Multiobjective duality for convex semidefinite programming problems

Multiobjective duality for convex semidefinite programming problems G. Wanka, R.I. Bot¸ and S.M. Grad

Abstract. We treat some duality assertions regarding multiobjective convex semidefinite programming problems. Having a vector minimization problem with convex entries in the objective vector function, we establish a dual for it using the so-called conjugacy approach. In order to deal with the duality assertions between these problems we need to study the duality properties and the optimality conditions of the scalarized problem associated to the initial one. Using these results we present the weak, strong and converse duality assertions regarding the primal problem and the dual we obtained for it. Keywords: multiobjective duality, semidefinite programming, Pareto-efficiency, convex optimization AMS subject classification: Primary 49N15, secondary 90C22, 90C25, 90C29

1. Introduction This paper presents some duality assertions regarding the multiobjective semidefinite programming problems. The duality model we are considering here has been introduced by W. Fenchel and R.T. Rockafellar and it consists in attaching to an optimization problem another problem, called its dual, by means of perturbation functions. This dual problem is important, because its solutions may reveal us in certain conditions the solutions of the initial problem. More on this subject may be found in [2], [6], [10], [11]. We deal further with semidefinite programming problems, namely, optimization problems with positive semidefinite constraints. The duality for the single objective linear case has already been presented in many papers, such as [1], [4], [5], [8], [9]. The word ”multiobjective” appears in the title because we consider multiple vector valued objective functions. We treat the duality properties of these multiobjective functions considering the so-called Pareto-efficiency. The approach we use to treat the multiobjective dual problems has been introduced in the articles [10] and [11]. G. Wanka: Chemnitz University of Technology, Faculty of Mathematics, D-09107 Chemnitz, Germany, e-mail: [email protected] R.I. Bot¸: Chemnitz University of Technology, Faculty of Mathematics, D-09107 Chemnitz, Germany, e-mail: [email protected] S.M. Grad: ”Babe¸s-Bolyai” University Cluj-Napoca, Faculty of Mathematics and Computing Science, RO-3400 Cluj-Napoca, Romania, e-mail: [email protected] ISSN 0232-2064 / $ 2.50

c Heldermann Verlag Berlin

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G. Wanka et. al.

We begin with a problem of minimizing a vector function with convex entries subject to positive semidefinite inequality constraints. This vector minimization is considered using the so-called Pareto-efficiency and proper efficiency, whose definitions are reminded here. Then we take the scalarized problem associated to it and we calculate its dual. From the duality assertions and optimality conditions, obtained further, we are able to extract duality properties regarding the primal multiobjective semidefinite optimization problem and its dual. Next we present the weak, strong and converse duality assertions regarding these problems. Finally, we derive as special cases the dual problems of the multiobjective semidefinite programming problem with linear objective function and of the multiobjective fractional programming problem with linear inequality constraints. The last one is presented also as a special case of the problem treated in [12].

2. Problem formulation Let us consider the following multiobjective semidefinite programming problem with convex objective functions and convex constraints (P )

v − min f (x), x∈X

where f = (f1 , f2 , ..., fk )T , X = {x = (x1 , x2 , ..., xm ) ∈ Rm : F (x) ≥S+n 0}, and F (x) = F0 +

m X

xi · F i .

i=1

m

For each j = 1, ..., k, fj : R → R is a real-valued convex function, and also Fi ∈ S n , i = 0, ..., m. Here we have denoted by S n the linear subspace of the symmetric n × n matrices with real entries, i.e. S n = {A ∈ Rn×n | A = AT }, n and by S+ the cone of the symmetric positive semidefinite n × n matrices with real entries, i.e. n S+ = {A ∈ S n | xT · A · x ≥ 0, ∀x ∈ Rn },

which introduces the so-called L¨owner partial order on S n n A ≥S+n B ⇔ A − B ∈ S+ , A, B ∈ S n .

So our constraint F (x) ≥S+n 0 means actually that F (x) is a symmetric positive n we consider the scalar product from S n semidefinite matrix. Also, on S+ hA, Bi =

n X

i,j=1

Aij · Bij = T r AT · B , A, B ∈ S n ,

Multiobjective duality for semidefinite programming

3

where T r(A) denotes the trace of the matrix A and ”· ” is the well-known product of matrices. To deal with the dual properties of the problem (P ), using the method introduced in [10], we need to reformulate the feasible set by introducing a new function g:R

m

n

→ S , g(x) := −F0 +

m X

xi · (−Fi ).

i=1

In this circumstance, othe feasible set of the problem (P ) may be written as X = n x ∈ Rm : g(x) ≤S+n 0 .

There are several notions of solutions for this type of problem, but we use here so-called Pareto-efficient and properly efficient solutions. Let us remind these notions.

