### Objective optimization function of adaptive management

Aiming at the multi-objective optimization of virtual machine placement technology, the particle swarm optimization (PSO) method is used to construct an adaptive management target optimization model of virtual machines in cloud computing [20, 21]. It is assumed that there are 8 migrating virtual machines and 5 physical machines. The particle population is adapted to obtain new individuals in the process of multi-objective optimization. The operator is:

$$ {p}_{\mathrm{id}}^{\mathrm{new}}=\left\{\begin{array}{cc}{p}_{\mathrm{id}}+m\left(\right)\left({X}_{\mathrm{max}}-{p}_{\mathrm{id}}\right)& \mathrm{if}\ m\left(\right)>0\ \\ {}{p}_{\mathrm{id}}+m\left(\right)\left({p}_{id}-{X}_{\mathrm{min}}\right)& \mathrm{if}\ m\left(\right)\le 0\end{array}\right. $$

(1)

In which, *X*_{max}and *X*_{min} are the maximum and minimum position value of particles. *m*() is the wavelet function of each migration virtual machine, which is defined as:

$$ m\left(\right)=\frac{1}{\sqrt{a}}{e}^{-\frac{\varphi^2}{2{a}^2}}\cos \frac{5\varphi }{a} $$

(2)

In which, *φ* ∈ [−2.5*a*, 2.5*a*], *a* ∈ [1, 10000]. Given the migration virtual machine and the destination host selected for each iteration, the wavelet high frequency mutation can make the algorithm easier to search in the solution space.

**Assumption 1:***f*(*D*(*x*, *ξ*)) ≤ *f*(*x*), and if the adaptive management crossover operator *ξ* ∈ *U* is placed in cloud computing, then *f*(*D*(*x*, *ξ*)) ≤ *f*(*x*), where *D* is a random optimization algorithm, *ξ* is the solution found by adaptive management of virtual machine placement in cloud computing, *U* is the feasible solution space, and *f* is the objective function.

**Assumption 2**: For the adaptive management Borel subset *A* of virtual machines placed in any cloud computing of U, if its management measure *v*(*A*) > 0, then \( \prod \limits_{k=0}^{\infty}\left(1-{u}_k(A)\right)=0 \), where *u*_{k}(*A*) is the adaptive optimization algorithm *D* for placement of virtual machines in cloud computing, *D* in the *k* iteration, the probabilistic measure of the solution found on *U*.

Theorem 1: *Suppose that the adaptive management objective function*\( {\left\{{x}_{g,k}\right\}}_{k=0}^{\infty } \)*of virtual machine placement in cloud computing is a measurable function, and the solution space*\( \underset{t\to \infty }{\lim }P\left[{x}_{g,k}\in {R}_{\varepsilon}\right]=1 \)*is a measurable set, and the solution sequence generated by the algorithm is P*[*x*_{g, k} ∈ *R*_{ε}]*, then x*_{g, k} ∈ *R*_{ε}*holds, where VMbest and PMbest represent the migration virtual machine and the destination host selected by each iteration, k is the evolutionary algebra, and x*_{g, k} ∈ *R*_{ε}*is the optimal solution found by the k generation.*

Record the ordered pair and update the current virtual machine migration scheme. The algorithm starts iterating through the steps described above until the optimal virtual machine migration scheme is obtained, and the standard PSO algorithm for adaptive management of virtual machine placement in cloud computing is described as:

$$ D\left({p}_g(t),{p}_i(t)\right)=\left\{\begin{array}{cc}{p}_g(t)& f\left({p}_g(t)\right)\le f\left({x}_i(t)\right)\\ {}{x}_i(t)& f\left({p}_g(t)\right)>f\left({x}_i(t)\right)\end{array}\right. $$

(3)

It is easy to prove that it satisfies the assumption 1 in the convergence criterion.

### Multi-objective optimization problem description

The objective optimization model of adaptive management of virtual machine placement in cloud computing is constructed by particle swarm evolution method [22, 23]. The global optimization control of adaptive management of virtual machine placement in cloud computing is carried out by introducing extremum perturbation operator. The particle size of the swarm is *N*, and the sample space of the swarm must contain *U*, that is \( U\subseteq \underset{i=1}{\overset{N}{\cup }}{M}_{i,k} \), where *M*_{i, k} is the support set of the sample space of particle ∀*A* ⊂ *U* in the *k* generation. There are *N* × *M* combinations of ordered pairs between *N* virtual machines and *v*(*A*) > 0 physical computers. For BV, the measure placed by virtual machines in cloud computing, each one corresponds only to the running cost of virtual machine *VMi* on the *PMj* of physical machines *Cost*(*VMi*, *PMj*), and its speed is even higher. The new formula is convergent, and the improved ICLPSO algorithm converges to the global optimal solution provided it is proved that the interval algebra of particle cloning is finite.

All approximate solutions of intermediate particles can be regarded as a point in state space *S2*. Let *s*_{i} = (*x*_{1}, *x*_{2}, …, *x*_{n}) ∈ *S*^{1} denote the first state in the state space *S1* in which the virtual machine is placed adaptively, and *f*(*s*_{i}) = (*f*(*x*_{1}), *f*(*x*_{2}), …, *f*(*x*_{n})) denote that the random variable A processes the state *s*^{∗} = {*x* ∈ *X*| *f*(*x*) = max *f*(*x*)} at the *k* generation, so that the control objective of adaptive management of the virtual machine placement can be achieved. Function *f*(*x*) is the fitness function on *X*, and *s*^{∗} = {*x* ∈ *X*| *f*(*x*) = max *f*(*x*)} is used to represent the optimal solution of the state space of adaptive management placed by the virtual machine.

Where *I* = {*i*| *s*_{i} ≥ *s*_{j}, ∀*s*_{j} ∈ *S*^{1}}, an ordered pair corresponding to the minimum value is selected among all the resulting function values [24, 25], where VMbest and PMbest represent the migration virtual machine and destination host selected by the algorithm for each iteration, such as VMbest and PMbest satisfies:

$$ f\left({x}_1\right)=f\left({x}_2\right)=\dots =f\left({x}_n\right)={f}^{\ast } $$

(4)

The amount of state migration placed by virtual machines in cloud computing is *s*_{i} ∈ *s*^{∗}. Let the transition probability of stochastic process {*A*_{k}} be:

\( {p}_{ij}(k)=p\left\{{A}_{k+1}^j/{A}_k^i\right\}\ge 0 \), where

$$ {p}_{ij}(k)=p\left\{{A}_{k+1}^j/{A}_k^i\right\}=\sum \limits_{s_c\in {S}^2}p\left\{{C}_k^l/{A}_k^i\right\}p\left\{{C}_{k+1}^l/{A}_k^i{C}_k^l\right\} $$

(5)

In which, *i* ∈ *I*, *j* ∉ *I*, for any physical machine migration variable *L*, there are:

$$ p\left\{{A}_{k+1}^j/{A}_k^i{C}_k^l\right\}=\sum \limits_{s_c\in {S}^2}p\left\{{C}_k^l/{A}_k^i{B}_k^b\right\}p\left\{{A}_{k+1}^j/{A}_k^i{B}_k^b{C}_k^l\right\} $$

(6)

The update of virtual machine migration scheme will cause a state change. It is necessary to recalculate the affected element values in cost matrix CostMatrix for multi-objective optimization control [26, 27].