Stability radius of a vector integer linear programming problem: case of a regular norm in the space of criteria

Abstract

A multicriteria integer linear programming problem of finding a Pareto set is considered. The set of feasible solutions is supposed to be finite. The lower and upper achievable bounds for the radius of stability are obtained using a stability criterion and the Minkowski-Mahler inequality and assuming that the norm is arbitrary in the space of solutions and is monotone in the space of criteria. Bounds for the radius of stability in spaces with the Holder metric are given in corollaries.Keywords: vector integer linear programming problem, Pareto set, Slater set, stability, radius of stability, perturbing matrix, norm of vector, space of criteria, space of solutions.

Stability radius of a vector investment problem with Savage’s minimax risk criteria

“…As usually [2,6,[9][10][11], by the radius of stability of a problem Z R s ( ), s ³ 1, against perturbations of the parameters of a vector criterion we will mean the number…”

“…The majority of the results in this field is related to deriving formulas or estimates for the stability radius of vector problems of Boolean and integer programming with linear criteria [6,[9][10][11][12]. In the present paper, we will obtain the lower-and upper-bound attainable estimates for the stability radius of a vector Boolean problem with bottleneck criteria, i.e., of a portfolio optimization problem with Savage's minimax risk criteria.…”

Multicriterial investment problem in conditions of uncertainty and risk

“…Similar to [6,8,10,13,14] the stability radius of the efficient portfolio is the number where The set is called the set of perturbing matrices.…”

“…Earlier studies of quantitative characteristics of stability of efficient solutions were performed mainly for linear multicriterial problems, such as Boolean [6,7] and integer [8][9][10] programming, finite games [11,12], and a number of Boolean problems with nonlinear criteria [13][14][15].…”

Stability analysis and regularization of uncertain linear multi-objective integer optimization problems

“…Recently, a local quasi-stability concept has been introduced by Emelichev et al (2010), where uncertainty lies in the criteria space C. Here, the proposed approach considers uncertainty from (1) the criteria space C and the constraint space (A, b) separately and (2) all the input data (C, A, b) at the same time. Therefore, this new post-optimality method is superior to previous attempts as it helps build the decision maker's insight as to the impact of uncertainty on the efficient solution and better reflects the actual decision making process.…”