Definition 1. An element x ¯ ∈ X is said to be Pareto-efficient with respect to (P ) if from f (x) ≤Rk+ f (¯ x), x ∈ X , follows f (x) = f (¯ x). Definition 2. An element x ¯ ∈ X is called properly efficient with respect to (P ) k k P P if there exists λ = (λ1 , . . . , λk )T ∈ int(Rk+ ) such that λi fi (¯ x) ≤ λi fi (x), for all i=1

x ∈ X.

i=1

Remark 1. We denote by ”≤Rk+ ” the partial ordering induced by the non-negative orthant Rk+ = {x = (x1 , x2 , ..., xk )T : xi ≥ 0, i = 1, ..., k} on Rk . Hence int(Rk+ ) = λ = (λ1 , ..., λk )T ∈ Rk+ : λi > 0, i = 1, ..., k .

Remark 2. A properly efficient element is also a Pareto-efficient one, with respect to the optimization problem (P ).

3. The scalarized problem In order to deal with the properly efficient solutions of (P ) we consider the scalarized problem attached to it k X λi fi (x), (Pλ ) inf x∈X

i=1

where λ = (λ1 , . . . , λk )T ∈ int(Rk+ ) and then study its duality properties according to the mentioned approach. To be able to do this, let us consider first a general semidefinite optimization problem (Pg ) where f˜ : Rm → R is a convex function.

inf f˜(x),

x∈X

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G. Wanka et. al.

To obtain the desired dual for (Pg ) we use the method described in [10]. So let us consider the perturbation function m

m

n

Φ : R × R × S → R, Φ(x, p, Q) =

f˜ (x + p) , +∞,

if g(x) ≤S+n Q, otherwise.

Its conjugate function is Φ∗ (x∗ , p∗ , Q∗ ) =

{hx∗ , xi + hp∗ , pi + hQ∗ , Qi − Φ(x, p, Q)}

sup

m

x,p∈R , g(x)−Q≤S n 0 +

=

sup

x,p∈Rm , g(x)−Q≤S n 0 +

n

o ˜ hx , xi + hp , pi + hQ , Qi − f (x + p) . ∗

∗

∗

∗

It is well-known that the space S n is self-dual, i.e. (S n ) = S n . In [13] there is ∗ n n n = S+ . This last property will be proved that the cone S+ is also self-dual, i.e. S+ used later. The dual of (Pg ) is obtained (cf. [2]) calculating the expression Pg∗

sup {−Φ∗ (0, p∗ , Q∗ )} .

∗

p ∈Rm , Q∗ ∈S n

To ease our calculation we introduce the following new variables r := x + p, S := Q − g(x). The expression of the conjugate function of Φ becomes Φ∗ (x∗ , p∗ , Q∗ ) =

sup

x,r∈Rm , S≥S n 0 +

∗

= sup hQ , Si + sup S≥S n 0 +

r∈Rm

n

n

hx∗ , xi + hp∗ , r − xi + hQ∗ , S + g(x)i − f˜(r)

o

o ˜ hp , ri − f (r) + sup {hx∗ , xi − hp∗ , xi + hQ∗ , g(x)i} . (3.1) ∗

x∈Rm

As required above, we take x∗ = 0, Φ∗ (0, p∗ , Q∗ ) = sup hQ∗ , Si + f˜∗ (p∗ ) + sup {hQ∗ , g(x)i − hp∗ , xi} , x∈Rm

S≥S n 0 +

where f˜∗ (p∗ ) = sup r∈Rm

n

hp∗ , ri − f˜(r)

o

is the conjugate function of f˜ at p∗ . It follows

Φ∗ (0, p∗ , Q∗ ) = sup hQ∗ , Si + f˜∗ (p∗ ) S≥S n 0 +


+ sup x∈Rm

(

−

m X

xi hQ∗ , Fi i − hQ∗ , F0 i −

x∈Rm

xi p∗i

i=1

i=1

+ sup

m X

(

∗

−hQ , F0 i −

m X

)

= sup hQ∗ , Si + f˜∗ (p∗ ) S≥S n 0 +

)

xi (hQ∗ , Fi i + p∗i )

i=1

+f˜∗ (p∗ ) − hQ∗ , F0 i + sup

x∈Rm

(

−

m X

5

= sup hQ∗ , Si S≥S n 0 +

)

xi (hQ∗ , Fi i + p∗i ) .

i=1

For the two suprema encountered above, we have ∗

sup hQ , Si = S≥S n 0 +

0, +∞,

if Q∗ ≤S+n 0, otherwise,

0, +∞,

if hQ∗ , Fi i + p∗i = 0, i = 1, ..., m, otherwise.

