Comparative Analysis of Different Types of Stability with Respect to Constraints of a Vector Integer-Optimization Problem

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The paper analyzes five types of stability against perturbations of initial data in criterion functions and constraints of vector integer quadratic optimization problems. Necessary and sufficient conditions are proved for all the types of stability. The relationship among stability with respect to changes of vector criterion coefficients, stability with respect to changes of initial data in constraints, and stability with respect to vector criterion and constraints is established.Keywords: multicriterial discrete optimization, quadratic functions of partial criteria, stability with respect to vector criterion and constraints, perturbations of initial data.The present paper continues the studies [1, 2] concerned with the five well-known types of stability of vector discrete-optimization problems. It pertains to the series of studies on correctness, stability of discrete optimization problems conducted under the guidance of the Academician I. V. Sergienko at the V. M. Glushkov Institute of Cybernetics since the 80s of the last century. The paper [1] presents the results from analysis of stability against perturbations of initial data in quadratic criterion functions of vector problems of discrete optimization on a finite feasible set. The paper [2] compares the same five types of stability against perturbations of initial data in constraints for vector optimization problems on integer points of a polyhedron. Here, we will consider necessary and sufficient conditions of stability against variations in initial data occurring in both the vector criteria and constraints of vector integer quadratic optimization problems.

supporting

confidence: 62%

Stability of a vector integer quadratic programming problem with respect to vector criterion and constraints

Lebedeva

Сергиенко

2006

Self Cite

“…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].…”

mentioning

confidence: 99%

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

Emelichev

Kuzmin

2005

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...

“…An efficient solution of a vector ILP problem is stable if and only if it is strictly efficient. Following [4,5,7], we call the problem Z C n ( )T 2 -stable if there exists at least one stable efficient solution of the problem Z C n ( ) and T 4 -stable if all the efficient solutions of the problem Z C n ( ) are stable. We denote the set of all strictly efficient solutions (the Smale set) of the problem Z C n ( ) by S C n ( ).…”

Section: Corollarymentioning

confidence: 99%

“…COROLLARY 6 [2,4,5,16]. For any n ³ 1, the vector ILP problem Z C n ( ) is T 4 -stable if and only if we have P C S C n n ( ) ( ) = .…”

Section: Corollarymentioning

confidence: 99%

“…The theoretic investigations pursued in the Cybernetic Institute of NASU are oriented toward the obtaining of fundamental results and simultaneously are of obvious practical importance connected with the solution of problems of modelling economic, technological, and social processes under conditions in which the discrepancy and incorrectness of initial information should be efficiently taken into account. Among the recent achievements, we note a number of results of a qualitative theory that are obtained, in particular, in [5][6][7]. In these works, a thorough comparative analysis of five types of stability with respect to the constraints of a vector integer-optimization problem and also an analysis of the same five types of stability with respect to the functional of an integer-optimization problem with quadratic partial criteria are carried out.…”

Section: Introductionmentioning

confidence: 99%

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Stability radius of an efficient solution of a vector problem of integer linear programming in the Gölder metric

Emelichev

Kuzmin

2006

A vector (multicriterion) problem of integer linear programming is considered on a finite set of feasible solutions. A metric l p , 1£ £¥ p , is defined on the parameter space of the problem. A formula of the maximum permissible level of perturbations is obtained for the parameters that preserve the efficiency (Pareto optimality) of a given solution. Necessary and sufficient conditions of two types of stability of the problem are obtained as corollaries.