NONLINEAR PROGRAMMING UNDER GENERALIZED (α, ρ) − (η, θ)-TYPE I INVEXITY

(1)

NONLINEAR PROGRAMMING UNDER GENERALIZED (α, ρ) − (η, θ)-TYPE I INVEXITY

ANDREEA M ˘AD ˘ALINA STANCU and I.M. STANCU-MINASIAN

We establish Karush-Kuhn-Tucker sufficiency optimality criteria under generalized (α, ρ)−(η, θ)-type I invexity conditions for general continuous-time programming programs with nonlinear operator equality and inequality constraints.

AMS 2010 Subject Classification: 90C30, 90C46, 90C48.

Key words: (α, ρ)−(η, θ)-type I invexity, continuous-time nonlinear programming, optimality conditions.

1. INTRODUCTION

Continuous-time programming was first introduced by Bellman [1] in the treatment of production and inventory “bottleneck” problems. Tyndall [9]

extended Bellman’s theory and obtained existence and duality theorems for a class of continuous linear programming problem. Levinson [3] generalized and simplified some results of Tyndall. Some classes of continuous-time linear and nonlinear programming problems have been studied in different settings and under various assumptions.

Singh and Farr [7] established the optimality criteria of Kuhn-Tucker and Fritz John type without assuming differentiability of functions assuming that the functions involved are either convex or concave.

In [6] Singh weakened the convexity (concavity) restrictions and assumed that the objective function involved pseudoconvexity “almost everywhere” and the constraints involved quasi-convex and quasi-concave functions. He established a sufficient optimality criterion in continuous-time programming.

In [8] Suneja et al. obtained optimality conditions and duality results for a general class of continuous-time nonlinear fractional programming via subdifferentiability and generalized Charnes-Cooper transformation.

Zalmai [10] established the Kuhn-Tucker saddlepoint and stationary- point optimality conditions and a Lagrangian duality theory for a general class of continuous-time nonlinear programming problem.

REV. ROUMAINE MATH. PURES APPL.,56(2011),2, 169–179

(2)

In [5] Nobakhtian established optimality conditions for nonsmooth multiobjective continuous-time problems under variousρ-invexity restrictions on the components of the functions describing constraint and the objectives functions.

2. CONTINUOUS-TIME PROGRAMMING PROBLEMS WITH OPERATOR CONSTRAINTS

Consider the following continuous-time programming problem with nonlinear operator equality and inequality constraints

(P) subject to

minimizeφ(x) = Z T

0

f(x) (t)dt, g(x)(t)≤0 for allt∈[0, T], h(x)(t) = 0 for all t∈[0, T], x∈Wⁿ[0, T]≡W_2,1ⁿ [0, T],

whereWⁿ[0, T] is the Hilbert space of all absolutely continuousn-dimensional vector functions t → x(t) ∈ Rⁿ (n-dimensional Euclidean space) defined on the compact interval [0, T] ⊂ R with Lebesgue square-integrable derivative

˙

x(t) = dx(t)/dt (so RT

0 kx(t)k˙ ²dt = RT 0

P_n

i=1( ˙x_i(t))²dt < ∞), and with the inner product (·|·) defined by

(x|y) =hx(0), y(0)i+ Z _T

0

hx(t),˙ y(t)i˙ dt, withha, bi=Pn

i=1a_ib_i fora, b∈Rⁿ; andf, g(with componentsg₁, g₂, . . . , g_p), andh(with componentsh1, h2, . . . , hq) are nonlinear continuously Fr´echet differentiable operators fromWⁿ[0, T] intoC[0, T],C^p[0, T], andC^q[0, T], respectively, withC^r[0, T] denoting the space of all continuousr-dimensional vector functions defined on [0, T].

In Problem (P) we use the Hilbert spaceWⁿ[0, T].According to Zalmai [11], this particular functions space has certain properties which are desirable from the standpoint of existence theory and computational considerations.