and sup x∈Rm

(

−

m X

∗

xi (hQ , Fi i +

p∗i )

i=1

)

=

As the above infinite values are not relevant for our supremum problem (P g∗ ), the dual becomes n o sup Pg∗ −f˜∗ (p∗ ) + hQ∗ , F0 i , ∗ ∗ m Q∗ ≤S n 0,p∗ =(p∗ 1 ,p2 ,...,pm )∈R , +

∗ p∗ i =−T r(Q ·Fi ),i=1,...,m

which, denoting Q := −Q∗ , may be written after some transformations also as (Pg∗ )

sup Q≥S n 0 +

n

o ∗ ˜ −f (T r(Q · F1 ), T r(Q · F2 ), ..., T r(Q · Fm )) − T r(Q · F0 ) .

(3.2)

Remark 3. (Pg∗ ) is the dual problem obtained using the conjugacy approach to our semidefinite optimization problem with general convex objective function (P g ). For f˜(x) = hc, xi, x ∈ Rm , it becomes (Pc∗ )

sup

{−T r(Q · F0 )}.

Q≥S n 0, +

T r(Q·Fi )=ci ,i=1,...,m

This is exactly the dual problem already obtained for the linear case in the literature (see [1], [8], [9], [13]). So the dual of (Pλ ) looks like (Pλ∗ )

sup

∗

m

Q≥S n 0,p ∈R , +

p∗ =(T r(Q·F1 ),...,T r(Q·Fm ))

(

−

k X i=1

λ i fi

!∗

)

(p∗ ) − T r (Q · F0 ) ,

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G. Wanka et. al.

which may be turned, using the formula (cf. [6]) k X

λ i fi

i=1

!∗

∗

(p ) = inf

(

k X

(λi fi ) (˜ pi ) :

i=1

into (Pλ∗ )

∗

(

sup

m

Q≥S n 0,p˜i ∈R ,i=1,...,k, +

p˜i = p

i=1

−

k X

∗

)

,

)

(λi fi )∗ (˜ pi ) − T r (Q · F0 ) .

i=1

k

P

k X

p˜i =(T r(Q·F1 ),...,T r(Q·Fm ))

i=1

∗

λi fi∗

Knowing that (λi fi ) (˜ pi ) = dual problem may be simplified to (Pλ∗ )

1 ˜i λi p

sup

Q≥S n 0,pi ∈Rm ,i=1,...,k, +

k P

and denoting pi :=

(

−

k X

1 ˜i , i λi p

= 1, ..., k, the

)

λi fi∗ (pi ) − T r (Q · F0 ) .

i=1

λi pi =(T r(Q·F1 ),...,T r(Q·Fm ))

i=1

4. Duality for the scalarized problem The weak and strong duality assertions hold for the linear problem as it is proved in [9]. The scalarized problem we are currently treating is a natural extension of the linear problem, so similar duality properties are to be formulated for it. As the proof of the weak duality theorem is trivial we do not mention it here. Theorem 1. There holds weak duality between (Pλ ) and (Pλ∗ ), i.e. inf(Pλ ) ≥ sup(Pλ∗ ). In order to prove the strong duality theorem we have to introduce the Slater Constraint Qualification (SCQ) there exists x0 ∈ Rm such that F (x0 ) >S+n 0 .

By ” >S+n ” we have denoted the partial ordering on S n introduced by the set of the n symmetric positive definite n × n matrices, which actually coincides with int(S + ) (cf. [8]). We formulate the strong duality theorem for (Pg ) from which we obtain the one regarding (Pλ ). Theorem 2. Let be inf (Pg ) > −∞ and (SCQ) be fulfilled. Then the dual problem has an optimal solution and there is strong duality between (Pg ) and (Pg∗ ), i.e.

(Pg∗ )

inf(Pg ) = max(Pg∗ ).


7

Proof. The convexity of f and g ensures the convexity of Φ. The constraint qualn ). ification (SCQ) being fulfilled, there exists x0 ∈ Rm such that F (x0 ) ∈ int(S+ 0 Next we prove that the function Φ(x , ·, ·) is continuous at (0, 0). Proposition 2.3 and Theorem 4.1 in [2] imply in this case the existence of an optimal solution of (P g∗ ) and state the equality of the optimal objective values of (Pg ) and (Pg∗ ). Therefore let be ε > 0. The function f˜ being continuous over Rm , there exists V1 an open neighborhood of 0 in Rm such that for all p ∈ V1 , ˜ 0 0 ˜ f (x + p) − f (x ) < ε.

n Because g(x0 ) = −F (x) ∈ −int(S+ ), there exists V2 ⊆ S n an open neighborhood of 0 such that for all Q ∈ V2 , n g (x0 ) ∈ Q − S+ ⇔ g (x0 ) ≤S+n Q.

