CARATH´EODORY–JOHN-TYPE SUFFICIENCY CRITERIA IN CONTINUOUS-TIME NONLINEAR PROGRAMMING UNDER GENERALIZED (α, ρ) − (η, θ)-TYPE I INVEXITY

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CRITERIA IN CONTINUOUS-TIME NONLINEAR PROGRAMMING UNDER GENERALIZED

(α, ρ) − (η, θ)-TYPE I INVEXITY

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

We introduce the class of (α, ρ)−(η, θ)-pseudo strictly pseudo-type I invex functions. We establish Carath´eodory–John-type sufficient optimality conditions for a continuous-time programming problem with nonlinear operator equality and inequality constraints under generalized (α, ρ)−(η, θ)-type I invexity conditions.

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 linear programming was first introduced by Bellman [1]

in the treatment of production and inventory “bottleneck” problems. He formulated the dual problem, established a weak duality theorem and described sufficient optimality conditions. Tyndall [12] extended Bellman’s theory and obtained existence and duality theorems for a class of continuous linear programming problem. Levinson [4] generalized and simplified some results of Tyndall by considering the time-dependence matrices in which the functions from objective and constraints are continuous on [0, T].

Some classes of continuous-time linear and nonlinear programming problems have been studied in different settings and under various assumptions.

Singh and Farr [8] 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.

Singh [7] 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, assuming differentiability of functions.

MATH. REPORTS14(64),4 (2012), 345–354

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Suneja et al. [11] obtained optimality conditions and duality results for a general class of continuous-time nonlinear fractional programming via sub- differentiability and generalized Charnes-Cooper transformation.

Zalmai [14] 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.

Nobakhtian [6] established optimality conditions for nonsmooth multiobjective continuous-time problems under various ρ-invexity restrictions on the components of the functions describing constraint and the objectives functions.

Stancu-Minasian and T¸ igan [10] studied the stochastic continuous-time linear fractional programming problem. The minimum-risk approach is in- vestigated, both for linear-fractional problems and for problems with linear- fractional kernel.

Wen and Wu [13] considered a continuous-time linear fractional programming problem and proposed a Dinkelbach-type algorithm for solving it.

2. CONTINUOUS-TIME PROGRAMMING PROBLEMS WITH OPERATOR CONSTRAINTS

Consider the 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(R_T

0 kx(t)k˙ ²dt = R_T

0

Pn

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

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

0

hx(t),˙ y(t)i˙ dt, withha, bi=Pn

i=1aibi fora, b∈Rⁿ; andf, g(with componentsg1, g2, . . . , gp), and h (with components h₁, h₂, . . . , h_q) are nonlinear continuously Fr´echet differentiable operators fromWⁿ[0, T] intoC[0, T],C^p[0, T], andC^q[0, T], respectively, with C^r[0, T] denoting the space of all continuous r-dimensional vector functions defined on [0, T].

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In Problem (P) we use the Hilbert spaceWⁿ[0, T].According to Zalmai [15], 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 consid- eration, one can use a variety of other appropriated spaces in (P), for instance any Banach function spaces. In particular, instead of Wⁿ[0, T] one can use the space Lⁿ_∞[0, T] of all n-dimensional vector-valued Lebesgue measurable essentially bounded functions defined on the compact interval [0, T] of the real line R.

The optimization problem (P) have a range of applications in a great variety of optimal decision processes in the areas of economics, operations re- search and engineering. Furthermore, problems of this type are closely related to constrained variational problems. The relationship between mathematical programming and classical calculus of variations was explored and extended by Hanson [3].

The continuous-time programming problem (P) was first considered by Zalmai [15]. Using a general minmax approach, Zalmai [15] established saddle point optimality conditions and Lagrangian duality relation for (P) under generalized convexity hypotheses with and without differentiability assumptions.

In [16] 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 rea- ders can find in [5].

As a consequence of a general minmax approach developed in [15], Zalmai [17] 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 [18] Zalmai using a minmax duality approach have presented continuous-time analogues of the symmetric nonlinear programs originally studied by Dantzig et al. [2].

In [19] 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 model for a class of continuous-time multiobjective fractional programming problems with convex operator inequality and affine operator equality constraints.

In [20] 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.

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Recently, Andreea M˘ad˘alina Stancu and Stancu-Minasian [9] introduced new classes of functions called (α, ρ)−(η, θ)-type I invex and (α, ρ)−(η, θ)- pseudo (strictly) quasi-type I invex functions. They obtained sufficient Karush- Kuhn-Tucker optimality conditions for problem (P) under the assumptions involving such functions.