However, depending on the nature of the optimization problem under conside- ration, one can use a variety of other appropriate spaces in (P), for instance any Banach function spaces. In particular, instead ofWⁿ[0, T] one can use the space Lⁿ_∞[0, T] of all Lebesgue measurable essentially boundedn-dimensional vector functions defined on the compact interval [0, T] of the real lineR.

The optimization problem (P) has a range of applications in a great variety of optimal decision processes in the areas of economics, operations research and engineering.

(3)

The continuous-time programming problem (P) was first considered by Zalmai [11]. Using a general minmax approach, Zalmai [11] established saddle point optimality conditions and Lagrangian duality relation for (P) under generalized convexity hypotheses with and without differentiability assumptions.

In [12] Zalmai formulated a number of sufficiency criteria for (P) under various generalized ρ-(η, θ)-invexity conditions and discussed the relevance of these results to a class of variational-type inequalities. Similar results the read- ers can find in [4].

As a consequence of a general minmax approach developed in [11], Zalmai [13] established optimality conditions and duality theorems for a continuous- time minmax programming problem with Fr´echet differentiable convex operator inequality and linear operator equality constraints.

In [14] Zalmai, by using a minmax duality approach, have presented continuous-time analogues of the symmetric nonlinear programs originally studied by Dantzig et al. [2].

In [15] Zalmai established necessary and sufficient saddle-point and stationary-point-type proper efficiency conditions and constructed both parametric and semiparametric Lagrangian and Wolfe type duality models for a class of continuous-time multiobjective fractional programming problems with convex operator inequality and affine operator equality constraints.

In [16] Zalmai established both parametric and parametric-free saddlepoint and stationary-point-type necessary and sufficient optimality conditions for a class of continuous-time generalized fractional programming problems with convex operator inequality and affine operator equality constraints.

In this paper we introduce new classes of functions that we call (α, ρ)− (η, θ)-type I invex and (α, ρ)−(η, θ)-pseudo (strictly) quasi-type I invex functions. Sufficient optimality conditions for the problem (P) are obtained under the assumptions involving such functions.

Let S = {x ∈ Wⁿ[0, T] : g(x)(t) ≤ 0, h(x)(t) = 0 for allt ∈ [0, T]} be the set of all feasible solutions of (P).

LetW₊^k[0, T] =

u∈W^k[0, T] :u(t)≥0, for all t∈[0, T] .

LetV be a nonempty open convex subset of a real Banach space X and letW ⊂V. Also, letf andgtwo Fr´echet differentiable functionsf, g:V →R.

Definition 2.1. A pair (f, g) is said to be (ρ₁, ρ₂)−(η, θ)-type I invex at y ∈V with respect toW if there existρ1, ρ2∈Rand operatorsη, θ:V×V → X such that for eachx∈W,

f(x)−f(y)≥Df(y)η(x, y) +ρ1kθ(x, y)k², (2.1)

−g(y)≥Dg(y)η(x, y) +ρ2kθ(x, y)k², (2.2)

where Df(y) denotes the Fr´echet derivative of f at y, and k k is the norm on X.

(4)

If (f, g) is (ρ1, ρ2)−(η, θ)-type I invex at all y ∈ V with respect to W, then we say that (f, g) is (ρ₁, ρ₂)−(η, θ)-type I invex onV with respect toW.

Definition 2.2. A pair (f, g) is said to be (ρ₁, ρ₂)−(η, θ)-pseudoquasi- type Iinvex aty∈V with respect toW if there existρ1, ρ2 ∈Rand operators η, θ:V ×V →X such that for each x∈W ,

Df(y)η(x, y)≥ −ρ₁kθ(x, y)k² ⇒f(x)≥f(y), (2.3)

0≤g(y)⇒Dg(y)η(x, y)≤ −ρ₂kθ(x, y)k². (2.4)

If (f, g) is (ρ₁, ρ₂)−(η, θ)-pseudoquasi-type I invex at all y ∈ V with respect to W, then we say that (f, g) is (ρ₁, ρ₂)−(η, θ)-pseudoquasi-type I invex on V with respect toW.