Consider V = V1 × V2 , that is a neighborhood of (0, 0) in Rm × S n . For all (p, Q) ∈ V we have |Φ (x0 , p, Q) − Φ (x0 , 0, 0)| = f˜ (x0 + p) − f˜ (x0 ) < ε, which actually means that Φ(x0 , ·, ·) is continuous at (0, 0).

k P λi fi , we obtain the strong duality assertion for the scalarized Considering f˜ = i=1

problem.

Corollary 1. If inf(Pλ ) > −∞ and (SCQ) holds, then the dual problem (Pλ∗ ) has an optimal solution and there is strong duality between (Pλ ) and (Pλ∗ ), i.e. inf(Pλ ) = max(Pλ∗ ). Further we need also the optimality conditions regarding (Pλ ) and its dual (Pλ∗ ), so we formulate and prove the following theorem. Theorem 3. (a) Let (SCQ) be fulfilled and let x ¯ ∈ X be a solution to (Pλ ). Then there exists ¯ an optimal solution (¯ p1 , ..., p¯k , Q) to (Pλ∗ ) satisfying the following optimality conditions (i) fi (¯ x) + fi∗ (¯ pi ) = h¯ pi , x ¯i, i = 1, ..., k, ¯ · F (¯ (ii) T r Q x) = 0.

¯ feasible to (P ∗ ) satisfying (i) and (ii). Then x (b) Let x ¯ ∈ X and p¯1 , ..., p¯k , Q ¯ λ ∗ ¯ turns out to be an optimal solution to (Pλ ), p¯1 , ..., p¯k , Q an optimal solution to (Pλ ) and the strong duality between (Pλ ) and (Pλ∗ ) is true, k X i=1

λi fi (¯ x) = −

k X i=1

¯ · F0 . λi fi∗ (¯ pi ) − T r Q

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G. Wanka et. al.

¯ to Proof. (a) As (SCQ) is fulfilled, there exists an optimal solution (¯ p 1 , ..., p¯k , Q) such that k k X X ¯ · F0 . λi fi∗ (¯ pi ) − T r Q λi fi (¯ x) = −

(Pλ∗ )

i=1

i=1

This may be also written in the following way k X i=1

¯ · F0 = 0. λi (fi∗ (¯ pi ) + fi (¯ x)) + T r Q

Adding and subtracting in the left-hand side the term k X

λi (fi∗ (¯ pi ) + fi (¯ x) − h¯ pi , x ¯i) +

i=1

As

*

k X i=1

λi p¯i , x ¯

+

*

k X

λi p¯i , x ¯

i=1

k P

λi p¯i , x ¯

i=1

+

we get

¯ · F0 = 0. + Tr Q

¯ · F1 ), ..., T r(Q ¯ · Fm )), x = h(T r(Q ¯i =

m X

¯ · Fi ), x ¯i T r(Q

i=1

the previous relation becomes k X

λi (fi∗ (¯ pi ) + fi (¯ x) − h¯ pi , x ¯i) +

m X

¯ · Fi ) + T r(Q ¯ · F0 ) = 0, x ¯i T r(Q

i=1

i=1

which is equivalent to k X

¯ F (¯ λi (fi∗ (¯ pi ) + fi (¯ x) − h¯ pi , x ¯i) + hQ, x)i = 0.

(4.1)

i=1

¯ ≥S n 0, we have As x ¯ ∈ X , there follows F (¯ x) ≥S+n 0. Also knowing that Q + ¯ F (¯ hQ, x)i ≥ 0.

(4.2)

fi∗ (¯ pi ) + fi (¯ x) − hpi , x ¯i ≥ 0, i = 1, ..., k,

(4.3)

From Young’s inequality it stands

and, as λi > 0, i = 1, ..., k, k X

λi (fi∗ (¯ pi ) + fi (¯ x) − h¯ pi , x ¯i) ≥ 0.

i=1

Therefore

k X

¯ F (¯ λi (fi∗ (¯ pi ) + fi (¯ x) − h¯ pi , x ¯i) + hQ, x)i ≥ 0.

(4.4)

i=1

By (4.1) there follows that the inequalities encountered in (4.2) and (4.3) must be fulfilled as equalities. So the optimality conditions (i) and (ii) are verified. (b) All the calculations from (a) may be carried out in the reverse direction.