Our purpose in the present paper is to introduce the class of (α, ρ)− (η, θ)-pseudo strictly pseudo type I invex functions. Carath´eodory–John-type sufficient optimality conditions for a continuous-time programming problem with nonlinear operator equality and inequality constraints are obtained under the assumptions involving such functions.

LetS ={x∈Wⁿ[0, T] :g(x)(t)≤0, h(x)(t) = 0 for all t∈[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.

The following two definitions are from [9].

Definition 2.1. A pair (f, g) is said to be (ρ1, ρ2)−(η, θ)-type Iinvex 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.

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 I invex at y ∈ V with respect to W if there exist ρ₁, ρ₂ ∈ R and operators η, θ:V ×V →X such that for each x∈W ,

(2.3) Df(y)η(x, y)≥ −ρ₁kθ(x, y)k²⇒f(x)≥f(y), (2.4) 0≤g(y)⇒Dg(y)η(x, y)≤ −ρ₂kθ(x, y)k².

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.

Furthermore we introduce the following

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Definition 2.3. A pair (f, g) is said to be (ρ1, ρ2)−(η, θ)-pseudopseudo- 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), (2.5)

Dg(y)η(x, y)≥ −ρ₂kθ(x, y)k² ⇒g(y)≤0.

(2.6)

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

If the second (implied) inequality in (2.6) is strict (x6=y), then we say that (f, g) is (ρ₁, ρ₂)−(η, θ)-pseudo strictly pseudo-type I invex aty∈V with respect to W.

3. SUFFICIENCY CRITERIA

In this section, we establish various Carath´eodory–John-type sufficient optimality conditions for (P) under generalized (α, ρ)−(η, θ)-type I invexity conditions.

Let {I₁, . . . , I_M} and {J₁, . . . , J_M} be two partitions of the index sets {1,2, . . . , p} and {1,2, . . . , q}, respectively; thus Ir ⊂ {1,2, . . . , p}, for each r = 1, . . . , M, I_r ∩I_s = ∅ for each r 6= s, and

M

S

r=1

I_r = {1,2, . . . , p}; and similarly for{J₁, . . . , J_M}.

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 Im=∅orJm=∅ form >min{M₁, M2}. In addition, we shall make use of the operators Γ_m(·, u, v) : Wⁿ[0, T] → C[0, T] defined for fixed u and v by

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= 1, . . . , M.

Theorem 3.1. Let x^∗∈S and assume that f, g and h are continuously Fr´echet differentiable at x^∗,and there exist u^∗₀ ∈R₊ ≡[0,∞), u^∗ ∈W₊^p[0, T] and v^∗ ∈W^q[0, T], (u^∗₀, u^∗, v^∗) 6= 0,such that

Z T 0

u^∗₀Df(x^∗) +

p

X

i=1

u^∗_i(t)Dg_i(x^∗) +

q

X

j=1

v^∗_j(t)Dh_j(x^∗)

η(x, x^∗)(t)dt≥0 (3.1)

for all x∈S,

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

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

(i)for each m∈N ⊂ {1,2, . . . , M}, (φ,Γm(·, u^∗, v^∗))is ( ˆρ,ρ¯m)−(η, θ)- pseudo strictly pseudo-type I invex at x^∗ with respect to S;

(ii)for each m∈ {1,2, . . . , M} \N, (φ,Γ_m(·, u^∗, v^∗)) is ( ˆρ,ρ¯_m)−(η, θ)- pseudoquasi-type Iinvex at x^∗ with respect to S;

(iii) u^∗₀, u^∗_i, i ∈ I_m, v^∗_j, j ∈ J_m, m ∈ N are not all zero and u^∗₀ρˆ+ T

M

P

m=1

¯ ρm≥0.

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

Proof. Assume for a contradiction that x^∗ is not optimal for (P) so that there is a feasible solution ¯x6=x^∗ such that

(3.3) φ(¯x)< φ(x^∗).

Since u^∗_i ≥ 0,i= 1, . . . , p, using (3.2) and the feasibility of ¯x and x^∗, we get that for each m∈N we have

(3.4) Γm(x, u^∗, v^∗) (t)≤0 = Γm(x^∗, u^∗, v^∗) (t) for all t∈[0, T].

