Theory and Methodology
Symmetric duality with pseudo-invexity in variational problems

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Abstract

Weak and strong duality results are established under pseudo-invexity hypotheses for symmetric dual variational problems. Self-dual problems and static symmetric dual programs are included as special cases.

Introduction

Duality in nonlinear programming is usually treated using the scheme of Wolfe [11], in which the formation of a dual involves the introduction of new variables corresponding to primal constraints. Thus, except in linear programs, the primal can not be obtained by forming the dual of the dual.

The concept of symmetric dual programs, in which the dual of the dual equals the primal, was introduced and developed in Refs. 3, 2. Mond and Hanson [5] extended symmetric duality to variational problems, giving continuous analogues of the previous results.

Assumptions common to these works are those of convexity and concavity. Since the identification of invex functions in Ref. [4], pseudo-invex functions in Refs. 6, 8, 9, many results which formerly required concavity have been extended using invexity and pseudo-invexity, including the variational problems in Ref. [7] and in Refs. 6, 8, 9, respectively. In this paper, we apply pseudo-invexity to symmetric dual variational problems of Smart and Mond [10]. The special case of self-dual variational problems, along with the reduction to static symmetric dual programs when there is no time dependency is presented.

Section snippets

Notation

Consider the real scalar function f(t,x,x′,y,y′), where t∈[to,tf], x and y are functions of t with x(t)∈Rn and y(t)∈Rm, and x′ and y′ denote the derivatives of x and y, respectively, with respect to t. Assume that f has continuous fourth-order partial derivatives w.r.t. x,x′,y,y′·fx and fx′ denote the gradient vectors of f w.r.t. x and x′ i.e.fx≡∂f∂x1,…,∂f∂xnTandfx′≡∂f∂x1′,…,∂f∂xn′T,where T denotes transpose of a matrix. Similarly, fy and fy′ denote the gradient vectors of f w.r.t. y and y′.

The

Symmetric duality

Let S be a nonempty subset of a normed linear space X, the positive dual cone of S (denoted by S§) is defined byS§=x§∈X§:x§(x)⩾0,x∈S,where X§ denotes the space of all continuous linear functionals on X, and x§(x)=(x§,x).

Let S1, S2 be closed convex cones with nonempty interiors in Rn and Rm, respectively.

We consider the problem of finding functions x:[to,tf]→Rn and y:[to,tf]→Rm, with (x′(t),y′(t)) piecewise smooth on [to,tf], to solve the following pair of optimization problems:
(P) Minimize∫totf

Self-duality

Assume m=n,f(t,x,x′,y,y′)=−f(t,y,y′,x,x′) i.e. (f is skew-symmetric) ∀(x(t),y(t)),t∈[to,tf] and that xo=yo,xf=yf. It follows that (D) may be rewritten as a minimization problem:
(D′) Maximize∫totf−f(t,y,y′,x,x′)+x(t)Tfx(t,y,y′,x,x′)−x(t)Tddtfx′(t,y,y′,x,x′)dt,subject to:x(to)=xo,x(tf)=xf,y(to)=xo,y(tf)=xf−fx+ddtfx′∈S1§which becomes
(D′) Minimise∫totff(t,y,y′,x,x′)−x(t)Tfx(t,y,y′,x,x′)+x(t)Tddtfx′(t,y,y′,x,x′)dt,subject to:x(to)=xox(tf)=xf,y(to)=xoy(tf)=xf−fx+ddtfx′∈S1§.(D′) is formally identical

Static symmetric dual programs

If the time dependency of problems (P) and (D) is removed and f is considered to have domain Rn×Rm, we obtain the symmetric dual pair given by
(SP) Minimizef(x,y)−yTfy(x,y)subject to:−fy∈S2§.(SD) Maximizef(x,y)−xTfx(x,y)subject to:fx∈S§1.

Definition. The function f:Rn×Rm→R is pseudo-invex w.r.t. η:Rn×Rn→Rn if for each y∈Rm, η:Rn×Rn→Rn such thatηT(x,u)fx(u,y)⩾0→f(x,y)−f(u,y)⩾0∀x,u∈Rn,−f is pseudo-invex w.r.t. ξ in y if for each x∈Rn,∃ ξ:Rm×Rm→Rm such that−ξ(v,y)Tfy(x,y)⩾0→−f(x,v)+f(x,y)⩾0.Note

Acknowledgements

The authors thank the referees for their valuable suggestions which improved the presentation of the paper.

References (11)

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