If the second (implied) inequality in (2.3) is strict (x6=y), then we say that (f, g) is (ρ1, ρ2)−(η, θ)-pseudo strictly quasi-type I invex at y∈V with respect to W.

3. SUFFICIENCY CRITERIA

In this section, we establish various Karush-Kuhn-Tucker sufficient optimality conditions for (P) under (α, ρ)−(η, θ)-type I invexity conditions.

Theorem 3.1. Let x^∗ ∈S and assume that f, g and h are continuously Fr´echet differentiable at x^∗, and that there exist u^∗ ∈ W₊^p[0, T] and v^∗ ∈ W^q[0, T]such that

Z T 0

Df(x^∗) +

p

X

i=1

u^∗_i(t)Dgi(x^∗) + (3.1)

+

q

X

j=1

v_j^∗(t)Dhj(x^∗)

η(x, x^∗) (t)dt≥0 for all x∈S, u^∗_i(t)gi(x^∗) (t) = 0for all t∈[0, T], i= 1, . . . , p, (3.2)

where η is an operator from Wⁿ[0, T]×Wⁿ[0, T]into Wⁿ[0, T]. Further, assume that:

(i) for each j= 1, . . . , q, f, v^∗_j(t)h_j

is ( ˆρ,ρ˜_j)−(η, θ)-type Iinvex at x^∗ with respect to S;

(ii)for each i= 1, . . . , pand j = 1, . . . , q, g_i, v_j^∗(t)h_j

is ( ¯ρ_i,ρ˜_j)−(η, θ)- type Iinvex at x^∗ with respect to S;

(iii) ˆρ+

p

P

i=1

u^∗_i(t) ¯ρ_i+

q

P

j=1

˜

ρ_j ≥0 for all t∈[0, T].

Then x^∗ is an optimal solution of (P).

(5)

Proof. Letx be an arbitrary feasible solution of (P).

As g_i, v^∗_j(t)h_j

, i= 1, . . . , p and j = 1, . . . , q, is ( ¯ρ_i,ρ˜_j)−(η, θ)-type I invex at x^∗ with respect toS,we have

g_i(x)−g_i(x^∗)≥Dg_i(x^∗)η(x, y) + ¯ρ_ikθ(x, x^∗)k², i= 1, . . . , p, (3.3)

−v_j^∗(t)h_j(x^∗)≥v_j^∗(t)Dh_j(x^∗)η(x, x^∗) + ˜ρ_jkθ(x, x^∗)k², j= 1, . . . , q.

(3.4)

Sinceu^∗_i ≥0, it follows by (3.3) that Z T

0

u^∗_i(t)gi(x) (t) dt− Z T

0

u^∗_i(t)gi(x^∗) (t)dt≥

≥ Z T

0

h

u^∗_i(t)Dgi(x^∗)η(x, x^∗) (t) +u^∗_i (t) ¯ρikθ(x, x^∗)k²i dt.

By summation of both sides from i= 1, . . . , p, we have Z T

0 p

X

i=1

u^∗_i(t)gi(x)(t)dt− Z T

0 p

X

i=1

u^∗_i(t)gi(x^∗) (t)dt≥ (3.5)

≥ Z T

0

" _p X

i=1

u^∗_i(t)Dgi(x^∗)η(x, x^∗) (t) +

p

X

i=1

u^∗_i(t) ¯ρikθ(x, x^∗)k²

# dt.

Integrating the inequalities (3.4) and taking summation of both sides from j = 1, . . . , q, we have

Z T 0

"

−

q

X

j=1

v^∗_j(t)Dhj(x^∗)η(x, x^∗) (t)

