
A×B if g(:; y) is convex on A for each xed y∈B, and g(x; :) is convex on B for each xedx∈A. A mathematical programmingproblem involvingminimizinga biconvex function under linear constraints was introduced rst by Al-Khayyal and Falk [1] in 1983. In recent years, mathematical programming problems where the objective function and=or constraints are biconvex functions have been studied by several authors [10,11,19]. A solution approach based upon a primal-dual relaxation has been developed in [10] for this class. In this section, we formulate the problem of optimizinga convex function over the weakly ecient set of a multicriteria ane fractional program as a special convex program with an additional biconvex constraint.
To be precise, we suppose that the criterion functions in the multicriteria programming problem (VP) are given by
F(x) =
A1x+s1
B1x+t1
; : : : ;Apx+sp Bpx+tp
;
where Ai; Bi are n-dimensional vectors, si; ti are real numbers for all i= 1; : : : ; p. As usual we assume that
Bix+ti¿0 for all x∈X and all i= 1; : : : ; p. It is well known that an ane fractional function is both
pseudoconvex and pseudoconcave.
By denition, the weakly ecient set of Problem (VP) can be given by
WE(F; X) ={x∈X|@y∈X:F(x)−F(y)¿0}:
Since X is compact, WE(F; X) is closed and connected [8,14,18]. Thus, an optimal solution of (P) always exists. The followingtheorem due to Malivert [15] will be useful for our purpose.
Theorem 2.1 (Malivert [15]). A vector x∈X is weakly ecient if and only if there exist real numbers
i¿0; i= 1; : : : ; p; not all zeros such that
p
i=1
i[(Bix+ti)Ai−(Aix+si)Bi](y−x)¿0 ∀y∈X:
By dividingeach i by pj=1j¿0 we may assume that pj=1j= 1. So if
:=
= (1; : : : ; p)|¿0; p
j=1
j= 1
;
then
WE(F; X) =
x∈X| ∃∈;
p
i=1
i[(Bix+ti)Ai−(Aix+si)Bi](y−x)¿0∀y∈X
:
Thus (P) can be formulated as the followingsemi-innite programmingproblem:
(IP)
min f(x)
subject to x∈X; ∈;
p
i=1
Dene the function g:×X →R by setting, for each (; x)∈×X,
Proposition 2.1. (i) g is a continuous biconvex function on ×X. (ii) g(; x) +TCx¿0 for all(; x)∈×X.
(iii) Problem(IP) is equivalent to the problem
(P)
min f(x)
subject to g(; x) +TCx60; x∈X; ∈:
Proof. (i) SinceX is compact and the functionp
i=1i[(Bix+ti)Ai−(Aix+si)Bi]yis continuous, the continuity
of g follows immediately from the maximum theorem [4].
Let 0661; 1; 2 ∈. Then 1+ (1−)2∈. For each xedx∈X we have
For simplicity of notation let Li
= max
These inequalities can be rewritten as
we have
p
i=1
i(tiAi−siBi)x6min y∈X
p
i=1
i[(Bix+ti)Ai−(Aix+si)Bi]y:
Usingagain the denition of g(; x) and the matrix C we can write the last inequality as g(; x) +TCx60.
Corollary 2.1. For each ∈ the function g(; x) +TCx is ane on the set
X:={x∈X|g(; x) +TCx60}:
Proof. For simplicity of notation let
h(; x) :=g(; x) +TCx:
By Proposition 2.1, h(; :) is convex on X and h(; x)¿0 for all (; x)∈×X. ThusX is convex, and
X={x∈X|g(; x) +TCx= 0}:
Let x; y∈X, 06t61. Since X is convex,tx+ (1−t)y∈X. Then
0 =h(; tx+ (1−t)y)6th(; x) + (1−t)h(; y) = 0:
Hence
h(; tx+ (1−t)y) =th(; x) + (1−t)h(; y):
Remark. To evaluate the function h at each (; x) we have to solve the problem
min
y∈X p
i=1
i[(Bix+ti)Ai−(Aix+si)Bi]y:
This is a linear program if X is a polyhedral convex set given by the format traditional in linear program-ming.
3. Solution method
In this section we shall describe a decomposition method for approximatinga globally optimal solution of Problem (P) with f beinga continuous convex function on X. As usual, for a given ¿0, we call a point
x an -optimal solution to Problem (P) if x is feasible and f(x)−f∗6(|f(x)|+ 1), where f∗ denotes the
optimal value of (P).
In view of Proposition 2.1, minimizinga convex function over the weakly ecient set of a multicriteria ane fractional programmingproblem amounts to solvingthe biconvex program (P). This problem can be treated by existingmethods of biconvex programming[10,19]. Below we propose a decomposition algorithm for solvingProblem (P) which takes into account the specic structure of the biconvex-constrained function and the simplex appearingin this problem. The proposed algorithm is a branch-and-bound procedure usinga Lagrangian bounding operation and a simplicial subdivision.
