DUALITY FOR A NONDIFFERENTIABLE MATHEMATICAL PROGRAMMING CLASS

DUALITY FOR A NONDIFFERENTIABLE MATHEMATICAL PROGRAMMING CLASS

VASILE PREDA, MIRUNA BELDIMAN and ELENA CRISTINA BAIBARAC

Some new important results have been recently proved concerning higher-order duality for some mathematical programming classes (Yang, Teo and Yang (2004), Mishra and Rueda (2002)). Following this line, we consider some new necessary optimality conditions and duality results for a nondifferentiable mathematical programming class.

AMS 2000 Subject Classification: 90C29, 90C30, 90C46.

Key words: multiobjective programming, optimization.

1. INTRODUCTION

In this paper we consider a general class of nondifferentiable mathe- matical programming problems

(P) minf(x) +

s

P

j=1

x^TB_jx¹₂ subject to x∈X₀,

where X₀ = {x ∈ Rⁿ | g(x) ≥ 0}, f : Rⁿ → R and g : Rⁿ → R^m are twice differentiable functions, andB_j,j = 1, s,aren×npositive semi-definite (symmetric) matrices. Hereg= (g₁, . . . , g_m)^T. A feasible solution of (P) is an element x⁰ ∈X0.

This class arises naturally in finance when one measures the risk of a portfolio by its variance-covariance matrix, in stochastic programming un- der chance constraints, and in location theory. Thus, some special cases of (P), with f nondifferentiable, have appeared in, for example, [11, 12]. The root terms appear, for example, in the following context. One reasonable for- mulation of stochastic linear programming problem leads to a deterministic nonlinear programming problem, where the nonlinearity occurs in the objec- tive function as the sum of square roots of positive semidefinite quadratic forms. In Francis and Cabot [5] is given an application of (P) to the problem

MATH. REPORTS10(60),4 (2008), 375–384

of minimizing a cost function which includes costs directly proportional to Euclidean distances.

Fors= 1 we obtain the class of nondifferentiable mathematical program- ming problems defined by Mond [11]. In this case, under some assumptions of convexity, Mond gave duality results for a dual of Wolfe type. Some gen- eralizations of these results were given, for example, by Chandra, Craven and Mond [4], Preda [14], Zhang and Mond [19], Preda and Koller [15], Mishra and Rueda [9, 10]. Also, some second order duality or higher-order duality results for Mond’s problem were given in Zhang and Mond [19], Bector and Chandra [1] (according to [10]), Preda and Koller [15], Zhang [20] and Mishra and Rueda [9, 10].

The study of higher-order duality is significant due to the computational advantage over first-order duality as it provides higher bounds for the value of the objective function when approximations are used (Mangasarian [7], Yang [18]). Mangasarian [7] and Hanson [6] had also indicated, for example, that an advantage of second-order duality when applicable, is that if a feasible solution in the primal problem is given and first-order duality conditions do not apply, then we can use second-order duality to provide a lower bound of the value in the primal problem. See also Mishra [8] and Yang [17, 18].

This paper is organized as follows. In Section 2 a general higher-order dual of Mond-Weir type to problem (P) is defined and some definitions of (ρ, ρ⁰)-higher-order invexity and (ρ, ρ⁰)-generalized higher-order invexity are given. Relative to (P), in Section 3, some necessary optimality conditions are given.

In Section 4, relative to the general higher-order dual of Mond-Weir type given in Section 2, weak duality, strong duality and strict converse duality results are established.

2. PRELIMINARIES AND SOME DEFINITIONS

In this section we introduce a general Mond-Weir type (see [13]) higher- order dual to problem (P) and give some definitions of higher-order invexity and generalized higher-order invexity types.

Let h, k₁, . . . , k_m : Rⁿ×Rⁿ → R, differentiable functions relative to each argument. In the following, the operator ∇ is taken relative to the first argument and the operator ∇_p is taken relative to the second argument.