# dt≥ (3.6)

≥ Z T

0

" _q X

j=1

v_j^∗(t)h_j(x^∗) (t) +

q

X

j=1

˜

ρ_jkθ(x, x^∗)k²

# dt.

From (3.1), (3.5) and (3.6), it follows that for allx∈S, Z T

0

Df(x^∗)η(x, x^∗) (t) + ˆρkθ(x, x^∗)k²

dt≥

− Z _T

0 p

X

i=1

u^∗_i(t)Dg_i(x^∗)η(x, x^∗) (t)dt−

− Z _T

0 q

X

j=1

v_j^∗(t)Dh_j(x^∗)η(x, x^∗) (t)dt+ Z _T

0

ˆ

ρkθ(x, x^∗)k²dt≥

≥ − Z T

0 p

X

i=1

u^∗_i(t)g_i(x)(t)dt+

(6)

+ Z T

0

" _p X

i=1

u^∗_i(t)g_i(x^∗) (t) +

p

X

i=1

u^∗_i(t) ¯ρ_ikθ(x, x^∗)k²

# dt+

+ Z T

0





q

X

j=1

v_j^∗(t)hj(x^∗) (t) +

q

X

j=1

˜

ρjkθ(x, x^∗)k²+ ˜ρkθ(x, x^∗)k²



dt≥0 (since x, x^∗ ∈S, u^∗(t)≥0 for all t∈[0, T], (3.2) holds and (iii)), i.e., (3.7)

Z T 0

h

Df(x^∗)η(x, x^∗) (t) + ˆρkθ(x, x^∗)k²i

dt≥0.

As f, v_j^∗(t)hj

,j= 1, . . . , q,is ( ˆρ,ρ˜j)−(η, θ)-type I invex at x^∗ with respect toS, we have

f(x)−f(x^∗)≥Df(x^∗)η(x, x^∗) + ˆρkθ(x, x^∗)k², (3.8)

−v^∗_j(t)h_j(x^∗)≥Dv^∗_j(t)h_j(x^∗)η(x, x^∗) + ˜ρ_jkθ(x, x^∗)k², j= 1, . . . , q.

(3.9)

From (3.7) and (3.8) it results Z T

0

f(x)(t)dt− Z T

0

f(x^∗)(t)dt≥ Z T

0

h

Df(x^∗)η(x, x^∗)(t) + ˆρkθ(x, x^∗)k²i dt≥0, i.e.,

f(x)≥f(x^∗). So, x^∗ is an optimal solution of (P).

Remark 3.2. In Theorem 3.1, condition (3.1) can be replaced by Z T

0

"

Df(x^∗) +

p

X

i=1

u^∗_i(t)Dgi(x^∗) +

q

X

j=1

v_j^∗(t)Dhj(x^∗)

#

z(t)dt= 0 for all z ∈ Wⁿ[0, T], without having to make any significant changes in the proof.

Let{I₀, I1, . . . , IM} and {J₀, J1, . . . , JM} be two partitions of the index sets {1,2, . . . , p} and {1,2, . . . , q}, respectively; thus I_r ⊂ {1,2, . . . , p}, for each r = 0, . . . , M, Ir∩Is=∅ for each r 6=s, and

M

S

r=0

Ir ={1,2, . . . , p}; and similarly for{J₀, J1, JM}.

Furthermore, ifM₁ is the number of partitioning subsets of {1,2, . . . , p}

and M2 is the number of partitioning subsets of {1,2, . . . , q}, then M = max{M₁, M₂} and I_m=∅orJ_m =∅ form >min{M₁, M₂}. In addition, we shall make use of the function Λ0(·, u, v) :Wⁿ[0, T]→ R, and the operators

(7)

Γm(·, u, v) :Wⁿ[0, T]→C[0, T] defined for fixedu and vby Λ0(x, u, v) =

Z T 0

"

f(x)(t) +X

i∈I0

ui(t)gi(x)(t) +X

j∈J0

vj(t)hj(x)(t)

# dt, Γ_m(x, u, v) (t) = X

i∈I_m

u_i(t)g_i(x)(t) + X

j∈J_m

v_j(t)h_j(x)(t), m= 0,1, . . . , M.

In Theorem 3.1 we impose (α, ρ) −(η, θ)-type I invexity assumptions on the individual operators. In the next theorem we impose (α, ρ)−(η, θ)- type I invexity assumptions on certain combinations on the objective and constraint operators.

