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Generalized Duality for a Nondifferentiable Control Problem

I. Husain*, Vikas K. Jain, Abdul Raoof Shah

Department of Mathematics, Jaypee University of Engineering and Technology, Guna, India

Abstract: A generalized dual to a control problem containing support functions is formulated and various duality theorems are established under generalized convexity hypotheses. This dual model represents the combination of Wolfe and Mond-Weir type dual models to the control problem and hence it is described as a generalized dual. Some special cases are obtained. A close relationship of duality results with those of the nonlinear programming problems involving support functions is indicated.

Keywords: control problem, support function, optimality conditions, generalized duality, converse duality, nonlinear programming

Received March 13, 2014; Revised May 20, 2014; Accepted June 30, 2014

Citation: I. Husain, Vikas K. Jain, Abdul Raoof Shah, Generalized Duality for a Nondifferentiable Control Problem. American Journal of Applied Mathematics and Statistics, 2 (4). 193-202. January 2014. doi:10.12691/ajams-2-4-3

1. Introduction

In [3], Husain et al. considered the following control problem containing support functions:

(CP):

subject to

[picture] | (1)

[picture] | (2)

[picture] | (3)

where

(i) is a differentiable state vector function with its derivative and is a smooth control vector function,

(ii) denotes an n-dimensional Euclidean space and is a real interval.

(iii) and are continuously differentiable.

(iv) and are the support functions of the compact sets and respectively.

Denote the partial derivatives of where by and

where superscripts denote the vector components. Similarly we have and Designate by The space of continuously differentiable state functions such that and and is equipped with the norm and , the space of piecewise continuous control vector functions having the uniform norm The differential equation (2) with initial conditions expressed as may be written as where being the space of continuous function from to defined as In the derivation of the optimality conditions, some constraint qualification to make the equality constraints locally solvable is needed. For this the derivative of (say) with respect to namely are required to be surjective. Husain et al. [3] established the following Fritz type necessary conditions for the control problem (CP):

Proposition 1. (Fritz John Necessary Conditions): If is an optimal solution of (CP) and the derivative is surjective, then there exists Langrange multipliers and piecewise smooth and such that for all t,

As in [5], Husain et al. [3] pointed out if the optimal solution for (CP) is normal, then the Fritz John type optimal conditions reduce to the following Karush-Kuhn-Tucker optimal conditions:

Proposition 2. If is an optimal solution and is normal and is surjective, there exist piecewise smooth with and such that

[picture] | (4)

[picture] | (5)

[picture] | (6)

[picture] | (7)

[picture] | (8)

[picture] | (9)

[picture] | (10)

Using the Karush-Kuhn-Tucker type necessary optimality conditions, Husain et al. [3] constructed the following Wolf type dual control problem to (CP) and proved various duality results:

(WCD): Maximize

subject to

The problem (WCD) is a dual to (CP) assuming that is pseudo convex in for all and

Husain et al. [4] further weakened the generalized convexity for duality by constructing a Mond-Weir type dual to (CP) given below.

(M-WCD): Maximize

subject to

where

(i) and are compact sets in , and

(ii) and are piecewise smooth functions and

Husain et al. [4] proved duality theorems for the problem (CP) and (M-WCD) under the assumptions of pseudoconvexity of for and quasi convexity of and for all

We review some well known facts about a support function for easy reference. Let be a compact convex set in . Then the support function of denoted by is defined as

A support function, being convex and everywhere finite, has a subdifferential in the sense of convex analysis, that is, there exists z such that for all The subdifferential of is given by Let be normal cone at a point Then if and only if or equivalently, is in the subdifferential of at

In this paper, we propose a generalized dual to (CP) and prove various duality theorems under appropriate generalized convexity assumption. From our duality results, special cases are deduced and it is shown that our results derived in this research can be considered as dynamic generalization of those of nonlinear programming problems having support functions.

2. Generalized Duality

Let with and and with and .

We propose the following generalized dual to the problem (CP) and prove various duality theorem under appropriate generalized convexity condition:

(GCD):

subject to

[picture] | (11)

[picture] | (12)

[picture] | (13)

[picture] | (14)

[picture] | (15)

[picture] | (16)

[picture] | (17)

Theorem 1 (Weak duality): let be feasible for (CP) and with and feasible for (GCD). If for all feasible

is pseudoconvex, and and are quasiconvex, then

Proof: Since is feasible for (CP) and is feasible for (GCD), we have

and

By the quasiconvexity of and the above inequality respectively yields,

and

Hence

and

Combining the above inequalities and the using equality constraints (12) and (13), we have

This, because of pseudoconvexity of

at ,we have

Using and , together with feasibility of for (CP) in the above inequality, we have

yielding

Theorem 2 (Strong Duality): if is an optimal solution of (CP) and is normal , then there exist piecewise smooth and such that is feasible for (GCD), and the corresponding values of (CP) and (GCD) are equal. If the hypotheses of Theorem 1 hold, then is an optimal solution of (GCD).

Proof: Since is an optimal solution of (CP) and is normal , then from Proposition 2, there exist piecewise smooth , and such that conditions (4)-(10) hold. So is feasible for (GCD) and in view of conditions (4), (5), (6), (9) and (10), the equality of the objective functionals follows. If is pseudoconvex, and and are quasiconvex for all and , then from Theorem 1 must be an optimal solution of (GCD).