3.1. Lagrangian bounding operation
Dene the function :→R by setting
Problem (P) can then be rewritten as
min{()|∈}: (MP)
Note that since X; are closed, and f and g are continuous on X and ×X, respectively, the function, by the maximum theorem [4], is continuous on .
In view of Corollary 2.1, the function g(; x) +TCxis ane on the feasible domain of Problem (P). So
in an important case when X is polyhedral, the feasible domain of Problem (P) is polyhedral too.
The followingproposition gives a relationship between Problems (P), (P) and (MP). The proof of this proposition is obvious from the results of the precedingsection.
Proposition 3.1. A point (∗
; x∗
) is optimal to Problem (P) if and only if x∗
is optimal to (P); ∗
is optimal to (MP); and f∗=f(x
∗
) =(∗
).
Note that unlike general mathematical programming problems having nonconvex feasible domains, a feasible point of Problem (P) can be computed by solvinga standard convex program. In fact, if ∈ and x is
an optimal solution of the convex problem
min{g(; x) +TCx|x∈X};
then (; x) is feasible for (P). Hence x is feasible for (P). So upper bounds forf∗ can be computed by
existing methods of convex programming. As the algorithm (to be described below) executes, more and more feasible points can be found, and thereby upper bounds for f∗ can be iteratively improved.
We now compute a lower bound for f∗ by using Lagrangian duality. It is well recognized [9] that the
duality gap obtained by solving Lagrangian dual is often reduced to zero in the limit by appropriate renement of the partition sets. Let S be a fully dimensional subsimplex of . Let V(S) denote the vertex-set of S. Consider Problem (P) restricted to S, i.e.,
(PS)
f∗(S) := min f(x)
subject to g(; x) +TCx60
x∈X; ∈S:
Let L(; ; x) be the Lagrangian function of this problem. That is
L(; ; x) =f(x) +(g(; x) +TCx): (1)
Dene the function m(; ) as
m(; ) = min
x∈X{f(x) +(g(; x) + TCx)}:
From the well-known Lagrangian duality theorem we have
m(; )6() ∀¿0; ∀∈S: (2)
Since, by Corollary 2.1, g(; x) +TCx is ane on the feasible domain of Problem (P
), we have
sup
¿0
m(; ) =(): (3)
Let
uS() = min
∈S m(; ): (4)
From (2) it follows that
uS() = min ∈S m(; )
6min
∈S() =f∗(S) ∀
Hence
is concave in . Thus the function
uS() = min ∈S m(; )
is concave on R+, because it is the minimum of a family of concave functions.
Since X and S are compact and the function l(:; x; :) is bilinear on S ×X), by the well-known minimax
theorem [16] one can interchange the min–max and max–min operations. Thus
uS() = min x∈X
f(x) + max
y∈X min∈S ‘(; x; y)
:
Observingthat, for each xed x and y, the function ‘(·; x; y) is linear in we have
min
∈S ‘(; x; y) = min∈V(S)‘(; x; y):
For each j∈V(S), let
uj() := min
x∈X maxy∈X{f(x) +‘( j; x; y)}:
Then
uS() = min j∈V(S)uj()
= min
j∈V(S)minx∈X maxy∈X{f(x) +‘( j; x; y)};
which proves the lemma.
Remark. In view of Lemma 3.1, computingthe lower bound (S) amounts to maximizingthe concave function uS() on R+. To evaluate uS() at each¿0 we have to solve standard minimax subproblems, one
for each vertex of S.
3.2. Simplicial bisection
At each iteration k of the algorithm to be described below a subsimplex of the simplex will be bisected into two subsimplices in such a way that as the algorithm executes the obtained lower and upper bounds tend to the same limit. We shall use the simplicial bisection via a longest edge. This bisection is widely used in global optimization (see e.g. [12,13]). This simplicial bisection can be described as follows. Let Sk be a fully
dimensional subsimplex of to be bisected at iteration k. Let vk andwk be two vertices of S
k such that the
edge joining these vertices is longest. Let uk be another point on this edge. Thusuk=t
kvk+ (1−tk)wk with
0¡ tk¡1. BisectSk into two subsimplicesSk1 andSk2, whereSk1 andSk2 are obtained fromSk by replacing
vk and wk, respectively, by uk. It is well known [12,13] that S
k=Sk1∪Sk2, and that if {Sk} is an innite
sequence of nested simplices generated by this simplicial bisection process such that 0¡ 0¡ tk¡ 1¡1 for
every k, then the sequence {Sk} shrinks to a singleton.
Now we are in a position to describe the algorithm.