We put k(u, p) = (k₁(u, p), . . . , k_m(u, p))^T, where the symbol ^T denotes the transpose. Also, let I_α⊆ {1,2, . . . , m}, α= 0, r,with

r

[

α=0

I_α ={1,2, . . . , m} and I_α∩I_β =∅ forα6=β.

Relative to (P) we consider the general Mond-Weir type higher-order dual (HGD) below.

(HGD) : maxf(u) +h(u, p) +u

s

P

j=1

Bjwj−p^T∇_ph(u, p)−

−P

i∈I₀

y_ig_i(u)− P

i∈I₀

y_ik_i(u, p) +p^T∇_p

P

i∈I₀

y_ik_i(u, p)

subject to:

(2.1)∇_ph(u, p) +

s

P

j=1

Bjwj =∇_p y^Tk(u, p) (2.2) P

i∈Iα

y_ig_i(u)+ P

i∈Iα

y_ik_i(u, p)−p^T∇_p

P

i∈Iα

y_ik_i(u, p)

≤0, α= 1, r (2.3)w_j^TBjwj ≤1, j ∈ {1,2, . . . , s}

(2.4)y ≥0,

where u, w1, . . . , ws,p∈Rⁿ and y∈R^m.

Ifs= 1,withB =B₁,we get the (NDHGD) dual considered by Mishra and Rueda [10] and studied by Zhang [20]. Also, we note that ifs= 1,I0=∅ and I_α ={1,2, . . . , m} for some α ∈ {1,2, . . . , r} then, with B =B₁,(HGD) becomes the dual (NDHD), a Mond-Weir type higher-order dual to (P), see Zhang [20] and Mishra and Rueda [10]. For s = 1 (with B = B₁), I₀ = {1,2, . . . , m}andI_α=∅forα∈ {1,2, . . . , r}, (HGD) becomes a Mangasarian type higher-order dual [7] given by Zhang [20]. Further, we note that if (2.5) h(u, p) =p^T∇f(u) + 1

2p^T∇²f(u)p and

(2.6) k_i(u, p) =p^T∇g_i(u) +1

2p^T∇²g_i(u)p, i= 1, m,

then (NDHD) becomes a second-order Mond-Weir type dual considered, ac- cording to [9], by Bector and Chandra [1]. Also, in this case, for h(u, p) and k(u, p),there was given a second-order Mangasarian type [7] dual to (P). Fi- nally, we note that for s= 1 and h and k1, . . . , km given by (2.5) respectively (2.6), the general dual (HGD) becomes the second-order dual defined by Zhang and Mond [19].

In Section 4 some duality results are established relative to (P) and (HGD) under (ρ, ρ⁰)-higher-order invexity assumptions, defined in the following.

Letρ∈R, ρ⁰ = (ρ⁰₁, . . . , ρ⁰_m)∈R^m and d:Rⁿ×Rⁿ→R+.

Definition 2.1. The objective function f and the constraint functions gi, i = 1, m, are said to be (ρ, ρ⁰)-higher-order type I at u with respect to a

function η if

f(x) +x^T

s

X

j=1

B_jw_j−f(u)−u^T

s

X

j=1

B_jw_j ≥

≥η(x, u)^T

∇_ph(u, p) +

s

X

j=1

Bjwj

+h(u, p)−p^T(∇_ph(u, p)) +ρd²(x, u) for all x, and

−g_i(u)≤η(x, u)^T∇_pki(u, p) +ki(u, p)−p^T(∇_pki(u, p))−ρ⁰_id²(x, u), i= 1, m.

Definition 2.2. The objective function f and the constraint functions gi, i= 1, m, are said to be (ρ, ρ⁰)-higher-order pseudo-quasi type I at u with respect to a function η if

η(x, u)^T

∇_ph(u, p) +

s

X

j=1

Bjwj

≥ −ρd²(x, u)⇒

⇒f(x) +x^T

s

X

j=1

Bjwj −f(u)−h(u, p)−u^T

s

X

j=1

Bjwj +p^T(∇_ph(u, p))≥0 for all x, and

−g_i(u)≥ki(u, p)−p^T ∇_pki(u, p)

⇒η(x, u)^T∇_pki(u, p)≥ρ⁰_id²(x, u), i= 1, m.

