On strong quasistability of a vector problem on substitutions

Computer Science Journal of Moldova, vol.9, no.1(25), 2001 On strong quasistability of a vector problem on substitutions V.A. Emelichev V.G. Pakhilka Abstract A type of the stability of the Pareto, Smale, and Slater sets for a problem of minimizing linear forms over an arbitrary set of substitutions of the symmetric group is investigated. This type of stability assumes that at least one substitution preserves corresponding efficiency for ”small” independent perturbations of coefficients of the linear forms. Quantitative bounds of such a type of stability are found. In the paper [1], two types of stability for a vector integer linear programming (ILP) problem are investigated. This problem consists in finding the Pareto set. Note that these types of stability are first introduced for a scalar trajectory problem in [2]. In [1], a formula for the strong quasistability radius is deduced and a necessary and sufficient condition of such a stability for a vector ILP problem is obtained. The aim of this paper is to extend these results to vector combinatorial problems of finding the Pareto, Smale, and Slater sets among substitutions of the symmetric group. 1 Preliminaries Let m, n ∈ N, m ≥ 2, A = [aij ]n×m and B = [bij ]n×m be the pair of real matrices (throughout the paper, N denotes the set of natural numbers). Let Sm be the symmetric group of substitutions acting on c °2001 by V.A.Emelichev, V.G.Pakhilka 71 V.A.Emelichev, V.G.Pakhilka the set Nm = {1, 2, ..., m}. On a nonempty set of substitutions T ⊆ Sm , we specify the vector criterion f (t, A, B) = (f1 (t, A1 , B1 ), f2 (t, A2 , B2 ), ..., fn (t, An , Bn )) → min t∈T with partial criteria of the form fi (t, Ai , Bi ) = m X aij bit(j) , i ∈ Nn , j=1 1 2 ... m where t = ( t(1) t(2) ... t(m) ). Here and subsequently, a lower index at a matrix (vector) points to the corresponding row (component) of the matrix (vector). For example, Ai = (ai1 , ai2 , ..., aim ). In this context, traditional definitions (see for instance [3]) of the set of strongly efficient substitutions (Smale set), set of truly efficient substitutions (Pareto set), and set of weakly efficient substitutions (Slater set) have, respectively, the form: Tkn (A, B) = {t ∈ T : τk (t, A, B) = ∅}, k ∈ N3 , where τ1 (t, A, B) = {t0 ∈ T \ {t} : q(t, t0 , A, B) ≥ 0(n) }, τ2 (t, A, B) = {t0 ∈ T : q(t, t0 , A, B) ≥ 0(n) , q(t, t0 , A, B) 6= 0(n) }, τ3 (t, A, B) = {t0 ∈ T : ∀i ∈ Nn (qi (t, t0 , Ai , Bi ) > 0)}, q(t, t0 , A, B) = (q1 (t, t0 , A1 , B1 ), q2 (t, t0 , A2 , B2 ), ..., qn (t, t0 , An , Bn )), qi (t, t0 , Ai , Bi ) = fi (t, Ai , Bi ) − fi (t0 , Ai , Bi ), i ∈ Nn , 0(n) = (0, 0, ..., 0) ∈ Rn . It follows directly from these definitions that T1n (A, B) ⊆ T2n (A, B) ⊆ T3n (A, B). (1) For any number k ∈ N3 , let us denote by Zkn (A, B) the problem of finding the set of efficient substitutions Tkn (A, B). As we assumed 72 On strong quasistability of a vector problem . . . the nonemptiness of the set T , it is evident that T2n (A, B) 6= ∅ and T3n (A, B) 6= ∅ for any A, B ∈ Rnm . Notice that the set of strong efficient substitutions T1n (A, B) can be empty. In the sequel, speaking about the problem Z1n (A, B) we suppose that T1n (A, B) 6= ∅. Obviously, if we go over the single-criterion case (n = 1, A, B ∈ Rm ), the sets T21 (A, B) and T31 (A, B) coincide and turn into the set of optimal substitutions whereas our problem turns into a well-known scalar problem of minimizing linear forms over an