9

5. The multiobjective dual problem Now we are ready to introduce the multiobjective dual problem to the primal problem (P ),   −f1∗ (p1 ) − kλ1 1 T r (Q · F0 ) + t1   .. (D) v − max  , . (p,Q,λ,t)∈Y −fk∗ (pk ) − kλ1 k T r (Q · F0 ) + tk where

p = (p1 , p2 , ..., pk ) , pi ∈ Rm , i = 1, ..., k, Q ∈ S n , T

λ = (λ1 , λ2 , ..., λk ) ∈ Rk , t = (t1 , t2 , ..., tk )T ∈ Rk , and Y=

(

(p, Q, λ, t) : λ ∈ int(Rk+ ),

k X

λi ti = 0, Q ≥S+n 0,

i=1

k X

λi p i =

i=1

T r (Q · F1 ) , T r (Q · F2 ) , ..., T r (Q · Fm )

)

.

As (D) is a maximum vector optimization problem, we have to specify that we consider here the so-called Pareto-efficiency in the sense of maximum to distinguish its solutions. We recall its definition. ¯ t¯ ∈ Y is said to be Pareto-efficient for (D) if ¯ λ, Definition 3. An element p¯, Q, for each j = 1, ..., k, and for (p, Q, λ, t) ∈ Y, from 1 ¯ · F0 + t¯j ≤ −fj∗ (pj ) − 1 T r (Q · F0 ) + tj , −fj∗ (¯ pj ) − ¯ T r Q kλj k λj there follows 1 ¯ · F0 + t¯j = −fj∗ (pj ) − 1 T r (Q · F0 ) + tj . −fj∗ (¯ pj ) − ¯ T r Q kλj k λj

Further we formulate and prove the weak duality assertion for the multiobjective problems (P ) and (D). Theorem 4. There is no x ∈ X and no (p, Q, λ, t) ∈ Y such that fi (x) ≤ −fi∗ (pi ) −

1 T r (Q · F0 ) + ti , ∀i = 1, ..., k, kλi

and, for at least one j ∈ {1, ..., k}, fj (x) < −fj∗ (pj ) −

1 T r (Q · F0 ) + tj . kλj

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Proof. Let us assume the contrary, i.e. that there exist some x and (p, Q, λ, t) feasible to our problems fulfilling the conditions mentioned above. Then assembling the relations given in the hypothesis we obtain k X

λi fi (x)
S+n 0. If x ¯ is properly efficient to (P ), then there exists a Pareto-efficient ¯ t¯) ∈ Y to (D) and strong duality between (P ) and (D) is fulfilled, i.e. ¯ λ, solution (¯ p, Q, 1 ¯ · F0 + t¯i , i = 1, ..., k. fi (¯ x) = −fi∗ (¯ pi ) − ¯ T r Q k λi

Proof. The element x ¯ being properly efficient to (P ) implies that there exists a T m ¯ λ ∈ int(R+ ) such that x ¯ solves (Pλ¯ ), i.e. k X i=1

¯ i fi (¯ λ x) = min x∈X

k X

¯ i fi (x) . λ

i=1

(SCQ) is fulfilled, so there exists an optimal solution to Pλ¯∗ satisfying the optimality ¯ and define conditions in Theorem 3. Let us denote it by (¯ p1 , ..., p¯k , Q) 1 ¯ · F0 ∈ R, j = 1, ..., k. t¯j := p¯Tj x ¯ + ¯ Tr Q k λj


11

It follows + * k k k k X X X X 1 T ¯ j p¯j , x ¯j ¯ j p¯ x ¯ j t¯j = ¯ · F0 ¯ · F0 = λ ¯ + T r Q λ T r Q λ λ j ¯+ ¯j kλ j=1 j=1 j=1 j=1 k X ¯ Fj i¯ ¯ F0 i = hQ, ¯ F (¯ = hQ, xj + hQ, x)i = 0. j=1

¯ t¯ belongs to the set Y. Let us show ¯ λ, So far we have proved that the element p¯, Q, the remaining requirement, namely, that for all i = 1, ..., k, 1 ¯ · F0 ) + t¯i . fi (¯ x) = −fi∗ (¯ pi ) − ¯ T r(Q k λi According to Theorem 3(i) we have 1 ¯ · F0 + t¯i = −f ∗ (p¯ ) − 1 T r Q ¯ · F0 −fi∗ (¯ pi ) − ¯ T r Q i i ¯ k λi k λi 1 ¯ · F0 = −fi∗ (p¯ ) + p¯Ti x +¯ pTi x ¯ + ¯ Tr Q ¯ = fi (¯ x). i k λi ¯ t¯ is Pareto-efficient to ¯ λ, With the weak duality (cf. Theorem 4) follows that p¯, Q, (D). We can also formulate the converse duality theorem, whose proof is not mentioned here (cf. [10]).