Similarly, it follows that for each m∈ {1, . . . M} \N,we have

(3.5) Γm(x, u^∗, v^∗) (t)≤0 = Γm(x^∗, u^∗, v^∗) (t) for all t∈[0, T].

As for each m ∈N ⊂ {1, . . . , M}, (φ,Γ_n(·, u^∗, v^∗)) is ( ˆρ,ρ¯_m)−(η, θ)-pseudo strictly pseudo-type I invex at x^∗ with respect to S, we have from (3.3) and (3.4) that

(3.6) Dφ(x^∗)η(¯x, x^∗)<−ˆρkθ(¯x, x^∗)k² and

(3.7)

DΓ_m(x^∗, u^∗, v^∗)η(¯x, x^∗) (t)≤ −ρ¯_mkθ(¯x, x^∗)k² for all t∈[0, T], m∈N, with strict inequality holding if u^∗_i, i ∈ I_m, v_j^∗, j ∈ J_m, m ∈ N, are not all zero.

Similarly, as

(φ,Γ_m(·, u^∗, v^∗)), m∈ {1,2, . . . , M} \N

is (ρ,bρ¯_m)−(η, θ) pseudo quasi-type I invex at x^∗ with respect to S, we have the inequality (3.6) and

(3.8) DΓ_m(x^∗, u^∗, v^∗)η(¯x, x^∗) (t)≤ −¯ρ_mkθ(¯x, x^∗)k² for all t∈[0, T], m∈ {1,2, . . . , M} \N.

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Since u^∗_i, i ∈ Im, v_j^∗, j ∈ Jm, m ∈ N, are not all zero, the inequalities (3.6)–(3.8) and iii) implies that

Z T 0



u^∗₀Df(x^∗) +

p

X

i=1

u^∗_i(t)Dgi(x^∗) +

q

X

j=1

v_j^∗(t)Dhj(x^∗)



η(x, x^∗) (t)dt=

= Z _T

0



u^∗₀Df(x^∗)+

M

X

m=1

X

i∈I_m

u^∗_i(t)Dg_i(x^∗)+X

j∈J_m

u^∗_i(t)Dh_j(x^∗)



η(¯x, x^∗)(t)dt

<

u^∗₀ρb+T

M

X

m=1

¯ ρ_m

kθ(¯x, x^∗)k² <0

which is a contradiction to (3.1). Hence x^∗ is an optimal solution of (P).

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

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

0



u^∗₀Df(x^∗) +

p

X

i=1

u^∗_i(t)Dg_i(x^∗) +

q

X

j=1

v^∗_j(t)Dh_j(x^∗)



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

The proofs of the next three theorems are similar to that of Theorem 3.1.

Theorem 3.3. Let x^∗ ∈S and assume that f, g, h, u^∗ and v^∗ are as in Theorem 3.1. Assume that:

(i)For eachm∈N1 ⊂ {1, . . . , M},

φ,RT

0 Γm(·, u^∗, v^∗) (t)dt

is( ˆρ,ρ¯m)−

(η, θ)-pseudopseudo-type Iinvex at x^∗ with respect to S.

(ii) For each m ∈ N2 ≡ {1,2, . . . , M}\N₁, (φ,Γm(·, u^∗, v^∗) (t)dt) is ( ˆρ,ρ¯_m)−(η, θ)-pseudoquasi-type I invex at x^∗ with respect to S.

(iii) u^∗₀ρb+ P

m∈N1

¯

ρ_m+T P

m∈N2

¯ ρ_m ≥0.

(i) For each m∈N1 ⊂ {1, . . . , M}, (φ,Γm(·, u^∗, v^∗))is ( ˆρ,ρ¯m)−(η, θ)- pseudo strictly pseudo-type I invex at x^∗ with respect to S.

(ii)For each m∈N2 ≡ {1,2, . . . , M} \N₁,

φ,RT

0 Γm(·, u^∗, v^∗) (t)dt

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

(iii) u^∗₀ρb+T P

m∈N1

¯

ρm+ P

m∈N2

¯ ρm ≥0.

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(i)For eachm∈N₁ ⊂ {1, . . . , M}, φ,RT

0 Γ_m(·, u^∗, v^∗) (t)dt

is( ˆρ,ρ¯_m)−

(η, θ)-pseudo strictly pseudo-type Iinvex at x^∗ with respect to S.

