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Abstract
A marginal value formula has recently been introduced for bi-convex mathematical programming models on a particular region of stability. In this paper, first we show that the formula remains valid for convex models and on every region of stability, where a certain point to set mapping is lower semicontinuous. These significantly weaker assumptions enable us to use the formula in new situations, such as interactive methods of multiobjective optimization. One term in the formula contains the directional derivative of the optimal solution function. We find sufficient conditions for the upper Lipschitzian behaviour of the solution set on regions of stability. Finally, we give the marginal value estimates for the case when the saddle points are not unique.
Our approach is different from the ones described in the literature because we work with arbitrary stable perturbations and do not assume any constraint qualification. Instead of employing the Fritz John condition we use complete characterizations of optimality.
†Research partly supported by Natural Sciences and Engineering Research Council of Canada. It has been completed during the second author’s visit to the Department of Computational & Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa.
‡Contribution of this author is part of his Ph.D. thesis in Applied Mathematics at McGill University.
†Research partly supported by Natural Sciences and Engineering Research Council of Canada. It has been completed during the second author’s visit to the Department of Computational & Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa.
‡Contribution of this author is part of his Ph.D. thesis in Applied Mathematics at McGill University.
Notes
†Research partly supported by Natural Sciences and Engineering Research Council of Canada. It has been completed during the second author’s visit to the Department of Computational & Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa.
‡Contribution of this author is part of his Ph.D. thesis in Applied Mathematics at McGill University.