Stability Radius of a Lexicographic Optimum of a Vector Problem of Boolean Programming

Abstract

A Boolean problem of vector lexicographic optimization is considered. Its partial criteria are projections of linear functions on the nonnegative orthant. A formula is obtained for calculation of the limit level of perturbations of the parameter space of the problem with a metric l 1 that preserve the lexicographic optimality of a given solution.Keywords: lexicographic optimum, projection on the nonnegative orthant, stability radius.In past decades, widespread use of discrete optimization models in economy, control, and design stimulated the concentration of the attention of many experts on the study of various aspects of stability (see, for example, [1][2][3][4][5][6][7]) and other problems of parametrical and postoptimal analysis [8-10] of scalar and vector (multicriterion) problems of discrete optimization. In this case, by the stability of a problem we usually understood the existence of a vicinity (in the parameter space of the problem) such that any "perturbed" problem with parameters from this vicinity has some invariant property with respect to the initial one [11,12], and by the stability of a fixed solution of a problem we understood the property of preservation of the corresponding efficiency (optimality) of this solution under given perturbations [7,13].The present work continues the investigations of stability of solutions of vector discrete problems with various kinds of partial criteria and principles of optimality, which were initiated in [13][14][15][16][17][18][19][20]. Here, a formula is derived for the stability radius of a lexicographic optimum of a Boolean problem with a vector criterion that is the projection of the vector of linear functions on the nonnegative orthant of the criterion space. We consider the case where the metric l 1 [13] is specified in the parameter space. As a consequence, the necessary and simultaneously sufficient condition of stability of a lexicographic optimum is given. The results are announced in [21]. We note that a formula for the stability radius of such an optimum was earlier obtained in [22] for a vector linear combinatorial problem with the Chebyshev metric in the perturbing parameter space.We assume that m ³ 1, n ³ 2 , A i is the ith row of a real matrix A a ij m n, . . . , ) 1 2 R , X n n Í = E { } 0 1 , , | | X ³ 2 , and x x x x n T = ( , , . . . , ) 1 2 . We introduce the operator of projection of a vector a a a a m m = Î ( , , , ) 1 2 K R on the nonnegative orthant R + = m { } y y i N m i m Î ³ Î R : , 0 as follows: a a a a a m + + + + + = = [ ] ( , , . . . , ) 1 2 , where a a a i i i + + = = [ ] max , { } 0 , i N m m Î ={ } 1 2 , , . . . , . Thus, this operator is a positive cut-off of the vector being projected.We note that the operations of projection on simplest convex sets such as an nonnegative orthant, a hyperplane, a half-space, a linear variety, etc. underlie various variants of Fejer iterative processes used for the numerical analysis of properly and improperly posed linear programming problems and also consistent and inconsistent systems of l...

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