We note that fors= 1, ρ= 0 andρ⁰= 0 we obtain higher-order invexity and generalized higher-order invexity considered in Zhang [20] and Mishra and Rueda [10].

3. NECESSARY OPTIMALITY CONDITIONS

Letx⁰ be a feasible solution of (P). We define the setsS={1,2, . . . , s}, B(x⁰) =

j ∈S|x^0TB_jx⁰ >0 ,B(x⁰) =

j ∈S|x^0TB_jx⁰ = 0 ,and Z(x⁰) =n

z∈Rⁿ|z^T∇g_i(x⁰)≥0, ∀i∈M(x⁰) and

z^T∇f(x⁰) + X

j∈B(x⁰)

z^TB_jx⁰ (x^0TBjx⁰)¹²

+ X

j∈B(x⁰)

z^TB_jz¹₂

<0o ,

where M(x⁰) ={i|1≤i≤m, gi(x⁰) = 0}.

We note that ifs= 1,the set Z(x⁰) is the set defined by Mond [11].

Now, we consider the following result given by Sinha [16].

Lemma 3.1([16]). Let Abe a p×nmatrix,c∈Rⁿand let Dj, j= 1, s, be n×nsymmetric positive semidefinite matrices. Then

(3.1) Ax≥0, x∈Rⁿ implies c^Tx+

s

X

j=1

x^TDjx¹₂

≥0

if and only if there exist y∈R^p, y≥0, and w_j ∈Rⁿ, j∈S, such that Aw_j ≥0, w_j^TD_jw_j ≤1, j ∈S,

A^Ty=c+

s

X

j=1

D_jw_j. (3.2)

We note that if Dj = 0, for any j = 1, s, Lemma 3.1 reduces to the Farkas lemma.

In the following we shall use the generalized Schwarz inequality (3.3) x^TDw≤ x^TDx¹₂

w^TDw¹₂ ,

where D is n×n symmetric positive semidefinite matrice. We have equality in (3.3) if Dx=λDw for someλ∈R, λ≥0.Using (3.3) we get

(3.4) w^TDw≤1 impliesx^TDw≤ x^TDx¹₂ .

Now, we have the following necessary optimality conditions for x⁰ to be an optimal solution of (P).

Theorem 3.1 If x⁰ is an optimal solution of (P) and Z(x⁰) =∅, then there exist y ∈ R^m, y ≥ 0, and wj ∈ Rⁿ, j ∈ S such that y^Tg(x⁰) = 0, ∇y^Tg(x⁰) = ∇f(x⁰) +

s

P

j=1

B_jw_j, w^T_jB_jw_j ≤1and x^0TB_jx⁰¹₂

=x^0TB_jw_j for j ∈S.

Proof. Let x⁰ be an optimal solution of (P) and Z(x⁰) = ∅. Then for z^T∇g_j(x⁰)≥0, j ∈M(x⁰) we have

z^T

∇f(x⁰) + X

j∈B(x⁰)

Bjx⁰ (x^0TBjx⁰)¹²

+ X

j∈B(x⁰)

z^TBjz¹₂

≥0.

In Lemma 3.1 take

A= ∇g_j(x⁰)

j∈M(x⁰), c=∇f(x⁰) + X

j∈B(x⁰)

Bjx⁰ (x^0TB_jx⁰)¹²

, D_j =B_j, j ∈B(x⁰).

