Summary
The concept of duality plays an important role in mathematical programming and has been studied extensively in a finite dimensional Eucledian space, (see e.g. [13, 4, 6, 8]). More recently various dual problems with functionals as objective functions have been studied in infinite dimensional vector spaces [5, 7, 1, 10, 12].
In this note we consider a nonlinear minimization problem in a partially ordered Banach space. It is assumed that the objective function of this problem is given by a (nonlinear) operator and that its feasible domain is defined by a system of (nonlinear) operator inequalities. In analogy to the finite dimensional case we associate with this minimization problem a dual maximization problem which is defined in the Cartesian product of certain Banach spaces. It is shown that under suitable assumptions the main results of the finite dimensional duality theory can be extended to this general case. This extension is based on optimality conditions obtained in [11].
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References
Dieter, U.: Optimierungsaufgaben in topologischen VektorrÄumen I: DualitÄtstheorie. Z.Wahrscheinlichkeitstheorie verw. Geb. 5, 89–117 (1966).
Dieudonné, J.: Foundations of modern analysis. New York: Academic Press 1960.
Dunford, N., and J. T. Schwartz: Linear operators, Part I. New York: Interscience Publishers 1966.
Hanson, M. A.: A duality theorem in nonlinear programming with nonlinear constraints. Austral. J. Statist. 3, 64–72 (1961).
—: Infinite nonlinear programming. J. Austral. math. Soc. 3, 294–300 (1963).
Huard, P.: Dual programs. IBM J. Res. Develop. 6, 137–139 (1962).
Kelly, R. J., and W. A. Thompson Jr.: Quadratic programming in real Hubert spaces. SIAM J. appl. Math. 11, 1063–1070 (1963).
Mangasarian, O. L.: Duality in nonlinear programming. Quart. appl. Math. 20, 300–302 (1962).
—: Pseudo-convex functions. SIAM J. Control 3, 281–290 (1965).
Ritter, K.: Duality for nonlinear programming in a Banach space. SIAM J. appl. Math. 15, 294–302 (1967).
—: Optimization theory in linear spaces, Part III: Mathematical programming in partially ordered Banach spaces. Math. Ann. 184, 133–154 (1970).
Van Slyke, R. M., and R. J. Wets: A duality theory for abstract mathematical programs with applications to optimal control theory. Boeing Scientific Research Laboratories Document D1-82-0671 (1967).
Wolfe, P.: A duality theorem for nonlinear programming. Quart. appl. Math. 19, 239–244 (1961).
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Ritter, K. Dual nonlinear programming problems in partially ordered Banach spaces. Z. Wahrscheinlichkeitstheorie verw Gebiete 14, 257–263 (1970). https://doi.org/10.1007/BF00533663
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DOI: https://doi.org/10.1007/BF00533663