Theorem 3.3. Let x^∗∈S and assume that f, g and h are continuously Fr´echet differentiable at x^∗,and there exist u^∗∈W₊^p[0, T]and v^∗ ∈W^q[0, T] such that (3.1) and (3.2) hold. Furthermore, we assume that the following conditions are satisfied:

(i) for each m= 1,2, . . . , M, (Λ0(·, u^∗, v^∗),Γm(·, u^∗, v^∗))is ( ˆρT,ρ¯m)− (η, θ)-pseudoquasi-type I invex at x^∗ with respect to S,

(ii) ˆρ+

M

X

m=1

¯ ρ_m ≥0.

Proof. Let x ∈S be an arbitrary feasible solution of (P). Since u^∗_i ≥0, i= 1, . . . , p, using (3.2), we get that for eachm= 1,2, . . . , M we have

(3.10) Γ_m(x, u^∗, v^∗) (t)≤0 = Γ_m(x^∗, u^∗, v^∗) (t) for allt∈[0, T].

As (Λ₀(·, u^∗, v^∗),Γ_m(·, u^∗, v^∗)),m= 1, . . . , M,is ( ˆρT,ρ¯_m)−(η, θ)-pseudoquasi- type I invex at x^∗ with respect to S, it follows that

DΛ₀(x^∗, u^∗, v^∗)η(x, x^∗)≥ −ρTˆ kθ(x, x^∗)k² (3.11)

⇒Λ0(x, u^∗, v^∗)≥Λ0(x^∗, u^∗, v^∗),

0≤Γm(x^∗, u^∗, v^∗) (t)⇒DΓm(x^∗, u^∗, v^∗)η(x, x^∗)≤ (3.12)

≤ −ρ¯mkθ(x, x^∗)k², m= 1, . . . , M.

Now, by combining the inequalities (3.10) and (3.12), we obtain

(3.13) DΓm(x^∗, u^∗, v^∗)η(x, x^∗) (t)≤ −ρ¯mkθ(x, x^∗)k², m= 1, . . . , M.

The inequality (3.1) can be written as (3.14) DΛ₀(x^∗, u^∗, v^∗)η(x, x^∗)≥ −

Z T 0

M

X

m=1

DΓ_m(x^∗, u^∗, v^∗)η(x, x^∗) (t)dt.

(8)

From (3.12), (3.14) and (ii), we get DΛ0(x^∗, u^∗, v^∗)η(x, x^∗)≥ −

Z T 0

M

X

m=1

DΓm(x^∗, u^∗, v^∗)η(x, x^∗) (t)dt≥ (3.15)

≥ Z T

0 M

X

m=1

¯

ρ_mkθ(x, x^∗)k² =

M

X

m=1

Z T 0

¯

ρ_mkθ(x, x^∗)k² ≥ −ˆρTkθ(x, x^∗)k², i.e., DΛ₀(x^∗, u^∗, v^∗)η(x, x^∗)≥ −ˆρTkθ(x, x^∗)k².

From (3.11) and (3.15), it follows that

Λ0(x, u^∗, v^∗)≥Λ0(x^∗, u^∗, v^∗).

Using feasibility of x and x^∗,nonnegativity ofu^∗, and (3.2) it results φ(x)≥ φ(x^∗).Therefore, x^∗ is an optimal solution of (P).

Remark 3.4. In Theorem 3.3, condition (i) can be replaced by (i) for eachi= 1, . . . , M, Λ₀(·, u^∗, v^∗),RT

0 Γ_m(·, u^∗, v^∗) (t)dt

is ( ˆρ,ρ¯_m)−

(η, θ)-pseudoquasi-type I invex at x^∗ with respect toS.

The proof of the next theorem is similar to that of Theorem 3.3.

Theorem 3.5. Let x^∗ ∈S and assume that f, g, h, u^∗ and v^∗ are as in Theorem 3.1. Assume that the following conditions are satisfied:

(i) for each m ∈ N₁ ⊂ {1, . . . , M},(Λ₀(·, u^∗, v^∗),Γ_m(·, u^∗, v^∗)) is ( ˆρT,ρ¯m)−(η, θ)-pseudoquasi-type I invex at x^∗ with respect to S.