(If B(x⁰) =∅orB(x⁰) =∅, the corresponding terms do not appear.) There- fore, there exist scalars yj ≥ 0, j ∈ M(x⁰), and vectors wj, j ∈ B(x⁰), such that

X

j∈M(x⁰)

yj∇g_j(x⁰) =∇f(x⁰) + X

j∈B(x⁰)

Bjx⁰ (x^0TBjx⁰)¹²

+ X

j∈B(x⁰)

Bjwj

and w_j^TB_jw_j ≤ 1, j ∈ B(x⁰). Since g_j(x⁰) = 0 for j ∈ M(x⁰), we have y_jg_j(x⁰) = 0 for j ∈ M(x⁰). If j /∈ M(x⁰), we put y_j = 0. Then we have y^Tg(x⁰) = 0.

Now, for j∈B(x⁰),let

wj = x⁰ (x^0TBjx⁰)¹²

.

For j ∈ B(x⁰) we have w^T_jBjwj = 1 and x^0TBjwj = x^0TBjx⁰¹₂

. Also, for j ∈B(x⁰) we have x^0TB_jx⁰ = 0, which impliesB_jx⁰= 0, x^0TB_jx⁰¹₂

= 0 = x^0TB_jw_j.Therefore, the theorem is proved.

Remark. Relative to the assumption Z(x⁰) = ∅, we note that in the case s= 1, Mond and Schechter [12] and Wolkowitz [21] gave some constraint qualifications which imply Z(x⁰) =∅.

Now, for (P) it is easy to reformulate similar constraint qualifications such that Z(x⁰) =∅.

4. DUALITY RESULTS

In this section, relative to (P) and (HGD), we consider weak duality, strong duality and strict converse duality results.

Theorem 4.1 (Weak duality). Let η:Rⁿ×Rⁿ→Rⁿ such that for all x∈X₀ and a feasible solution (u, y, w₁, . . . , w_s, p) of (HGD) we have

(4.1) η(x, u)^T

"

∇_ph(u, p) +

s

X

j=1

Bjwj− ∇_p

X

i∈I₀

yiki(u, p) #

≥ −ρd²(x, u)⇒

⇒f(x) +x^T

s

X

j=1

B_jw_j−

f(u) +u^T

s

X

j=1

B_jw_j−X

i∈I₀

y_ig_i(u)

−

−

h(u, p)−X

i∈I0

y_ik_i(u, p)

+p^T

"

∇_ph(u, p)− ∇_p

X

i∈I0

y_ik_i(u, p) #

≥0,

(4.2) −X

i∈Iα

y_ig_i(u)−X

i∈Iα

y_ik_i(u, p) +p^T

"

∇_p

X

i∈Iα

k_i(u, p) #

≥0⇒

⇒η(x, u)^T

"

∇_p

X

i∈I_α

y_ik_i(u, p) #

≥ −ρ_αd²(x, u), α= 1, r and

(4.3) ρ+

r

X

α=1

ρα ≥0.

Then inf (P)≥sup (HGD).

Proof. For a feasible solution (u, y, w1, . . . , ws, p) we have X

i∈Iα

yigi(u) +X

i∈Iα

yiki(u, p)−p^T

"

∇_p

X

i∈Iα

yiki(u, p) #

≤0.

for any α= 1, r.Thus, by (4.2),

(4.4) η(x, u)^T

"

∇_p

X

i∈Iα

y_ik_i(u, p) #

≥ρ_αd²(x, u)

while by I_α∩I_β =∅forα6=β, Sm

α=0

I_α={1,2, . . . , m},we obtain

(4.5) η(x, u)^T

"

∇_p

X

i /∈I0

yiki(u, p) #

≥

r

X

α=1

ραd²(x, u).

Also, by the feasibility of (u, y, w₁, . . . , w_s, p) for (HGD) we have

∇_ph(u, p) +

s

X

j=1

Bjwj =∇_p y^Tk(u, p) .

Then by (4.5) and by I_α∩I_β =∅ forα6=β, Sm

α=0

I_α ={1,2, . . . , m},

η(x, u)^T

"

∇_ph(u, p) +

s

X

j=1

B_jw_j− ∇_p

X

i∈I₀

y_ik_i(u, p) #

≥

r

X

α=1

ρ_αd²(x, u).

