Duality for a Control Problem Involving Support Functions

Mond-Weir type duality for control problem with support functions is investigated under generalized convexity conditions. Special cases are derived. A relationship between our results and those of nonlinear programming problem containing support functions is outlined.

• Control Problem
• Support Function
• Generalize Convexity
• Converse Duality
• Nonlinear Programming

1. Introduction and Preliminaries

Consider the following control problem containing support functions introduced by Husain et al. [1]

subject to

where

1) is a differentiable state vector function with its derivative and is a smooth control vector function.

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

3), and are continuously differentiable.

4) and, are the support function of the compact set K and respectively.

Denote the partial derivatives of f where by f_t, f_x and f_t,

where superscript denote the vector components. Similarly we have h_t, h_x, h_u and g_t, g_x, g_u. X is the space of continuously differentiable state functions. Such that and and are equipped with

the norm and U, 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 I to Rⁿ defined as In the derivation of these optimality condition, some constraint qualification to make the equality constraint locally solvable [2] and hence the Fréchét derivative of (say) with respect to namely are required to be surjective. In [1] , Husain et al. derived the following Fritz john type necessary optimality for the existence of optimal solution of (CP).

Proposition 1. (Fritz John Condition): If is an optimal solution of (CP) and the Fréchét derivative Q' is surjective, then there exist Langrange multipliers and piecewise smooth, , and such that for all t,

As in [3] , Husain et al. [1] 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 Q' is surjective, there exist piecewise smooth with, , and, such that

Using the Karush-Kuhn-Tucker type optimality condition given in Proposition 2, Husain et al. [1] presented the following Wolfe type dual to the control problem (CP) and proved usual duality theorem under the pseudo-

convexity of for all, and,.

(WCD): Maximize

subject to

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 such that for all x. The subdif-

ferential of is given by Let be normal

cone at a point Then if and only if or, equivalently, is in the subdifferential of s at

In order to relax the pseudoconvexity in [1] , Mond-Weir type dual to (CP) is constructed and various duality theorems are derived. Particular cases are deduced and it is also indicated that our results can be considered as the dynamic generalization of the duality results for nonlinear programming problem with support functions.

2. Mond-Weir Type Duality

We propose the following Mond-Weir type dual (M-WCD) to the control problem (CP):

Dual (M-WCD): Maximize

subject to

Theorem 1. (Weak Duality): Assume that

(A₁): is feasible for (CP),

(A₂): is feasible for the problem (M-WCD),

(A₃): for is pseudoconvex, and

(A₄): for all and are quasiconvex at

Then

Proof: Since we have

and

Combining these inequalities with (14) and (15) respectively, we have

and

These, because of the hypothesis (A₄) yields

Combining (19) and (20) and then using (12) and (13), we have

This, due to the pseudoconvexity of for implies

Since the above inequality gives

yielding

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

Proof: Since is an optimal solution of (CP) and is normal, it follows by Proposition 2 that there exist piecewise smooth and. satisfying for all the conditions (4)-(10) are satisfied. The conditions (4)-(6) together with (9) and (10) imply that is feasible for (M-WCD). Using we obtain,

The equality of the objective functionals of the problems (CP) and (M-WCD) follows. This along with the hypotheses of Theorem 1, the optimality of for (M-WCD) follows.

The following gives the Mangasarian type strict converse duality theorem:

Theorem 3. (Strict Converse Duality): Assume that

(A₁): is an optimality solution of (CP) and is normal;

(A₂): is an optimal solution of (M-WCD),

(A₃): in strictly is pseudoconvex for all and

(A₄): for all and are quasi convex.

Then i.e. is an optimal solution of (CP).

Proof: Assume that and exhibit a contradiction. Since is an optimality solution of

(CP). By Theorem 2 there exist with such that

is an optimal solution of (M-WCD).

Thus

Since is feasible for (CP) and for (M-WCD), we have

and

These, because of the hypothesis (A4) imply the merged inequality

This, by using the equality constraints (12) and (13) of (M-WCD) gives

By the hypothesis (A₂), this implies

(using (21)). Consequently, we have

Since for and for this yields,

This cannot happen. Hence

3. Converse Duality

The problem (M-WCD) can be written as the follows:

Maximize:

subject to

where

Consider and as defining a map- pings and respectively where is the space of piecewise smooth, V is space of piececewise smooth, W^j is the space of piecewise of smooth W^j, B¹ and B² are Banach spaces. and with Here some restrictions are required on the equality constraints. For this, it suffices that if the derivatives and

have weak * closed range.

Theorem 4. (Converse Duality): Assume that

(A₁): and h are twice continuously differentiable.

(A₂): is an optimal solution of (CP).

(A₃): and have weak * closed ranges.

(A₄): for some, and

(A₅): 1) The gradient vectors and are linearly independent, or

2) The gradient vectors and are linearly independent.

(A₆):

Proof: Since is an optimal solution of (M-WCD), by Proposition 1 there exists and and piecewise smooth functions , , such that

Multiplying (24) by and summing over i and then integrating using (28), we have

which can be written as,

Multiplying (25) by and then integrating and using (29), we have

This implies

or

Using the equality constraints (12) and (13) of the problem (M-WCD) in (22) and (23), we have

Combining (35) and (36), we have

This by premultiplying by and then integrating, we have

Using (33) and (34), we have

This because of hypothesis (A₄) implies

Using gives

This, because of hypothesis (A₅) implies

Assume (37) gives from (24) it follows Consequently we have

contradicting (32). Hence and The relations (26) and (27) gives

and

yielding and.

From (24), we have

and

From (25), we have

and

The feasibility of for (CP) follows from (38) and (40).

Consider

(by using along (39) and (41)).

This along with the generalized convexity hypotheses implies that is an optimal solution of (M-WCD).

4. Special Cases

Let for and be positive semidefinite matrices and continuous on I. Then

where

and

where

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

Primal (CP₀): Minimize

subject to

(M-WCD₀): Maximize

subject to

The above pair of nondifferentiable dual control problem has not been explicitly reported in the literature but the duality amongst (CP₀) and (M-WCD₀) readily follows on the lines of the analysis of the preceding section.

5. Related Nonlinear Programming Problems

If the time dependency of the problem (CP) and (M-WCD) is removed, then these problems reduce to the following problem (NP), its Mond-Weir dual (M-WND):

Primal (NP₀): Minimize

subject to

Dual (M-WND₀): Maximize

subject to

The above nonlinear programming problems with support functions do not appear in the literature. However, if and are replaced by and respectively in (NP₀), then problems reduced to following studied by Hussain et al. [4] .

(P₁): Minimize

subject to

(M-WCD): Maximize

subject to

6. Conclusion

Mond-Weir type duality for a control problem having support functions is studied under generalized convexity assumptions. Special cases are deduced. The linkage between the results of this research and those of nonlinear programming problem with support functions is indicated. The problem of this research can be revisited in multiobjective setting.

References

1. Husain, I., Ahmad, A. and Shah, A.R. (2014) On a Control Problem with Support Functions. (Submitted for Publication)

2. Craven, B.D. (1978) Mathematical Programming and Control Theory. Chapman and Hall, Landon. http://dx.doi.org/10.1007/978-94-009-5796-1

3. Mond, B. and Hanson, M. (1968) Duality for Control Problem. SIAM Journal on Control, 6, 114-120. http://dx.doi.org/10.1137/0306009

4. Husain I., Abha and Jabeen Z. ,et al. (2002)On Nonlinear Programming with Support Function Journal of Applied Mathematics and Computing 10, 83-99.