arbitrary set of substitutions (see, e.g., the monographs [4,5] and the review [6] with its bibliography). In the case when an optimal substitution t∗ of the problem Z21 (A, B) is unique, we obviously see that T11 (A, B) = T21 (A, B) = T31 (A, B) = {t∗ }. Otherwise the set of strongly efficient substitutions of the scalar problem is empty. As it was known before, to carry out the solution sensitivity analysis to variation of problem’s parameters is one of the important elements of solving practical optimization problems. In this paper, we study the behavior of the set of efficient substitutions for perturbation of elements of the matrix A. Now the question is: how much strongly can one vary these parameters independently from each other such that at least one substitution preserves corresponding efficiency in any perturbed problem? Such a type of stability of a vector problem is accepted to call strong quasistability. Note that the sense of this term is explained in [1]. A quantitative characteristic of similar stability naturally leads to the concept of the strong quasistability radius of the problem. Before giving the strong definition of such a radius, following [2] we introduce the following notation. For a substitution t ∈ Tkn (A, B), k = 1, 2, let Wkn (t, B) = {t0 ∈ T : I n (t, t0 , B) 6= ∅}; for a substitution t ∈ T3n (A, B), let W3n (t, B) = {t0 ∈ T : I n (t, t0 , B) = Nn }, where I n (t, t0 , B) = {i ∈ Nn : δi (t, t0 , Bi ) > 0}, 73 V.A.Emelichev, V.G.Pakhilka δi (t, t0 , Bi ) = m X |bit(j) − bit0 (j) |. j=1 Clearly, for any substitution t ∈ T1n (A, B), we have T \ {t} = W1n (t, B) = W2n (t, B) ⊇ W3n (t, B); (2) for t ∈ T2n (A, B), we have T \ {t} ⊇ W2n (t, B) ⊇ W3n (t, B). For any k ∈ N3 , we call a problem Zkn (A, B) nontrivial if the set Wkn (t, B) is not empty for any efficient substitution t ∈ Tkn (A, B). In the case when there exists a substitution t ∈ Tkn (A, B) such that Wkn (t, B) = ∅, a problem Zkn (A, B) is called trivial. In the above notation, we give the following evident properties. Property 1 . The problem Z1n (A, B) of finding the set T1n (A, B) is nontrivial if and only if |T | > 1. Property 2 . If |T | = 1, then any problems Z2n (A, B) and Z3n (A, B) are trivial. Property 3 . If ∃t0 ∈ T ∀i ∈ Nn (qi (t, t0 , Ai , Bi ) > 0), then t 6∈ T3n (A, B). Property 4 . Let a problem Zkn (A, B), k ∈ N3 , be nontrivial. Then we have ∀t ∈ Tkn (A, B) ∀t0 ∈ Wkn (t, B) (I n (t, t0 , B) 6= ∅). Property 5 . It follows that δi (t, t0 , Bi ) = 0 ⇒ qi (t, t0 , Ai , Bi ) = 0, I n (t, t0 , Bi ) = I n (t, t00 , Bi ) = ∅ ⇒ I n (t0 , t00 , Bi ) = ∅. 74 On strong quasistability of a vector problem . . . Property 6 . If I n (t, t0 , B) 6= Nn , then ∀i ∈ Nn \ I n (t, t0 , B) ∀A0i ∈ Rm (qi (t, t0 , Ai + A0i , Bi ) = 0). Property 7 . Let t ∈ Tkn (A, B), k = 2, 3, |T | > 1, and all elements of each row in the matrix B pairwise different. Then the following three assertions are true: • ∀t0 6= t (I n (t, t0 , B) = Nn ); • Wkn (t, B) = T \ {t}; • the problem Zkn (A, B) is nontrivial. For any natural number d, by the norm of a vector x = (x1 , x2 , . . . , xd ) ∈ Rd we mean the norm l∞ as follows: kxk = max{|xi | : i ∈ Nd }. By the norm of a matrix we mean the norm of the vector constrained from the elements of the matrix. Property 8 . If different substitutions t, t0 ∈ T , an index i ∈ Nn , and a vector A0i ∈ Rm are such that qi (t, t0 , Ai , Bi ) + kA0i kδi (t, t0 , Bi ) < 0, then it follows that qi (t, t0 , Ai + A0i , Bi ) < 0. Actually, on account of t (...truncated)