Now, by (4.3) we get η(x, u)^T

"

∇_ph(u, p) +

s

X

j=1

Bjwj− ∇_p

X

i∈I₀

yiki(u, p) #

≥ −ρd²(x, u).

Using this inequality and (4.1), we obtain (4.6) f(x) +x^T

s

X

j=1

B_jw_j −

f(u) +u^T

s

X

j=1

B_jw_j−X

i∈I0

y_ig_i(u)

−

−

h(u, p)−X

i∈I0

yiki(u, p)

+p^T

"

∇_ph(u, p)− ∇_p

X

i∈I0

yiki(u, p) #

≥0.

Again by the feasibility of (u, y, w1, . . . , ws, p) for (HGD), we havew^T_jBjwj ≤1 for any j ∈S.Thus, by (4.6) and the generalized Schwarz inequality (3.3),

f(x) +

s

X

j=1

x^TB_jx¹₂

≥f(u) +u^T

s

X

j=1

B_jw_j−X

i∈I₀

y_ig_i(u)+

+h(u, p)−X

i∈I0

y_ik_i(u, p) +p^T

"

∇_ph(u, p)− ∇_p

X

i∈I0

y_ik_i(u, p) #

. The proof is complete.

For the duality results below we suppose thath and ksatisfy some “ini- tial” conditions (defined by (4.7)) that were considered in Zhang [20] and Mishra and Rueda [10].

Theorem 4.2(Strong duality).Let x⁰ be a local or global optimal solu- tion of (P)with Z(x⁰) =∅and

(4.7) h(x⁰,0) = 0, k(x⁰,0) = 0, ∇_ph(x⁰,0) =∇f(x⁰), ∇_pk(x⁰,0) =∇g(x⁰).

Then there exist y∈R^m and w1, . . . , ws ∈Rⁿ such that x⁰, y, w₁, . . . , w_s, p= 0

is a feasible solution of (HGD) and the corresponding values of (P) and (HGD)are equal. If the weak duality Theorem4.1also hold, then (x⁰, y, w1, . . . , w_s, p= 0)is an optimal solution of (HGD).

Proof. We have Z(x⁰) =∅and x⁰ is an optimal solution of (P). Then, by Theorem 3.1 there exist y∈R^m and w₁, . . . , w_s∈Rⁿ such that

y^Tg(x⁰) = 0, ∇y^Tg(x⁰) =∇f(x⁰) +

s

X

j=1

Bjwj, w_j^TBjwj ≤1, and

x^0TB_jx⁰¹₂

=x^0TB_jw_j, j∈S, y≥0.

According to (4.7), we deduce that x⁰, y, w1, . . . , ws, p= 0

is a feasible solu- tion of (HGD) and the corresponding values of (P) and (HGD) are equal. The last part of this theorem follows by using the weak duality Theorem 4.1.

Now, we consider a strict converse duality of Mangasarian type [7].

Theorem 4.3(Strict converse duality). Let x⁰ be an optimal solution of (P) with Z(x⁰) = ∅ and assume that (4.7)is satisfied. Assume also that the hypotheses of the weak duality Theorem4.1are satisfied. If (x, y, w1, . . . , ws, p) is an optimal solution of (HGD) and

η(x, x)^T

"

∇_ph(x, p) +

s

X

j=1

Bjwj− ∇_p

X

i∈I₀

yiki(x, p) #

≥ −ρd²(x, x)⇒

⇒f(x) +x^T

s

X

j=1

B_jw_j −

f(x) +x^T

s

X

j=1

B_jw_j −X

i∈I₀

y_ig_i(x)

−

−

h(x, p)−X

i∈I0

y_ik_i(x, p)

+p^T

"

∇_ph(x, p)− ∇_p

X

i∈I0

y_ik_i(x, p) #

>0, for any x 6= x, then x⁰ = x, i.e., x is an optimal solution of (P) and the optimal values of the objective functions of (P) and (HGD) are equal.