Algorithm
Initialization: Fix a tolerance ¿0. Set S0:= and solve the univariate convex program (concave
maxi-mization)
(S0) := sup{uS0()|¿0};
to obtain a lower bound for f∗.
Choose ∈ and compute an upper bound for f∗ by solvingthe convex program
() := min{f(x)|g(; x) +TCx60; x∈X}:
If x is an optimal solution of this problem, then (; x) is feasible for Problem (P) (hencex ∈WE(F; X)).
Set 0:=(S0), and -0:=
{S0} if 0−0¿ (|0|+ 1); ∅ otherwise:
Let k←0 and go to Iteration k.
Iteration k (k= 0;1: : :).
Step k1 (selection):
(a) If -k=∅ then the algorithm terminates: xk is an -optimal solution and k is an -optimal value to
Problem (P).
(b) If -k =∅, then select Sk∈-k such that
k:=(Sk) = min{(S)|S ∈-k}:
Choose k∈S
k and compute (k) to obtain a new upper bound for f∗.
Step k2 (bisection): Bisect Sk into two simplices Sk1 and Sk2 by the simplicial bisection described above. Step k3 (bounding): Compute (Skj) (j= 1;2) by solving
(Skj) := sup{uSkj()|¿0}; (j= 1;2):
Step k4 (Updating): As(k), (S
k1) and (Sk2) are computed one or more new feasible points have been
found. Let xk+1 be the currently best feasible point (with respect to f) amongxk and the newly generated feasible points.
Set k+1 :=f( xk+1)
and
-k+1← {S ∈(-k\{Sk})∪ {Sk1; Sk2}:k+1−(S)¿ (|k+1|+ 1)}:
Increase k by 1 and go to iteration k.
Convergence Theorem. (i) If the algorithm terminates at iteration k then xk is a global -optimal solution to Problem (P).
(ii) If the algorithm does not terminate then k րf∗; k ցf∗ as k→+∞; and any cluster point of
the sequence {xk} is a global optimal solution to Problem (P).
Proof. (i) If the algorithm terminates at iterationk, then-k=∅. This implies thatk−k6(|k|+ 1). Since
k6f∗, andk=f( xk), it follows that f( xk)−f∗6(|f( xk)|+ 1). Hence xk is a global -optimal solution.
(ii) Since for every k we haveSk=Sk1∪Sk2, by the rule for computinglower bound(Sk) we have
k=(Sk)6(Sk+1) =k+1 ∀k:
Also, since k+1 is the currently smallest upper bound determined at Step k4, we have k+16k for every k.
Thus both ∗= limk and∗= limk do exist and satisfy
∗6f∗6∗: (5)
Suppose that the algorithm does not terminate. Then it generates an innite sequence of nested simplices that, for simplicity of notation, we also denote by {Sk}. Since the subdivision is the simplicial bisection, this
sequence shrinks to a singleton, say ∗∈
. By the rule for computinglower bound k we have
k= sup ¿0
min
∈Skm(; )
¿min
∈Skm(; ) ∀
¿0:
Notingthat the sequence {Sk} tends to ∗ as k→+∞we obtain
Since (k) is an upper bound determined at Step k1 and
k+1 is the currently smallest upper bound obtained
at Step k4, we have
k+16(k) ∀k:
Since k →∗, it follows from the continuity of
that
∗= limk6lim(k) =(∗): (7)
On the other hand, by the Lagrangian duality theorem for the convex problem determining (∗
) we have
sup
¿0
m(; ∗
) =(∗
):
Then from (6) and (7) it follows that
∗6( ∗
)6∗;
which together with (5) implies ∗=f∗=∗=( ∗
). Let x∗
be any cluster point of the sequence {xk}. By denition we have k=f( xk). Since k → f∗,
by the continuity of f we have f∗=f(x∗). Since xk ∈ WE(F; X) for all k and WE(F; X) is closed [8],
x∗∈WE(F; X). Hence x∗ is a globally optimal solution of Problem (P).
Remark. (1) When ¿0 the algorithm must terminate after a nite number of iterations. Indeed, if the algorithm does not terminate at iteration k, then k−k¿ (|k|+ 1). Since k−k → 0 as k → +∞,
it follows that when ¿0 the inequality k−k¿ (|k|+ 1) cannot happen with innitely many k. This
implies that the algorithm must terminate after a nite number of iterations.
(2) Since uS()6f∗(S) for all ¿0, instead of computing (S) = sup{uS()|¿0}, we can take (S) =
sup{uS()|¿0} −k, where k¿0 and k ց0 as k→+∞.
(3) Since the branchingoperation takes place in the criterion space, the proposed algorithm is expected to apply to problems where the number of the criteria is relatively small. The number of the decision variables may be larger.
Acknowledgements
The authors would like to express their gratitude to the referee for his useful comments and remarks on an earlier version of this paper which helped them very much to improve the paper.
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