(ii) for each m ∈ N₂ ≡ {1,2, . . . , M} \N₁, Λ₀(·, u^∗, v^∗),RT

0 Γ_m(·, u^∗, v^∗)(t)dt

, ( ˆρT,ρ¯_m)−(η, θ)-pseudoquasi-type I invex at x^∗ with respect to S.

(iii) ˆρT+ P

m∈N₁

¯

ρmT + P

m∈N₂

¯ ρm ≥0.

Note that Theorem 3.3 contains a number of special cases that can be easily identified by appropriate choices of the partitioning sets{I₀, I1, . . . , IM} and {J₀, J₁, . . . , J_M}. We shall state these cases as an corollary.

Corollary 3.6. Let x^∗ ∈ S and assume that f, g, h, are continuously Fr´echet differentiable at x^∗, and that there exist u^∗ ∈ W₊^p[0, T] and v^∗ ∈ W^q[0, T]such that (3.1)and (3.2)hold. Furthermore, we assume that any one of the following sets of hypothesis is satisfied:

(a) (i)

φ(x),

p

X

i=1

u^∗_i(t)gi(x)(t) +

q

X

j=1

v^∗_j(t)hj(x)(t)

!

is (ρT,b ρ)¯ −(η, θ)-pseudoquasi-type Iinvex at x^∗ with respect to S, (ii)ρb+ ¯ρ≥0.

(9)

(b) (i)For each m= 1, . . . , M, Z T

0

"

f(x)(t) +

p

X

i=1

u^∗_i(t)g_i(x)(t)

#

dt, X

j∈J_m

v^∗_j(t)h_j(x)(t)

!

is (ρT,b ρ¯m)−(η, θ)-pseudoquasi-type Iinvex at x^∗ with respect to S, (ii)ρb+

M

P

m=1

¯ ρm ≥0.

(c) (i)



 Z T

0

"

f(x)(t) +

p

X

i=1

u^∗_i(t)g_i(x)(t)

# dt,

q

X

j=1

v_j^∗(t)h_j(x)(t)





(d) (i)For each j= 1, . . . , q, Z T

0

"

f(x) (t) +

p

X

i=1

u^∗_i(t)gi(x)(t)

#

dt, v^∗_j(t)hj(x)(t)

!

is (ρT,b ρ¯_j)−(η, θ)-pseudoquasiinvex-type Iat x^∗ with respect to S, (ii)ρb+

q

P

j=1

¯ ρj ≥0.

(e) (i)For each m= 1, . . . , M,



 Z T

0



f(x)(t) +

q

X

j=1

v_j^∗(t)h_j(t)(t)



dt, X

i∈Im

u^∗_i(t)g_i(x)(t)





M

P

m=1

¯ ρm ≥0.

(f) (i)



 Z T

0



f(x)(t) +

q

X

j=1

v^∗_j(t)hj(x)(t)



dt,

p

X

i=1

u^∗_i(t)gi(x)(t)





(g) (i)For each i= 1, . . . , p, Z T

0

"

f(x)(t) +

m

X

m=1

v_j^∗(t)hj(t)(t)

#

dt, u^∗_i (t)gi(x)(t)

!

is (ρT,b ρ¯i)−(η, θ)-pseudoquasi-type Iinvex at x^∗ with respect to S,

(10)

(ii)ρb+

p

P

i=1

¯ ρi ≥0.

(h) (i)For each m= 1, . . . , M, Z T

0







f(x)(t)+

p

X

i=1

u^∗_ig_i(x)(t)+X

j∈J0

v_j^∗(t)h_j(x)(t)



dt, X

j∈Jm

v_j^∗(t)h_j(x)(t)





M

P

m=1

¯ ρm ≥0.