Theorem 3 (Strict converse duality): Let the problem (CP) have an optimal solution that satisfies the normality condition and be optimal solution of (GCD) if is strictly pseudoconvex, and and are quasi convex for all , then , i.e., is an optimal solution of (CP).

Proof: We shall assume that and exhibit a contradiction, since is an optimal solution of (CP), it follows from Theorem 2 there exist , and such that is an optimal solution of (GCD). Hence

together with the feasibility of for (CP) and for (GCD). for , we have

Also

These, because of quasiconvexity hypothesis and merging their implication and then using equality constraints of (GCD), we have

This, in view of the strict pseudoconvexity of

implies

This in view of (2) and (3), yields.

which gives

In view of and this implies

which is absurd. Hence

3. Converse Duality

In this section, we shall prove the converse duality under the assumption and are twice continuously differentiable. The problem (GCD) may be written in the following form:

Maximize

subject to

where

with etc.

Consider as defining a mapping and as defining a mapping where (i) and are Banach spaces,

(ii) and are spaces of piecewise smooth functions and .

In order to apply the results of [1], some restrictions are required on the equality constraints and It suffices if the derivatives and have weak closed range. In the following theorem, we write and

Theorem 4 (Converse Duality): Assume that

(C₁): and are twice continuously differentiable.

(C₂): and have weak closed range.

(C₃):

is pseudoconvex.

(C₄): and are quasiconvex,

(C₅): , where is an appropriate vector function, and

(C₆):

and

(C₇): are linearly independent.

Then is an optimal solution of (CP) and the optimal values of (CP) and (GCD) are equal.

Proof: Since is an optimal solution of (GCD), by Proposition 1 there exist Langranges multipliers piecewise smooth and such that

[picture] | (18)

[picture] | (19)

[picture] | (20)

[picture] | (21)

[picture] | (22)

[picture] | (23)

[picture] | (24)

[picture] | (25)

[picture] | (26)

[picture] | (27)

[picture] | (28)

[picture] | (29)

[picture] | (30)

[picture] | (31)

Multiplying (21) by and using (29), we have

From (27), we have

which can be written as

[picture] | (32)

Multiplying (23) by, we have

From (28), we have

(by integrating by parts)

which on using the hypothesis (C₅), given in the relation can be written in the matrix form as

[picture] | (33)

Using the equation constraints of (GCD), in (18) and (19) respectively. We have

[picture] | (34)

and

[picture] | (35)

Combining (34) and (35), we have

Ppremultiplying by this gives

This, on using (32) and (33) gives

In view of the hypothesis (C₄), this yields

[picture] | (36)

Using (36), we have

This because of (C₆), yields

If , then and from (20) and (21), we have .

Consequently, we have ensuing a contradiction to (31).

Hence , implying . From (20) together with (21) and (22) together with (23), we have

[picture] | (37)

[picture] | (38)

[picture] | (39)

[picture] | (40)

The relations (24)-(26), we have

[picture] | (41)

[picture] | (42)

From (37), (39) and (41), we have

and

implying that is feasible for (CP).

From (38), (40) and (42), we have

In view of the hypotheses (C₃) and (C₄), by Theorem 1, the optimality of for (CP) follows.

4. Special Cases

If and then (GCD) becomes (WCD) which is Wolfe type dual to (CD) under the pseudoconvexity of

If and (for some ), then (GCD) becomes (M-WCD) is a Mond-Weir type dual to (CP) if is pseudoconvex, and and are quasiconvex.

Let and be positive semi definite matrices and continuous on . Then where and

where

Replacing the support function by its corresponding square root of a quadratic form, we have

(CP₀): Minimize

subject to

(GCD₀): Maximize

subject to

These dual models are not explicitly reported in the literature. However, the duality relationship between (CD₀) and (GCD₀) can be established analogously to that of the problem of preceding section.

5. Nonlinear Programming Problem

If all the functions involved in the formulation of (CP) and (GCD) are independent of , these problem reduce to the following nonlinear programming problems with support functions which do not appear in the literature.

(NP): Minimize

subject to

(GND): Maximize

subject to

and

Ignoring and replacing and by and respectively, we get the following problems studied by Husain and Jabeen [2]:

Primal (P₁): Minimize

subject to

Dual (GD): Maximize

subject to

References

[1]   | S. Chandra, B.D. Craven, and I. Husain, ‘A class of nondifferential Control problems’, J. Optim. Theory Appl. 56 (1988), 227-243.
[2]   | I. Husain and Z. Jabeen, ‘Mixed duality for a Programming containing support functions’, J. Appl. Math & computing Vol. 15 (2004), No 1-2 pp. 211-225.
[3]   | I. Husain, A. Ahmad and Abdul Raoof Shah, ‘On a control problem with support functions’, submitted for publication.
[4]   | I. Husain, Abdul Raoof Shah and Rishi K. Pandey, ‘Duality for a control Problem with support function’, submitted for publication.
[5]   | B. Mond and M. Hanson, ‘Duality for control problem’, SIAM J. Control 6 (1968), 114-120.