We use the method of minimal vectors to prove that certain classes of positive quasinilpotent operators on Banach lattices have invariant subspaces. We say that a collection of operators F on a Banach lattice X satisfies condition ( ) if there exists a closed ball B(x0,r) in X such that x0 > 0 and kx0k > r, and for every sequence (xn) in B(x0,r) (0,x0) there exists a subsequence (xni) and a sequence Ki 2 F such that Kixni converges to a non-zero vector. Let Q be a positive quasinilpotent operator on X, one-to-one, with dense range. Denote hQ) = {T > 0 : TQ 6 QT}. If either the set of all operators dominated by Q or the set of all contractions in hQ) satisfies ( ), then hQ) has a common invariant subspace. We also show that if Q is a one-to-one quasinilpotent interval preserving operator on C0(), then hQ) has a common invariant subspace. Lomonosov proved in (Lom73) that if T is not a multiple of the identity and com- mutes with a non-zero compact operator K, then T has a hyperinvariant subspace, that is, a proper closed nontrivial subspace invariant under every operator S in the commutant {T} 0 = {S 2 L(X) : ST = TS}. There has been numerous extensions and generalizations of the result of Lomonosov. In particular, Abramovich, Aliprantis, and Burkinshaw produced several generalizations of Lomonosov's theorem for Banach lattice setting (AAB93, AAB94, AAB98), see also (AA02). In these generalizations commutation relations are substituted by a super-commutation relation ST 6 TS or ST > TS and domination 0 6 K 6 T. They proved a series of results of the fol- lowing type: if S is related to a compact operator via a certain rather loose chain of super-commutations and dominations, then S has an invariant subspace. Ansari and Enflo (AE98) have recently introduced the so-called technique of minimal vectors in order to prove the existence of invariant subspaces for certain classes of operators on a Hilbert space. The method was later modified so that it could be used in arbitrary Banach spaces in (JKP03, And03, CPS04, Tr04). In particular, the method of minimal vectors allows to prove Lomonosov-type results where a compact operator is replaced with a family of operators that mimic a compact operator.