The proof is along the usual lines of similar theorems (see, for example, Preda [14]).

Remarks. 1. Some of the ideas used in this paper can be applied to problems studied, for example, in [2, 3].

2. The results obtained can be easily reformulated in the frame of (F, ρ)- convexity type or some variations of this concept.

Acknowledgements.This work was partially supported by Grant PN II IDEI, code ID, no 112/01.10.2007.

REFERENCES

[1] C.R. Bector and S. Chandra, Second order duality with nondifferentiable functions.

Working Paper, 1997.

[2] M. Beldiman,Equilibrium problems with set-value mapping in Banach spaces. Nonlinear Anal.68(2008), 3364–3371.

[3] M. Beldiman, E. Panaitescu and L. Dogaru, Approximate quasi efficient solutions in multiobjective optimization.Bull. Math. Soc. Sci. Math. Roumanie51(99)(2008), 109–

121.

[4] S. Chandra, B.D. Craven and B. Mond,Generalized concavity and duality with a square root term. Optimization16(1985), 653–662.

[5] R.L. Francis and A.V. Cabot, Properties of a multifacility location problem involving Euclidean distances. Naval Res. Logist. Quart.19(1972), 335–353.

[6] M.A. Hanson, Second order invexity and duality in mathematical programming.

Opsearch30(1993), 311–320.

[7] O.L. Mangasarian, Second and higher order duality in nonlinear programming.

J.Math.Anal.Appl.51(1975), 607–620.

[8] S.K. Mishra, Second order generalized invexity and duality in mathematical program- ming. Optimization42(1997), 51–69.

[9] S.K. Mishra and N.G. Rueda,Higher order generalized invexity and duality in mathe- matical programming. J. Math. Anal. Appl.247(2000), 173–182.

[10] S.K. Mishra and N.G. Rueda, Higher order generalized invexity and duality in nondif- ferentiable mathematical programming. J. Math. Anal. Appl.272(2002), 496–506.

[11] B. Mond, A class of nondifferentiable mathematical programming problems. J. Math.

Anal. Appl.46(1974), 169–174.

[12] B. Mond and M. Schechter, On a constraint qualification in a nondifferentiable pro- gramming problem. Naval Res. Logist. Quart.23(1976), 611–613.

[13] B. Mond and J. Weir, Generalized convexity and higher order duality. J. Math. Sci.

16-81(1981–1983), 74–94.

[14] V. Preda,On generalized convexity and duality with a square root term.Math. Methods Oper. Res.36(1992), 547–563.

[15] V. Preda and E. Koller,On duality for a nondifferentiable programming problem with a square root term. Rev. Roumaine Math. Pures Appl.45(2000), 873–882.

[16] S.M. Sinha, An extension of a theorem of supports of a convex function. Management Sci.12(1966), 380–384.

[17] X.M. Yang, K.L. Teo and X.Q. Yang,Higher-order generalized convexity and duality in nondifferentiable multiobjective mathematical programming. J. Math. Anal. Appl.297 (1), (2004), 48–55.

[18] X. Q. Yang,Second order global optimality conditions for convex composite optimization.

Math. Programming81(1998), 327–347.

[19] J. Zhang and B. Mond, Duality for a nondifferentiable programming problem. Bull.

Austral. Math. Soc.55(1997), 29–44.

[20] J. Zhang,Generalized convexity and higher order duality for mathematical programming problems. Ph.D. Thesis, La Trobe University, Australia, 1998.

[21] H. Wolkowicz,An optimal condition for a nondifferentiable convex program. Naval Res.

Logist. Quart.30(1983), 415–418.

Received 8 January 2008 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

014700 Bucharest, Romania Romanian Academy Institute of Mathematical Statistics

and Applied Mathematics Calea 13 Septembrie nr. 13 050711 Bucharest, Romania