A general approach to studying the stability of a Pareto optimal solution of a vector integer linear programming problem

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.…”

“…Since the lower bound is considered, taking into account Remark 1, here the set (unlike the problem ) can contain the zero vector 0. The stability radius of the efficient solution x 0 to problem (3.2) is the quantity [8] where is the positive cut of the vector , i.e., , , . At the same time, it can easily be seen that is the lower bound of ϕ for m = 1.…”

“…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].…”

On the stability radius of an efficient solution of a multicriteria portfolio optimisation problem with the Savage criteria

“…A particular content of this notion depends on the choice of the set of parameters of the problem being subjected to disturbances and on the structure determining the relations of closeness on the set of initial data, that is, on the metrics defined on the space of parameters. Most of the results in this direction is connected with obtaining formulas or estimations for the radius of stability of efficient (Pareto-optimal) solutions of vector problems of discrete optimisation with linear criteria [9][10][11][12][13]. In this paper, we find lower and upper bounds of the radius for multicriteria Boolean problem with nonlinear criteria, namely, for the portfolio optimisation problem with Savage's minimax risk criteria [14].…”

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

“…Moreover, we will use the Minkowski-Mahler inequality to generalize the estimates obtained earlier to an arbitrary norm in the space of solutions and to a norm in the space of criteria that satisfies some weak conditions (such norms are, for example, all the Holder norms l p , 1£ £¥ p ). Note that the Minkowski-Mahler inequality was earlier applied in [10] to derive the lower and upper achievable estimates of the radius of strong stability (T 1 -stability as termed in [5][6][7][8]), and also in [11] to derive a formula for the radius of stability of the efficient solution of a vector ILP problem.…”