Proof. Each of the eight sets of conditions in Corollary 3.6 can be considered as a family of sufficient optimality conditions whose members can easily be identified by appropriate choices of the partitioning sets{I₀, I₁, . . . , I_M}and {J₀, J₁, . . . , J_M}. In Theorem 3.3, let a) I₁ ={1, . . . , p}, J₁ = {1, . . . , q}; b) I0={1, . . . , p},J0 =φ, c)I0={1, . . . , p},J1 ={1, . . . , q}; d)I0={1, . . . , p}, J_m = {m}, m = 1, . . . q; e) I₀ =φ, J₀ ={1, . . . , q}; f) I₁ = {1, . . . , p}, J₀ = {1, . . . , q}; g)Im ={m},m= 1, . . . , p,J0 ={1, . . . , q}; h) I0={1, . . . , p}.

4. CONCLUSIONS

In this paper we introduced new classes of functions that we called (α, ρ)−(η, θ)-type I invex and (α, ρ)−(η, θ)-pseudo (strictly) quasi-type I invex functions. Sufficient optimality conditions for a continuous-time programming problem with nonlinear operator equality and inequality constraints were obtained under the assumptions involving such functions.

REFERENCES

[1] R. Bellman,Dynamic Programming. Princeton Univ. Press, Princeton, N.J. 1957.

[2] G.B. Dantzig, E. Eisenberg and R.W. Cottle,Symmetric dual nonlinear programs. Pa- cific J. Math.15(1965), 809–812.

[3] N. Levinson,A class of continuous linear programming problems. J. Math. Anal. Appl.

16(1966),1, 73–83.

[4] C. Nahak, N. Behera and S. Nanda,Application of continuous-time programming problems to a class of variational-type inequalities. Math. Comput. Modelling 47 (2008), 11–12, 1332-1338.

[5] S. Nobakhtian,Nonsmooth multiobjective continuous-time problems with generalized invexity. J. Global Optim.43(2009),4, 593–606.

[6] C. Singh,A sufficient optimality criterion in continuous-time programming for generalized convex functions. J. Math. Anal. Appl.62(1978),3, 506–511.

(11)

[7] C. Singh and W.H. Farr, Saddle-point optimality criteria in continuous time programming without differentiability. J. Math. Anal. Appl.59(1977),3, 442–453.

[8] S. Suneja, C. Singh and R.N. Kaul,Optimality and duality in continuous-time nonlinear fractional programming. J. Austral. Math. Soc. Ser. B34(1992),2, 229–244.

[9] W.F. Tyndall,A duality theorem for a class of continuous linear programming problems.

SIAM J. Appl. Math.13(1965),3, 644–666.

[10] G.J. Zalmai,Optimality conditions and Lagrangian duality in continuous-time nonlinear programming. J. Math. Anal. Appl.109(1985),2, 426–452.

[11] G.J. Zalmai,Optimality conditions and duality for a class of continuous-time programming problems with nonlinear operator equality and inequality constraints. J. Math.

Anal. Appl.153(1990),2, 309–330.

[12] G.J. Zalmai, Generalized sufficiency criteria in continuous-time programming with application to a class of variational-type inequalities. J. Math. Anal. Appl.153(1990),2, 331–355.

[13] G.J. Zalmai,Optimality conditions and duality for a class of continuous-time generalized fractional programming problems.J. Math. Anal. Appl.153(1990),2, 356–371.

[14] G.J. Zalmai, Symmetric duality in continuous-time programming via minmax duality.

J. Math. Anal. Appl.153(1990),2, 372–390.

[15] G.J. Zalmai, Proper efficiency principles and duality models for a class of continuous- time multiobjective fractional programming problems with operator constraints. J. Stat.

Manag. Systems1(1998),1, 11–59.

[16] G.J. Zalmai, Optimality principles and duality models for a class of continuous-time generalized fractional programming problems with operator constraints. J. Stat. Manag.

Systems1(1998),1, 61–100.

Received 21 February 2011 Romanian Academy

“Gheorghe Mihoc–Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics

Calea 13 Septembrie no. 13 050711 Bucharest 5, Romania andreea madalina [email protected]

stancu [email protected]