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WOLFE TYPE HIGHER ORDER SYMMETRIC DUALITY UNDER INVEXITY

  • Verma, Khushboo (Department of Mathematics, Indian Institute of Technology) ;
  • Gulati, T.R. (Department of Mathematics, Indian Institute of Technology)
  • Received : 2013.03.06
  • Accepted : 2013.07.15
  • Published : 2014.01.30

Abstract

In this paper, we introduce a pair of higher-order symmetric dual models/problems. Weak, strong and converse duality theorems for this pair are established under the assumption of higher-order invexity. Moreover, self duality theorem is also discussed.

Keywords

1. Introduction

The concept of symmetric duality was first introduced by Dorn [8] for quadratic programming. Later, in non linear programming this concept was significantly developed by Dantzig et al. [7], Mond [15] and Bazarra and Goode [5]. Mangasarian [14] introduced the concept of second and higher-order duality for nonlinear programming problems, which motivated several authors to work on second order dualty [3,4,9,10,11]. Subsequently, higher-order symmetric duality for nonlinear problems has been studied in [1,2,12,18]. The study of second and higher-order duality is significant due to the computational advantage over the first-order duality as it provides tighter bounds for the value of the objective function when approximations are used. Mond and Zhang [16] discussed the duality results for various higher-order dual problems under invexity assumptions. For a pair of nondifferentiable programs, Chen [6] also discussed the duality theorems under higher-order generalized F-convexity. Yang et al. [19] obtained the duality results for multiobjective higher-order symmetric duality under invexity assumptions.

Recently, Ahmad [1] discussed higher-order duality in nondifferentiable Multiobjective Programming. In this paper, we introduce a pair of higher-order symmetric dual models/problems. Weak, strong and converse duality theorems for this pair are established under the assumption of higher-order generalized invexity. Moreover, self duality theorem is also discussed.

 

2. Preliminaries

Definition 2.1. A function ϕ : Rn ↦ R is said to be higher-order invexity at u ∈ Rn with respect to η : Rn × Rn ↦ Rn and h : Rn × Rn ↦ R, if for all (x, p) ∈ Rn × Rn,

Let ∇xyf denote the n×m matrix and ∇xyf denote the m×n matrix of second order derivative. Also ∇xxf and ∇yyf denote the n × n and m × m symmetric Hessian matrices with respect to x and y, respectively.

 

3. Higher order symmetric duality

We consider the following pair of higher order symmetric duals and establish weak, strong and converse duality theorems.

Primal Problem (WHP):

Minimize

L(x, y, p) = f(x, y) + h(x, y, p) −pT∇ph(x, y, p) − yT [∇yf(x, y) + ∇ph(x, y, p)] subject to

Dual Problem (WHD):

Maximize

M(u, v, r) = f(u, v) + g(u, v, r) − rT∇rg(u, v, r) − uT [∇uf(u, v) + ∇rg(u, v, r)] subject to

where f : Rn × Rm ↦ R, g : Rn × Rm × Rn ↦ R and h : Rn × Rm × Rm ↦ R are twice differentiable functions.

Theorem 3.1 (Weak Duality). Let (x, y, p) and (u, v, r) be feasible solutions for primal and dual problem, respectively. Suppose that

(i) f(., v) is higher-order invexity at u with respect to η1 and g(u, v, r),

(ii) −[f(x, .)] is higher-order invexity at y with respect to η2 and −h(x, y, p),

(iii) η1(x, u) + u + r ⪴ 0,

(iv) η2(v, y) + y + p ⪴ 0.

Then

Proof. It is given (x, y, p) is feasible for (WHP) and (u, v, r) is feasible for (WHD), therefore by the hypothesis (iii) and the dual constraints (4), we get

or

which on using the dual constraint (5) implies that

Now by the higher order invexity of f(., v) at v with respect to η1 and g(u, v, r), we get

Similarly, hypothesis (iv) along with primal constraints (1) and (3) yields

Therefore, by higher-order invexity of −[f(x, .)] at y with respect to η2 and −h(x, y, p), we obtain

Adding inequalities (9) and (11), we get

or

Thus the result holds.

Theorem 3.2 (Strong Duality). Let be a local optimal solution of (WHP). Assume that

(i) is negative definite,

(ii)

(iii)

(iv) Then (I) is feasible for (WHD) and

(II) Also, if the hypotheses of Theorem (3.1) hold for all feasible solutions of (WHP) and (WHD), then are global optimal solutions of (WHP) and (WHD), respectively.

Proof. Since is a local optimal solution of (WHP), there exist α, δ ∈ R, β, ξ ∈ Rm and μ,∈ Rn such that the following Fritz-John conditions [13,17] are satisfied at

Premultiplying equation (14) by

which along with equations (15), (16) yields

Now from equations (1), (19) and (20), we obtain

Using hypothesis (i) in inequality (21), we have

This, together with hypothesis (ii) and equation (14), yields

Now, we claim that α ≠ 0. Indeed if α = 0, then equations (22) and (23) give

Therefore equations (12), (13) and (23), imply μ = 0 and ξ = 0. Hence (α, β, δ, μ, ξ), a contradiction to (19). Thus

Using equations (13), (22) and (24), we have

Therefore hypothesis (iii) implies

Moreover, equation (12) along with (22), (26) and hypothesis (iv) yields

or

Also using equation (17)

Thus satisfies the constraints (4)-(6), that is, it is a feasible solution for the dual problem (WHD). Now using equations (19), (26), (27) and hypothesis(iv), we get

i.e

Finally, by Theorem (3.1), are global optimal solutions of the respective problems.

Theorem 3.3 (Strong Duality). Let be a local optimal solution of (WHP). Assume that

(i) is positive definite,

(ii)

(iii)

(iv) Then (I) is feasible for (WHP) and

(II)

Also, if the hypotheses of Theorem (3.1) are satisfied for all feasible solutions of (WHP) and (WHD), then are global optimal solutions of (WHD) and (WHP), respectively.

Proof. Follows on the line of Theorem (3.2).

 

4. Self Duality

A mathematical problem is said to be self dual if it formally identical with its dual, that is, the dual can be rewritten in the form of the primal. In general, (WHP) is not self-dual without some added restrictions on f, g and h. If f : Rn × Rm → R and g : Rn × Rm × Rn → R are skew symmetric, i.e

as shown below. By recasting the dual problem (WHD) as a minimization problem, we have Minimize

M(u, v, r) = −{f(u, v)+g(u, v, r)−rT∇rg(u, v, r)−uT [∇uf(u, v) + ∇rg(u, v, r)]} subject to

Now as f and g are skew symmetric, i.e

then the above problem rewritten as :

Minimize

M(u, v, r) = f(v, u) + g(v, u, r) − rT∇rg(v, u, r) − uT [∇uf(v, u) + ∇rg(v, u, r)] subject to

Which is identical to primal problem, i.e., the objective and the constraint functions are identical. Thus, the problem (WHP) is self-dual.

It is obvious that (x, y, p) is feasible for (WHP), then (y, x, p) is feasible for (WHD) and vice versa.

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