Theorem 6. Assume the constraint qualification (SCQ) being fulfilled and that for each λ ∈ int(Rn+ ) the following property holds (C)

inf

x∈X

k X

λi fi (x) > −∞ implies ∃xλ ∈ X : inf

x∈X

i=1

k X

λi fi (x) =

i=1

k X

λi fi (xλ ).

i=1

¯ t¯) of (D) we have ¯ λ, (a) Then for any Pareto-efficient solution (¯ p, Q,  ¯ · F0 + t¯1  p1 ) − kλ1¯ 1 T r Q −f1∗ (¯   .. k  ∈ cl f (X ) + R+  . ¯ · F0 + t¯k −fk∗ (¯ pk ) − kλ1¯ T r Q k

and there exists a properly efficient solution to (P ) x ¯ λ¯ ∈ X such that k X 1 ∗ ¯ ¯ ¯ λi fi (¯ xλ¯ ) + fi (¯ pi ) + ¯ T r Q · F0 − ti = 0. k λi i=1

(b) If, additionally, f (X ) is Rk+ -closed (i.e. f (X ) + Rk+ is closed), then there exists an x ¯ ∈ X properly efficient to (P ) such that k X i=1

and

¯ i fi (¯ λ xλ¯ ) =

k X

¯ i fi (¯ λ x),

i=1

1 ¯ · F0 + t¯i , i = 1, ..., k. fi (¯ x) = −fi∗ (¯ pi ) − ¯ T r Q k λi

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G. Wanka et. al.

6. Special cases 6.1 Special case I. Let us consider the initial vector minimization problem with the linear objective function f = (hc1 , xi, ..., hck , xi)T . We have  hc1 , xi   v − min  ...  .

(Pl )

x∈X



hck , xi

To be able to calculate the dual problem of (P ) using the method presented before, we have to determine the conjugate function of each of the linear functions f i (x) = hci , xi, i = 1, ..., k, fi∗ (pi )

= sup {hpi , xi − hci , xi} = sup {hpi − ci , xi} = x∈Rm

x∈Rm

0, +∞,

if pi = ci , otherwise.

By this, the dual problem of (Pl ) looks like  − kλ1 1 T r(Q · F0 ) + t1   .. v − max  , . (Q,λ,t)∈Yl − kλ1 k T r(Q · F0 ) + tk 

(Dl )

where Yl =

(

(Q, λ, t) : λ ∈

int(Rk+ ),

λi c i =

i=1

Let us denote di := ti −

i=1

λ i di +

1 kλi T r(Q

λi ti = 0, Q ≥S+n 0,

i=1

k X

k P

k X

1 kλi T r(Q

T r (Q · F1 ) , T r (Q · F2 ) , ..., T r (Q · Fm )

· F0 ), i = 1, ..., k. The condition

i=1

· F0 ) = 0, which implies k X

1 λi di = −k T r(Q · F0 ) = −T r(Q · F0 ). k i=1

So the dual problem of (Pl ) is

(Dl )

k P

 d1 . v − max  ..  , (Q,λ,d)∈Yl dk 

)

.

λi ti = 0 becomes

13


where Yl =

(

(Q, λ, d) : λ ∈

int(Rk+ ),

k X

λi di = −T r(Q · F0 ), Q ≥S+n 0,

i=1

k X

λi c i =

i=1

T r (Q · F1 ) , T r (Q · F2 ) , ..., T r (Q · Fm )

)

.

6.2. Special case II. The next optimization problem we treat is known as the multiobjective fractional program with linear inequality constraints. A generalized case is presented in [12], while its single objective scalar case is mentioned in [9], where it is treated by means of semidefinite programming. For  hc1 ,xi2  hd1 ,xi

(Pf )

 v − min  A·x≤ p b  R

+

.. . k

hc ,xi2 hdk ,xi

 , 

with A = (aij ) ∈ Rp×m , b ∈ Rp , cj , dj ∈ Rm , j = 1, ..., k, we assume that for all j = 1, ..., k, and each feasible x, hdj , xi > 0. To be able to deal with (Pf ) within the framework of the present paper we reformulate it as a semidefinite programming problem. First it is obvious that (P f ) may be written also as   y1  ..  (Pf ) v − min  . . A·x≤Rp b,

yk

+

hcj ,xi2 hdj ,xi

≤yj ,j=1,...,k

The system of constraints above is equivalent (cf. [3], [9], [13]) of the following matrix  0 0  diag(b − A · x)    0 H1 0 F (x, y) =   ..  . 0 0   0

with

Hj = As we have

0

yj hcj , xi hcj , xi hdj , xi

, j = 1, ..., k.

 P  p a1i xi  i=1    .. , A·x=   . P  p ami xi i=1

0

to the semidefiniteness 

0    0  ,   0  

Hk

14

G. Wanka et. al.

and cj = (cj1 , ..., cjm ), dj = (dj1 , ..., djm ), j = 1, ..., k, F (x, y) may be written as a sum of symmetric matrices in the following way 

 diag(−a1i , ..., −api )   0    m X  0 F (x, y) = xi    i=1 0    0   0 

..