(ii)For each m∈N₂ ≡ {1,2, . . . , M} \N₁, φ,RT

0 Γ_m(·, u^∗, v^∗) (t)dt is ( ˆρ,ρ¯m)−(η, θ)-pseudoquasi-type I invex at x^∗ with respect to S.

(iii) u^∗₀, u^∗_i, i ∈ I_m, v_j^∗, j ∈ J_m, m ∈ N are not all zero and u^∗₀ρb+

M

P

m=1

¯

ρm ≥ 0.

Note that Theorem 3.1 contains a number of special cases that can be easily identified by appropriate choices of the partitioning sets {I₁, . . . , I_M} and {J₁, . . . , J_M} and of the set N. We shall state some of these cases as a 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^∗₀ ∈ R₊ ≡ [0,∞), 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 (ρ,b ρ)¯ −(η, θ)-pseudo strictly pseudo-type I invex at x^∗ with respect to S, (ii)

u^∗₀, u^∗_i, i∈ {1, . . . , p}, v_j^∗, j ∈ {1, . . . , q}are not all zero and u^∗₀ρb+Tρ¯≥0;

(b) (i)

φ(x),

p

X

i=1

u^∗_i(t)gi(x)(t) +v₁^∗(t)h1(x)(t)

is (ρ,b ρ¯1)−(η, θ)-pseudo strictly pseudo-type I invex at x^∗ with respect to S, (ii) for each m= 2,3, . . . , M, (φ(x), v_m^∗(t)h_m(x)(t))

is (ρ,b ρ¯m)−(η, θ)-pseudoquasi-type Iinvex at x^∗ with respect to S, (iii) u^∗₀, u^∗_i, i∈ {1, . . . , p}, v₁^∗, are not all zero and u^∗₀ρb+T

M

X

m=1

¯ ρ≥0;

(c) (i)

φ(x), u^∗₁(t)g1(x)(t) +

q

X

j=1

v_j^∗(t)hj(x)(t)

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is (ρT,b ρ¯1)−(η, θ)-pseudo strictly pseudo-type I invex at x^∗ with respect to S, (ii) for each m= 2,3, . . . , M, (φ(x), u^∗_i(t)g_i(x)(t))

is (ρ,b ρ¯_m)−(η, θ)-pseudoquasi-type Iinvex at x^∗ with respect to S, (iii) u^∗₀, u^∗₁, v^∗_j, j∈ {1, . . . , q} are not all zero and u^∗₀ρb+T

M

X

m=1

¯ ρ≥0.

Proof. Each of the three 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 {I₁, . . . , I_M} and {J₁, . . . , JM}. In Theorem 3.1, let

a)I₁ ={1, . . . , p},J₁ ={1, . . . , q}and N ={1};

b) I1={1, . . . , p},Jm={m}, m= 1, . . . q and N ={1};

c)Im ={m},m= 1, . . . , p,J1={1, . . . , q} and N ={1}.

4. CONCLUSIONS

In this paper we introduced the class of (α, ρ)−(η, θ)-pseudo strictly pseudo-type I invex functions. Carath´eodory–John-type sufficient optimality conditions for a continuous-time programming problem with nonlinear operator equality and inequality constraints were obtained under the assumptions involving generalized (α, ρ)−(η, θ)-invex 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] M.A. Hanson,Bounds for functionally convex optimal control problems.J. Math. Anal.

Appl.8(1964), 84–89.

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

16(1966),1, 73–83.

[5] 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.

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

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

[8] 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.

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[9] Andreea M˘ad˘alina Stancu and I.M.Stancu-Minasian,Sufficiency criteria in continuous- time nonlinear programming under generalized (α, ρ)− (η, θ)-type I invexity. Rev.

Roumaine Math. Pures Appl.56(2011),2, 169–179.

[10] I.M. Stancu-Minasian and S¸t. T¸ igan, Continuous time linear-fractional programming.

The minimum-risk approach.RAIRO Oper. Res.34(2000),4, 397–409.

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

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

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

[13] C.-F. Wen and H.-C. Wu,Using the Dinkelbach-type algorithm to solve the continuous- time linear fractional programming problems.J. Global Optim.49(2011),2, 237–263.

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

[15] 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.

[16] 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.

[17] 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.

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

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

[19] 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.

[20] 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 13 June 2011 Romanian Academy

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

Casa Academiei Romˆane Calea 13 Septembrie nr. 13 050711 Bucharest 5, Romania andreea madalina s@yahoo.com

stancu minasian@yahoo.com