Inequality assumes that there exist

Inequality (7) assumes that there exist a finite infimum and finite suprema. The H set can be finite or infinite. holds for this x. However, h can depend on x. Taking the infimum and the suprema in inequality (7) eliminates this dependence and still cytoskeleton the inequality valid, as x ≥ 0. Let us note that the LP principle can be reformulated as the metaprinciple obtained from the logical condition that all Ax ≥ b inequalities should hold true. Additionally, particular cases of the DC principle are the LP principle and the DC metaprinciple: Cut (7) normally possesses the properties that are typical for all systems if they are known for only one of them among the system of linear inequalities (6) which holds true (see Theorems 1 and 2, presented below). However, exceptions are possible sometimes, when cut (7) proves ineffective in yielding such information (see the following example).
System (6) involves one inequality at values of h=1 and 2. Notably, in the first case (h=1), it is consistent, while in the second one it is not. Then, using the DC principle, we obtain the following inequalities: and Inequality (9a) is a relaxation of the trivial cut · x1 ≥ which holds true for all x1. Thus, it follows from analyzing inequalities (8) that inequalities should hold true (as · x1 ≥ 1 does not hold true, the inequality should hold). Valid cuts for the disjunctive S4set are exactly those for a closed convex cover clconvS4of the S4 set.
Converse propositions to the DC principle The below theorems formulate the conditions which, if satisfied, provide that the DC principle yields all valid cuts. The presented Theorems 1 and 2 are the converse propositions to the DC principle. Another converse was given in Refs. [2,5].
where (λA) is a jth component of the λ A vector. Taking the suprema and the infimum, we obtain what is required.□ Before we formulate and prove the next theorem, let us introduce some new concepts. For each h, h ∈ H, let us introduce a cone Then Eq. (11) will consist of a sum of sets (cones) defined as According to the DC principle, not all systems (6) must be consistent in order to produce all valid cuts.
holds true (the sum is interpreted as , if all systems (6), h ∈ H, are inconsistent). holds true for any cut πx ≥ π0 that is valid for all consistent systems ( ≥ ≥ 0). Then taking the suprema and the infimum, will, the same as with proving Theorem 1, conclude the proof. Starting a detailed proof of Theorem 2, let us note that if the system is consistent and the πx ≥ π0 inequality holds true, then we have the πx ≥ inequality for x ∈ . This follows from the system of inequalities (λA) ≤ π for some λ, λ ≥ and from the fact that x ≥ 0. But if system (6) is inconsistent and x ∈ , then, according to system (11), when for certain from , we obtain that System (11а) also holds true if all systems (6), h ∈ H, are inconsistent. Therefore, the inequality πx ≥ follows from the inequalities and, according to the LP principle, we obtain the θ multipliers, for which Finally, according to the Farkas–Minkowski lemma, precipitation follows from the inconsistency of system (6) that there exists a vector for which (see Ref. [5], p. 39) the inequalities hold true. But then for a sufficiently high value of the r variable, r ≥ 0, assuming that we have
□ Theorem 2 is proved. A corollary to Theorem 2 is that is that the C cone introduced by Eq. (10) has an interesting interpretation. This cone is often called a recession cone, or a cone of directions at infinity [1]. To be definite, let us assume that is selected in such a way that the system is consistent. Let x° satisfy the system (12), and let . Then for any λ, λ ≥ 0, the relation holds true, and consequently, also satisfies system (12).
Strengthening of the disjunctive cut The CSM.Let T be a non-empty set, and M be a monoid, i.e., T ⊆the inequalityalways follows for some matrices A, whereth column of the matrix A.