0

0

0

0

c1i

0

0

c1i

d1i

0

0

0

0

..

0

0

0

0

0

0

0

cki

.

0

0   0     0    0     cki   dki



0

    k X  + yj    j=1   

0



. 1 ..

. 0

      + diag(b) 0 ≥ p+2k 0. S+  0 0    

Let us denote by Fi the matrix multiplied above by xi , i = 1, ..., m, by Fm+j the one multiplied above by yj , namely, the (p + 2k) × (p + 2k) matrix with all entries equal to 0, but the one in the position (p + 2j − 1, p + 2j − 1) whose value is 1, j = 1, ..., k, and the last matrix by F0 . One may notice that all the matrices encountered above are symmetric and (P f ) has been written in the same form as the primal problem (P ). In order to determine the dual of (Pf ) we need to calculate the conjugates of the entries of the vectorial objective function. For the functions fj (x, y) = yj , j = 1, ..., k, the conjugates are

fj∗ (u, v) = sup {hu, xi + hv, yi − fj (x, y)} = sup x∈Rm , y∈Rk

=

0, +∞,

x∈Rm , y∈Rk

(

m X

ui x i +

i=1

if u = 0, vj = 1, vl = 0, l 6= j, otherwise.

The previous results lead us to the following dual to (Pf )

(Df )

 − kλ1 1 T r(Q · F0 ) + t1   .. v − max  , . (Q,λ,t)∈Yf − kλ1 k T r(Q · F0 ) + tk 

k X l=1

v l yl − y j

)

15


where Yf =

(

(Q, λ, t) : λ ∈ int(Rk+ ),

k X

λi ti = 0, Q ≥S p+2k 0, +

i=1

)

T r(Q · Fi ) = 0, i = 1, ..., m, T r(Q · Fm+j ) = λj , j = 1, ..., k . As the matrices Fi , i = 0, ..., m + k, are known and Q’s entries may be denoted by (qij ), i, j = 1, ..., p + 2k, we can develop a simpler shape of the dual problem. So let us calculate the values of the scalar products between Q and Fi , i = 0, ..., m + k. First we p P have T r(Q · F0 ) = qii bi . Then, for each i = 1, ..., m, i=1

T r(Q · Fi ) =

p X

−ali qll + 2

k X

qp+2j−1,p+2j cji

+

j=1

i=1

p X

qp+2j,p+2j dji

i=1

and we also have T r(Q · Fm+j ) = qp+2j−1,p+2j−1 , j = 1, ..., k. Because T r(Q · Fm+j ) = λj , one has λj = qp+2j−1,p+2j−1 , j = 1, ..., k. So the variables λj may be eliminated from the dual problem whose form becomes   h1 (Q, t)   .. (Df ) v − max  , . (Q,t)∈Yf

hk (Q, t)

where

hj (Q, t) = −

1 kqp+2j−1,p+2j−1

and Yf =

(

(Q, t) : Q = (qij ) ≥S p+2k 0, +

k X

p X

qii bi + tj , j = 1, ..., k,

i=1

qp+2j−1,p+2j−1 tj = 0,

j=1

qp+2j−1,p+2j−1 > 0, j = 1, ..., k, AT · (q11 , ..., qpp )T = 2

k X

qp+2j−1,p+2j cj +

j=1

k X i=1

In [12] there is obtained the following dual to the problem (Pf ) (Df0 )

v − max h0 (λ, δ, q s , q t ),

(λ,δ,q s ,q t )∈Yf0

)

qp+2j,p+2j dj .

16

G. Wanka et. al.

where



 −hδ1 , bi   .. h0 (λ, δ, q s , q t ) =  , . −hδk , bi

and Yf0 =

(

s t k k (λ, δ, q s , q t ) : λ ∈ int(R+ k ), q , q ∈ R+ , δ = (δ1 , ..., δk ), δj ∈ R ,

(qjs )2 ≤ 4qjt , j = 1, ..., k,

k X

λj δj ≥Rp+ 0,

j=1

k X

AT ·

j=1

λ j δj

!

+

k X

)

λj (qjs cj − qjt dj ) = 0 .

j=1

In order to find some connections between these two multiobjective dual problems we study the relation of inclusion between the image sets of their objective functions over the corresponding feasible sets. Therefore let be d ∈ h0 (Yf0 ). So there exists a tuple (λ, δ, q s , q t ) ∈ Yf0 such that d = h0 (λ, δ, q s , q t ). Let us consider !  k P    λ j δj , if u = v = 1, ..., p,    j=1  u     λj , if u = v = p + 2j − 1, j = 1, ..., k,      quv := λj qjt , if u = v = p + 2j, j = 1, ..., k,     λj qjs   if (u, v) = (p + 2j, p + 2j − 1) or (u, v) = −   2 ,   (p + 2j − 1, p + 2j), j = 1, ..., k,       0, otherwise, ! k k P P λj δj . Also let we have denoted the u-th entry of the vector where by λ j δj j=1 j=1 u * + k P λj δj , b , j = 1, ..., k. us introduce tj := − hδj , bi + kλ1 j j=1

Using the properties of the positive semidefinite matrices (cf. [3], [9], [13]) one may p+2k notice that Q = (quv )u,v=1,...,p+2k ∈ S+ . On the other hand, for each j = 1, ..., k, we have qp+2j−1,p+2j−1 > 0. Simple calculations give the following relations k X

qp+2j−1,p+2j−1 tj = 0

j=1

and T

T

A · (q11 , ..., qpp ) − 2

k X j=1

j

qp+2j−1,p+2j c −

k X i=1

qp+2j,p+2j dj = 0.


17

By these, (Q, t) ∈ Yf . For each component of the objective function h(Q, t), we have 1

(q11 , ..., qpp )T , b + tj kqp+2j−1,p+2j−1 + + * k * k X X 1 1 =− λj δj , b − hδj , bi + λj δj , b = −hδj , bi, j = 1, ..., k. kλj j=1 kλj j=1

hj (Q, t) = −

Hence d = h(Q, t) ∈ h(Yf ), which means that h0 (Yf0 ) ⊆ hf (Yf ). One may notice that the reverse inclusion does not hold. A detailed analysis of the relations between different duals introduced in the literature to a general convex multiobjective problem will be given in a forthcoming paper. Acknowledgement. R.I. Bot¸ acknowledges partial support from the Gottlieb Daimlerand Karl Benz-Stiftung, Germany, under 02-48/99.

References [1] Alfakih, A. and Wolkowicz, H.: Matrix Completion Problems. In: Handbook of Semidefinite Programming: Theory, Algorithms, and Applications (eds.: Wolkowicz, H., Saigal, R. and Vandenberghe, L.), Dordrecht: International Series In Operations Research And Management Science - vol.27, Kluwer Academic Publishers 2000, pp. 533 – 546. [2] Ekeland, I. and Temam, R.: Convex analysis and variational problems. Amsterdam: North-Holland Publishing Company 1976. [3] Helmberg, C.: Semidefinite Programming. Eur. J. Oper. Res. 137 (2002), pp. 461 – 482. [4] de Klerk, E., Terlaky, T. and Roos, K.: Self-Dual Embeddings. In: Handbook of Semidefinite Programming: Theory, Algorithms, and Applications (eds.: Wolkowicz, H., Saigal, R. and Vandenberghe, L.), Dordrecht: International Series In Operations Research And Management Science - vol.27, Kluwer Academic Publishers 2000, pp. 111 – 138. [5] Nesterov, Y. and Nemirovsky, A.: Interior Point Polynomial Methods in Convex Programming. Philadelphia: SIAM Publications - Studies in Applied Mathematics - vol.13 (1994). [6] Rockafellar, R.T.: Extension of Fenchel’s duality theorem for convex functions. Duke Math. J. 33 (1960), pp. 81 – 90. [7] Rockafellar, R.T.: Convex Analysis. Princeton: Princeton University Press 1970. [8] Shapiro, A. and Scheinberg, K.: Duality and Optimality Conditions. In: Handbook of Semidefinite Programming: Theory, Algorithms, and Applications (eds.: Wolkowicz, H., Saigal, R. and Vandenberghe, L.), Dordrecht: International Series In Operations Research And Management Science - vol.27, Kluwer Academic Publishers 2000, pp. 67 – 110. [9] Vandenbeghe, L. and Boyd, S.: Semidefinite Programming. SIAM Review 38 (1996), pp. 49 – 95. [10] Wanka, G. and Bot¸, R.I.: A New Duality Approach for Multiobjective Convex Optimization Problems. J. Nonlinear Convex Anal. 3 (2002), No. 1, pp. 41 – 57. [11] Wanka, G. and Bot¸, R.I.: Multiobjective duality for convex-linear problems II. Math. Methods Oper. Res. 53 (2000), No. 3, pp. 419 – 433.

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[12] Wanka, G. and Bot¸, R.I.: Multiobjective duality for convex ratios. J. Math. Anal. Appl. 275 (2002), No. 1, pp. 354 – 368. [13] Zowe, J. and Kochvara, M.: Semidefinite Programming. In: Modern Optimization and its Applications in Engineering (eds.: Ben-Tal, A. and Nemirovsky, A. ), Haifa: Minerva Optimization Center Summer School, Technion 2000.